Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

09.12.2019 Wien, Univ.Prof.i.R.Dr.phil.Alexander Leitsch Mitwirkung: Ao.Univ.Prof.Dipl.-Ing. Dr.techn.Christian Fermüller Betreuer/in: Betreuung neshitVrasri UnterschriftBetreuer/in UnterschriftVerfasser/in

______Comparison and Analysis of of Analysis and Comparison Constructive Theories SetTheories Constructive Karlsplatz 13 | 1040 Wien | +43-1-58801- | Wien | 1040 13 Karlsplatz zur Erlangung des akademischen Grades Erlangungzur des akademischen Grades der Technischender Universität Wien Logic andComputation an deran Fakultät fürInformatik im Rahmen des Studiums im Matrikelnummer 1225784 Matrikelnummer Diplom-Ingenieur Technische Universität Wien Wien Universität Technische DIPLOMARBEIT DIPLOMARBEIT RobertFreiman

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______0 | www.tuwien.at | www.tuwien.at

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

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Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

cnwegmns aufmerksamen Korrekturhinweisen zur Seite standen. un Kritik konstruktiver Input, positivem mit Zeit gesamten der während mir die bedanken, Leitsch, Fermü Christian Prof. Betreuern, meinen bei herzlich hiermit mich möchte Ich Acknowledgements Acknowledgements

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Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

und und wohlbegründetem Mengenbegriff vielversprechende Ausgangbasis für Anwendungen heraus. und Theorie gerechtfertigte konstruktiv als auch sondern Alltag, mathematischen den für i ZreoFanesh Mnelhe npretn Mengenlehren inspirierten Mengenlehre Zermelo-Fraenkelsche die durch die Mengenlehre,Brouwers ist werden,behandelt Arbeit dieser in die Theorien,wichtigsten Die lage der gesamten Mathematik geschuldet. weisern offenzulegen. Das spezielle Augenmerk auf Mengenlehre ist ihrer historischen Rolle a vergleic zu und präsentieren zu dadurch dasPotential der Anwendbarkeit automatischer in Deduktion und automatischen Theorembe- Weise und Art geschlossenen sich in und verständlichen ein in Mengenlehre konstruktiven der Ansätze verschiedene es, ist Arbeit vorliegenden der Ziel Das derreich automatischen derDeduktion automatischenund Theorembeweiser. welche Teile der Mathematik computational fassbar sind. Potentielle Anwendungen liegen dabei Untersu der in Ansatz vielversprechender ein daher ist basiert, Logik intuitionistischer auf die Mathematik,konstruktive intuitionistischerDie Logik. in Beweisbarkeit Berechenbarkeit zwischenund Seit der Entdeckung der Curry-Howard-Korrespondenz kennen dietiefliegendenwir Zusammenhänge Kurzfassung Theory Set Constructive of Analysis and Comparison wird eine sinnbewahrende Interpretation von Interpretation sinnbewahrende eine wird nbägg o enr auf einer von unabhängig Bar-Induktion entscheidbaren der Prinzip das dass zeigt, und stützt Semantik topologischer wendung Ver die auf sich der präsentiert, wird Beweis Ein untersuchen. zu Eigenschaften metamathematische wün konstruktivistisch geeignet, deshalb ist und auf Berechenbarkeit der Konzepten auf direkt baut Erstere Semantik. topologische und Realisierbarkeit durchgeführt: Methoden semantische zwe durch wird Mengenlehren Zermelo-Fraenkel konstruktiven der Analyse metamathematische Die chung liegt allerdings in der metamathematischen Analyse. Eindruck vom Arbeiten in den Theorien und deren Grenzen zu erhalten. Der struktivistischen Standpunkt aus motiviert. Einige grundlegende Resultate werden abgeleitet, um einen

Mengenlehre 퐌퐋 . In einem ersten Schritt werden die Theorien und ihre Axiomatisierungen vom kon vom Axiomatisierungen ihre und Theorien die werden Schritt ersten einem In . 퐈퐙퐅

aire Vrat dr rue’ce Mteai it Schlussendlich ist. Mathematik Brouwer’schen der Variante basierten – diskutiert. Dadurch stellt sich 퐂퐙퐅 ~ 3 3 ~ in 퐌퐋

– einer Theorie, mit konstruktiv sehr klarem sehr konstruktiv mit Theorie, einer 퐈퐙퐅 퐂퐙퐅 und nicht nicht nur besonders geeignet chwerpunkt der Untersu- 퐂퐙퐅 swe Martin-Löfs sowie , schenswerte dadurch als dadurch WS WS ls Grund- e und hen 2019 chung, im Be- /20 er - - i

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. of decidable bar induction from a variant of Brouwer’s mathematics formalized within formalized mathematics Brouwer’s of variant a from induction bar decidable of sirable. A proof using topological semantics is presented to obtain an independence proof of the de- constructively are that propertiesmetamathematical of investigation an for allows thus and theory tools: Realizability and topological semantics. The former builds directly on notions from computability Metamathematicalanalysisofconstructive Zermelo-Fraenkeltheories executed set istwosemantical by ducted by means ofmetamathematicalanalysis. con- is however, investigation, broadest The system. respective the within limitations and reasoning to usedget to inferredare results basic Somestandpoint. constructive the motivatedfrom are matizations clear and well-justifiedclear and notions ofsets meaning-persevering interpretation of

for applications. mathematical practice, but also vindicates its constructive nature and makes it a promising starti sty Zermelo axiomatic the theory, set Brouwerian are thesis the in discussed theories main The theory isdue to its historically proven relevance in providing the very foundation ofmathematics. placed significance particular The proving. theorem automated and deduction automated application for potential its demonstrate thereby and fashion self-contained and comprehensible a objectiveThe of this thesis is topresent and compare different approaches to constructive set theories in theorem proving. a and deduction automated of areas the in applications potential with mathematics classical being mathematics,of contentcomputational investigating the in tool fruitful a be to promises logic,intuitionistic on constructivebased Hence, logic. intuitionistic in provability and computability tween be- connections deep-lying the of know we Curry-Howard-correspondence the of discovery the Since Abstract Abstract le settheoriesle 퐈퐙퐅 and and 퐂퐙퐅 and Martin- and – 퐂퐙퐅 isdiscussed. makes This Löf’ssettheory into into 퐌퐋 ~ 4 4 ~

– a theory that is considered to give a constructively 퐌퐋 . In a first step,a firsttheoriestheirand theIn axio-. 퐂퐙퐅 not only not especially well-suited for 퐈퐙퐅 here on set on here . Finally, a Finally, . WS WS -Fraenkel- utomated principle ng point 2019 s in s /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Martin- 5 Metamathematical properties constructive of axiomatic set theories ...... 49 4 Constructivism and ...... 11 2 Introduction ...... 7 1 Co Abstract ...... 4 Kurzfassung ...... 3 Acknowledgements ...... 2 Erklärung ...... 1 Contents Theory Set Constructive of Analysis and Comparison

Axiomaticconstructive set theories 3 . Interpreting 5.3 Additional rules ...... 78 5.2 Formulating 5.1 . Further results ...... 67 4.7 Proof ofmetamathematical properties ...... 65 4.6 SoundnessTheorem ...... 57 4.5 simple A Completeness Theorem ...... 56 4.4 3.3 3.4 2.4 . Realizability of 4.3 Metamathematical properties ...... 50 4.2 Some aspects of computability theory ...... 49 4.1 Relation between the theories ...... 47 3.5 . Basicconcepts in 3.2 the Setting stage ...... 34 3.1 . Martin- 2.3 2.2 Informal ...... 11 2.1 ntents ...... 5 퐙퐅 퐈퐙퐅 퐈퐙퐅 Brouwer’s settheory

Löf’s vs

and and vs

퐈퐙퐅 퐂퐙퐅 Löf’s settheory settheory ...... 70 퐂퐙퐅 or problem the with trichotomy ...... 39 or tolive how without the powerset operation ...... 42 ...... 23 퐂퐙퐅 퐌퐋 퐂퐙퐅 ...... 70 in in 퐁퐒퐓 ...... 53 퐌퐋 ...... 16 ...... 36 ...... 81 ...... 20 –

퐈퐙퐅 and and ~ 퐂퐙퐅 5 5 ~ ...... 34 WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Topological Semantics and Independence of Bar induction ...... 87 6 Contents Index ...... 129 References ...... 123 Conclusion ...... 120 . Heyting-valued interpretation of 6.3 Heyting-valued semantics ...... 88 6.2 6.1 . Compatibility with the Weak Continuity Principle ...... 117 6.6 Compatibility with the Fan theorem ...... 114 6.5 Independence ofBar induction ...... 111 6.4 Bar induction in Brouwer’s mathematics

퐈퐙퐅 ...... 96 ~ ...... 87 6 6 ~ WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. theory isdue to its historically proven relevance in providing the very foundation ofmathematics. placed significance particular The proving. theorem automated and deduction automated application for potential its demonstrate thereby and fashion self-contained and comprehensible a objectiveThe of this thesis is topresent and compare different approaches to constructive set theories in color theorem Coqin (see [ the same cardinality as cardinality same the Continuum Hypothesis NuPRL and Coq (see, for example, [ ming languages like Agda and Idris and was able to provide concepts for automated proof assistants like

numbers In [ In Set theory str proving. For example, Martin- theorem automated and deduction automated of areas the in applications potential with mathematics conte computational the investigating in tool fruitful a be to promises logic, intuitionistic based being mathematics, constructive Hence, logic. intuitionistic in provability and computability between connections deep the of know we Curry-Howard-correspondence the of discovery the Since 1 Theory Set Constructive of Analysis and Comparison used informally up to that date. He proved that the set of natural numbers natural of set the that proved He date. that to up informally used been had that infinity, of notion the of analysis mathematical systematic a start to first the was Cantor Džamonja writes foundation of nearly all mathematical activity and is for the most part widely undisputed in this r

퐙퐅퐂 dissipated Grundlagenkrise the As particular were desperatein need of solid foundations. Mathematics plunged the into Russel’s paradox in 1901, however, itof discovery the With mathematics. contemporary of foundation a providing of capable is theory set …

Anecdotal evidence workingAnecdotal mathematicians that from suggests systemthis uctive set theory set uctive is viewed as moreviewed as thanfor whatis needs.axiom sufficient the mathematics of While 14 choice theor choice continuumhypothesis exciteoccasionalstill might some discussion,

a gathering together into a whole a aof gathering into definite, together objectsperception distinct of our or Introduction Introduction – witnessed its revival. Nowadays, revival. its witnessed ], Georg], Cantor defined sets as ℚ is equipotent and that the set of real numbers real of set the that and equipotent is [ 22 of our thought of our 퐌퐋 ]: ]: , which asserts that every infinite setofreal numbers countable, iseither i.e.,it has . Explicitly based on paradigms from programming, from paradigms on based Explicitly . ℕ , or has the same cardinality as cardinality same the has or , 27 ]) testifying to efficacythe of Löf’s intuitionistic type theory was intended as a formalization of the con- , Cantor’s set theory – theory set Cantor’s , 43 — soon became evident that mathematics in general and set theory in ], ], [ which elementsof the set. arecalled 2 퐙퐅퐂 ], ], [ 33

is considered as the “classical set theory” and as such as and theory” set “classical the as considered is ] , [ ~ 39 7 7 ~ ], ], [ BO ]). Famously, Gonthier gave a proof of the four- ℝ now formalized by Zermelo and Fraenkel as Fraenkel and Zermelo by formalized now 퐌퐋 ℝ . Already with the 1890ies it turned out that out turned it 1890ies the with Already . is uncountable. In 1878 he formulated the formulated he 1878 In uncountable. is -based proof-based assistants. 퐌퐋 ℕ and the set of rational of set the and has inspired program- inspired has Grundlagenkrise nt of classical of nt here on set on here WS WS 2019 ole. As the s in s /20 on .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. some natural number natural some [ ( common the rejects constructivism why clear, particularly makes point last The If If procedure is known to check whether or no or whether check to known is procedure [ standpoint: Mathematics is a function of human intellect. Particularly, constructivism criticizes (see 퐙퐅퐂 co constru of spirit the with incompatible is it but world, Platonist perfect, a in hold could mathematical practice, like Russian constructivism, Martin- constructivism, Russian like practice, mathematical concerning Nowadays, of the constructivist standpoint. how this might be accomplished and why it could be of interest, allow first an illustration of a few asp mathematics could not circumvent the development of an intuitionistic notion of set. Before a discussion the paradoxes. Due to the central role of set theory, however, it was clear to Brouwer that a rethinking of forentitiesmathematical complete as objects of collections infinite of acceptance the blamed tuitionism crifundamentala Besides sharply. theory set tor’s [ in showed Cohen example, For results. celebrated

structive mathematics. We start with a proposition nrdcin the at competed intuitionism and Hilbert, David by represented formalism, also logicism, Frege’s Besides Constructivism of course the in founded metamathematics, of discipline The Introduction 9 퐋퐄퐌 43 ] 휓 nsider nsider a simple example to illustrate the different roles of time and actual infinity in classical a ): ] Grundlagenkrise • • • is sufficiently non-trivial, all we can do is to calling the procedure to check procedureto the calling to is do can we all sufficientlynon-trivial, is ). All these currents, however, can be traced back to Brouwer’s intuitionism and rely on one major one on rely and intuitionism Brouwer’s to back traced be can however, currents, these All ). . Hence, Platonists like Gödel towards tend extending this system.

mathematicians in mostin mathematicians

). ). According to this rule, every mathematical statement iseither or true false ( objects are created by mental constructions of an(ideal) mathematician. Instead, objects. mathematical of existence transcendental of kind any rejects tionism mathematician’sThe describeandtruthstoadvanceworldthis of to is them.Obviousto job Platonism, for holding that mathematical objects exist in an ideal world, independen mistaken for mathematics itself, which is assumed free to of be any (formal) language. communicatingusefuloftool languageas formmathematicalsubjects. the But constructiv many Admittedly, rules. predefined to according strings with nipulations nothi to reduced be maymathematics that idea the defendingfor Formalism, term logical purely claims that logic is part of mathematics and on not the other way around. mathematics building of idea the favoring for Logicism, and with somewhat less assurance, also somewhat less with and its assurance, ‘‘necessity’’. . The intuitionistic school of thought, championed by Luitzen Brouwer, criticized Can- 푛 satisfying ahmtcl constructivism mathematical areasseem‘‘sufficiency’’ of the to happily accept 휓 ( 푛 ) : 푛 휓 ∃푛. t ( 휓 푛 ( ) ~ 푛 . How would a constructive proof of proof constructive a would How . ) tique of axiomatic formalizations of mathematics, in-mathematics, of formalizationsaxiomatic of tique 휓 holds. Our statement Our holds. 8 8 ~ e nesad smay f eea cret of currents several of summary a understand we 16 about natural numbers, such that for each ], that the that ], Löf’s type theory and others ( others and theory type Löf’s Hilbert’s program Hilbert’s continuum hypothesis continuum 휙 is the assertion that there is there that assertion the is , provided us with some with us provided , 휓 ( ng more but mere ma- mere but more ng law of excluded middle excluded of law 푛 alism shouldneverbealism ¬휙 ∨ 휙 ) for each numbereach for independent fromindependent

¬휙 ∨ 휙 . Constructivism s. t t of man and time. ctivism. Let us Let ctivism. ists see formalsee ists ). Indeed, this). see mathematical WS WS look like? look [ 푛 풁푭푪 65 ly, intui-ly, , a finite nd con- 2019 ], [ ], [

ects 34 41 /20 푛 ] ] , ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. an such found Having other. the after Being Being based on paradigms from programming, Martin- is possible tojustify large parts ofmathematics constructively. to credited is mathematics constructive for Brouwer. We will discuss some of his ideas in sections foundation 2.2 and6.1. In his book[ as theory set a building at attempt first The Constructive set theory Theory Set Constructive of Analysis and Comparison to the computer scientist. We have listed some successful implementations of

give rise toa the process and conclude and process the Another approach to constructive set theory is to restrict the successful system successful the restrict to is theory set constructive to approach Another theorythe in chapter5. nent such theory, such nent promi-mostthe motivateviewpoint. We willconstructive the justifiabletheoryis fromthat a obtaining is, for example, not clear a priori, whether aremetamathematically. constructiveobtained are only confirmationtheories these ultimate It The that tions in chapter 3. To put it positively: If a constructive proof of a statement of the form the of statement a of proof constructive a If positively: it put To proving introduction. We give a introduction to the main ideas of ideas main the to introduction a give We introduction. show that show cancomputability.thusnotionsWe realizability-semantics of discuss the willto referring directly we 4, Classically, the existence of a natural number natural a of existence the Classically, In In chapter 5, we show how currents. sumption of sumption 휓 gation ofgation provability thein constructive sense in section2.1. a metamathematical justification of Finally, we introduce topological semantics of semantics topological introduce we Finally, his theoremhis that every total function duction is too strong an assumption for Brouwer’s proof of the Fan 퐈퐙퐅 ( 푛 is compatible with Brouwerian mathematics. It turns out, that in this context, the schema of bar in- bar of schema the context, this in that out, turns It mathematics. Brouwerian with compatible is ) . This kind of reasoninginvesti-contradictionof a furtherkind by is constructively justifiable. notgive This We . ¬휙 ∨ 휙 퐈퐙퐅 ∃.휓 ¬∃푛. decision procedure and and . 퐈퐙퐅 퐂퐙퐅 ( 푛 , and its further restriction further its and , ) are suitable starting points for mathematics building on different constructivistdifferent on building mathematics for points starting suitable are into a contradiction and concluding with transcendental existence of an of existence transcendental with concluding and contradiction a into ¬휙 퐂퐙퐅 . Hence, in the second case, this method is not sufficient for constructivelysufficientfor not methodis this secondcase, the in Hence, . for checking which one of may be interpreted in 퐂퐙퐅 [ 0,1 푛 as arestriction of , we conclude we , ] 퐈퐙퐅 ℝ → implies unwanted instances of 푛 isuniformly continuous). 퐈퐙퐅 with property with ~ 퐂퐙퐅 in chapter 6. With this semantics we can show, that show, can we semantics this With 6. chapter in 9 9 ~ in section 2.4 and further investigate their connec-their investigate further and 2.4 section in 퐌퐋 Löf’s set theory 휙 . On the other hand, we could at no point stop point no at could we hand, other the On . in in a sense-preserving fashion. Thus, we obtain 휙 퐈퐙퐅 퐌퐋 or . in section 2.3 and a whole description of description whole a and 2.3 section in ¬휙 휓 ( holds holds in finitely many steps. 푛 -theorem (and in further consequence ) could be proved by leading the as- the leading by proved be could 퐌퐋 ¬휙 ∨ 휙 seems particularly interesting 퐌퐋 퐋퐄퐌 is given, this will always will this given, is 6 ], Bishop], showed that it at the beginning of this 퐙퐅퐂 . Therefore, in chapter , with the hope of hope the with , WS WS 2019 푛 푛 with /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Introduction cal reasoningcal and are marked with nature. To prevent confusion, we mark classical results with Throughout the thesis, we infer classical and constructive results as well as results of metamathematical Ⓜ . ~ 10 ~ ⓒ . Metamatheorems usually rely on classi- WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. surdity ofsurdity non-existence objectof an T course: of is mean we What existence. its guarantee not does alone object an of non-existence hence we cannot always explicitly construct the next step of our path. To put it bluntly: Th moment when wechoose path, but the word cannot be understood in the sense of constructivism: The crucial point is of quotation in written “construct” word the is Why is a descendant indefinitely. If If for all descendants infinite path infinite mind, algorithm, etc.) constructing etc.) algorithm, mind, by contradiction and aspecial form of proofcases. by The two most important classes of classical proofs that are problematic in a constructive setting 2.1.1 a of followinthe convergeallon they seem, constructivismof schoolsmaythe different However 2.1 as Martin- principles of the field. We will continue with a brief introduction can be seen as least common denominator and a good starting point to get used to concepts and ofversion settheory, that has not been advocated byanyinformal subcurrent of constructivism in particular, an but with start We theory. set constructive of versions some discuss will we following, the In 2 Theory Set Constructive of Analysis and Comparison ⓒ Let Proofby contradiction We will explain the motivation behind this rejection in the following by examples. accept not thus are and middle excluded of law the on rely rules both course, Of Proof

us informallyus discuss some classical proofs that would berejected byconstructivists. Proof by contradiction

König’s lemma us us consider König’s lemma:

Set Theory and Constructivism Constructivism and Set Theory statement of the form “there is “there form the of statement : Let :

Informal constructive set theory Some non-constructiveSome proofs and notions 푇 Löf’s set theory be such a tree. For a node a For tree. a such be :ℕ→푇 → ℕ 훼: 푑 of 2.1 ( 훼 푑 as follows: For follows: as ( KL of 푛 ¬¬휙 ) such such that 휙 ) 훼 훼 퐌퐋 : Each infinite, finitely branching contains tree an infinite path. ( ( 푛

1 + 푛 ) and the and Hilbert-style systems , the trees

) : Itisimpossible, in general decide, which ofthe 푇 푥 푑 푥 do together with a proof of the fact the of proof a with together is infinite. We set 훼 such that such ( es not provide us with aconstruction 푑 0 푇 of ) 푑 take the root. Having defined Having root. the take were finite, then so would be 푇 denote by denote ~ 퐴 ( 11 푥 marks? In the proof we seem to construct an infinite an construct to seem we proof the In marks? ) Proof by cases ” must be given by a method (construction of the of (construction method a by given be must ” ~ 훼 ( 푇 1 + 푛 푑 the subtree rooted at rooted subtree the 퐈퐙퐅 ) and and to 푑 =

Brouwer’s ideas on set theory as well 퐂퐙퐅 . Clearly, we can repeat this process 휓→휙 → ¬휓 휙 → 휓 퐴 ( . 푇 푥 훼 훼 ) ( ( . To get used to this idea, let idea, this to used get To . 푛 푛 ) 휙 , contradiction. Thus, there ) . suppose, 푑 ed by constructivists. by ed . We . 푇 푑 is indeed isindeed infinite,

푇 “ 훼 e absurdity of construct ( 푛 ) is infinite. is WS WS g: A proof A g: course the are proofs operating 2019 he ab- he ” an /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. bounded subset of subset bounded in a constructive setting. When formulating the least upper bound principle in constructive analysi KL finite, contradiction. for each ⓒ Again, classically both notions are equivalent, but every classical Definition 2.6 of least upperbound,ofleastversion2 Definition 2.6 Proof hence through for each upper bound Fan Theorem 2.3 (FT) imal. If 푣 Constructivism Theoryand Set f oe rahbe rm h ro wtot passing without root the from reachable nodes of paths paths Definition 2.5 of least upperbound,ofleastversion1: Definition 2.5 ( Let definitions: following the give we notions distinguishable constructively but equivalent classically of example an As constructivists. by rejected thus and contradiction by is proof the directions, both In through bounded paths not passing through alent toalent distinguishes classically equivalent notions. One example isthe Fan theorem, which equiv-is classically and theorems to become different in constructivism. To put it positively: Often, constructive cannot be expected to be true. Indeed, we will observe the phenomenon of classically equ however, converse, The setting. classical the to passing when preserved be can notions and theorems constru all that means This proof. constructive a accept will mathematicianclassical every usually which shows that 2.2Definition examples the theories classical a of subtheory as interpreted be constructivistmathematics can theories,first-order In ,< 푉, such such that Theorem2.4 → : FT ) beaposet. We saythat FT 훼 푇 푣 oflength : Let 0 ≠ 푧 König’s lemma. must be finite. As finite. be must 퐴 ∈ 푥 → 퐵 퐵 ator before . A bariscalled. A a KL 푇 푇 푣 and and is finite. Trivially, : Let beafinitely branching tree l and eithe : A subsetnodes A of : : König’s lemma isequivalent to 푧 ≥ 푣 퐵 푇 is a node of level ℝ r cannot be a barafter all. be an infinite, bean finitely branching tree infinite with no path. Let 푙 = 푥 has a least upper bound), one usually prefers version 2. An objection to version 1 version to objection An 2. version prefers usually one bound), upper least a has pass through : If 퐈퐙퐅 푢 훼 , wehave

( 퐵 푧 and or there is some ) isabar of afinitely branching tree . uniform uniform bar 푣 was arbitrary in level in arbitrary was 푢 퐂퐙퐅 is an 퐵 푙 ≤ 푢 is a bar and thus, by the Fan theorem, there is some 퐵 are subtheories of 퐵 of a tree a of 퐵 1 − 푧 ator before upper bound of , i.e. . , if there, if isa number , then all its successors 푇 푦 ∈ 퐴 퐴 ∈ 푦 ′ is infinite. By 푇 is calleda is et : For: the fan theorem. ~ For such thatsuch 퐵 훼 12

(

푉 ⊆ 퐴 be 푧 푉 ⊆ 퐴 푉 ⊆ 퐴 퐵 1 − 푧 ) ~ 퐙퐅퐂 a bar,but auniformnot Let bar. . Without loss of generality, suppose this . 퐵 bar of bar if if for all nt en uiom en ta tee r un- are there that means uniform being not

this shows that actually that shows this – we say that say we we say that say we König’s lemma, 푙 ≤ 푦 < 푥 we will discuss these theories later). Indeed, 푧 such that each such path 푇

푇 , then it is a uniform bar. iff each infiniteof each path iff 푑 proof have have the property that 퐴 ∈ 푥 . of this equivalence will not work 푙 푙 is the leastthebound upperof is is the leastthebound upperof is wehave 푇 ′ must have an infinite path, 푢 ≤ 푥 훼 퐵 oflength 푧 must be must be the set bethe of nodes ℕ ∈ 푧 . . 푇 푇 ′ ivalent notions be the subtree bethe 푇 finally passes finally 푑 mathematics such that all is finite and WS WS 푧 ≥ 0 푧 , i.e. , one (for one ismin- passes 2019 s (each ctive 퐴 퐴 푇 /20 iff iff is ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ordered pair oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison 1 Next, we consider an example from set theory. We call two sets two call We theory. set from example an consider we Next, mathematical truth that is rejected in constructivism. 퐵 = 퐴 rational

Proof ⓒ We consider the following simple example: Proofby cases(decidability) on this later. this is problematic according to some constructivists, in the case when this collection is infinite, but more The problem with this proof is that it does not declare a method how to decide to how method a declare not does it that is proof this with problem The could be that this definition i Proposition 2.8 Classical proof The problem with this proof is that we assume that we can decide whether or not not or decidewhether can we thatassume we that isproof this withproblem The On On the other hand, for The onlyThe two-element set both Constructive proof Constructive after any arbitrarily large number of steps. However, we are able to prove the proposition constructively: suchBut procedure a will never tell with certainty sets if are equal indeed, even theyif seem to coincide one. by one element each check to is equal are sets two the if checking of procedure possible only The any follow not do that numbers natural of sets infinite two Considergeneral: in case the 푐 = 푎 푐 another pair of classically equivalent definitions that turn out to be different in constructivism: Note that in this proof, we did not make any assumption on the decidability of set equality. We cont { Gelfond and Schneider proved independently that that independently proved Schneider Gelfond and { or 푎 Proposition 2.7 } , { : If : If 푑 = 푏 . Again, . ,푏 푎, if if they contain the same elements, i.e. 1 √ . In conducting these kinds of case distinctions, one must refer to the kind of transcendental of kind the to refer must one distinctions, case of kinds these conducting In . } 2 } √ . If If . = 2 isrational, put 〈 : If { { ,푏 푎, { ,푏 푎, 푐 = 푏 : If : If 푐 푏 = 푎 } : 〉 , : There are irrational numbers } { { 〈 ≔ is contained in both sides and hence,and sides both in contained is ,푑 푐, 푎 ,푏 푎, } then , is an element of element an is { , then { } 푏 ≠ 푎 〉 푎 } = } = , { 〈 ,푏 푎, 〈 ,푑 푐, 푑 = 푐 = 푏 = 푎 〈 ,푏 푎, , the only one-element set of both = 푏 = 푎 s ,푏 푎,

impredicative } 〉 〈 } 〉 , then ,푏 푎, 〉 . . Wehave the following simple result: = 〉 { and and { √ 푐 = 푎 푎 2 〈 } . Else, let ,푏 푎, } and and thus 〈 ad both and ,푑 푐, : The object and and 〉 ∀푥 and and 〉 ( is 퐵 ∈ 푥 ↔ 퐴 ∈ 푥 √ ~ ,푏 푎, 푑 = 푏 2 〈 = 푎 { ,푑 푐, √ ,푏 푎, 13 { 2 such that such 푎 is irrational, see [ see isirrational, } 〈 〉 ~ √ } . ,푏 푎, is the only element of and thus, and

푙 2 – is defined referring to a collection containing weconclude √ { 〉 2 ,푏 푎, and and and and } ) 〈 푎 ,푏 푎, . We define the set-theoretic (Kuratowski) = 푏 = 푏 〈 isrational. ,푑 푐, { { 퐴 푎 〉 푐 and and and } } √ 〉 26 or or = ae qa. If equal. are 2 푑 = 푏 ] . In . In both cases, { { 푐 퐵 〈 ,푏 푎, } ,푑 푐,

or (extensionally) equal (extensionally) . } 〈 〉 { ,푑 푐, must be = 푎 } { 〉 ,푑 푐, = whether or not not or whether . Hence { 푏 = 푎 푑 = 푏 } ,푑 푐, . In either case, either In . 푎 { 푏 푎 } isrational. . In either case, either In . } . But this is notis this But . 푑 = 푐 = 푏 = 푎 . Hence, then , apparent law.apparent WS WS and write and 〈 √ 2019 ,푏 푎, 푐 = 푎 2 inue √ = 푏 2 〉 /20 푙 ∎ ∎ is ∎ ∎ =

– . .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 푉 distinguishing it from classical set theory, as there is a global relation of equality defined on set constructive of feature unique a is equality an withequipped coming sets that writes Beeson to classical although demanding (ii) is certainly true for true certainly is (ii) demanding although debatable is however, this, of justificationconstructive (the well as relation global a be will there is the important objection that given a set a givenobjectionthatimportant the is there Although merely informal, we cannot leave this definition without remarks. Concerning (i) quences representquences thedifferencesametheir realif approaches equality of not not so much talk about “constructing” the set (defined according to the “equality” given by intuition) and inherits the global equality from equality global the inherits and intuition) by given “equality” the to according (defined and and definiti member One One example of a set is the set unique feature of (informal) constructive settheories all. after coincides. How would we interpret expressions like like expressions interpret we would How coincides. Following 2.1.2 Constructivism Theoryand Set 2 Definition 2.9 푓: Clearly, classically both notions coincide.Constructivelyhowever, aretheydifferent. example, For let be any be any unsolved mathematical problem, like Goldbach’s conjecture. whether todefine a function with domain Clearly, Brouwer would reject this, see [ see this, reject would Brouwer of all sets. This statement seems imprecise for two reasons: Firstly, in the theories { 휙 퐴 ,…,푛−1 − 푛 , … 0, (iii) (ii) (i) satisfying some property . 퐂퐙퐅

on of equality is “hidden” in the equivalence classes. We see that see We classes.equivalence the in “hidden” is equality of on The notion The setof (if required with regard to suitable restrictions on restrictions suitable to regard with required (if 퐴 ahmtc. o eape i dfnn te e o ra nmes itiiey to Cauchy- two intuitively, numbers, real of set the defining in example, For mathematics. on the set of real numbers in this way. Of course, 푛 is a subset of the finite set , the object the , Beeson’s analysis and prove that the equality defined (ii)in is anequivalence relation. say what has to be done to prove canonicaltwo members of say what has to be done in order constructto canonical members of } : A set 퐴 → . The set . The 퐴 푠 is called ( 푛 )

is canonical too. Two canonical members are equal if their “numbers of “numbers their if equal are members canonical Two too. canonical is 퐴 [ iscalled 4

62 휙 ℕ ], in order], todefin (Kuratowski) finite ( ] of natural numbers. Its canonical members are 푥 ) is acceptable in constructivist set theories such as Martin- { 0,1 = 퐴 subfinite } and and thus subfinite. But to show that { { :푥=0∨ 0 = 푥 푥: 0 } or 퐵 informal iff it is asubset ofa finite set.

but rather of “separating the e ~ 퐴 iff there is a number { e a set e aset , the set the , 0,1 14 } ( , i.e. we would to need solve or reject ~ 푃 ∧ 1 = 푥

10 constructive 푆 , one must, one 10 = 퐵 technically or 휙 ) { 0 2 5 + 3 :휙 퐴: ∈ 푥 . In constructing In . ) . } In .

formally Consider set the mathematics Although ? 푆 , one considers equivalence classes 푛 equal, together with a bijective function ( 푥 (ii) does not seem to be that a that be to seem not does (ii) ) } , i.e. the subset of elementsofsubsetthe i.e. , 푆 , whatdoesoneto , is 퐴 , lements of is finite we need to know 0 10 퐵 and and for each canonical – however, one would one however, 퐈퐙퐅 10 so it is for is it so

is and and o a canonical a not 퐴 ” with respect 퐂퐙퐅 the universe 푃 it seems that ). Secondly, ). Löf’s or . WS WS , equality informal 푉 2019 define theory

– the 푠 퐈퐙퐅 /20 se- ’s” 푃 푥

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. crucial in Martin- As another important example we may consider, given sets given consider, may we example important another As tively, the set the tively, ing to (i), or (i), to ing We thus say that yields some canonical member of member canonical some yields each each ℕ intuitionistic logic (from on: here “BHK”, is constructivism of spirit the in semantics hensible In constructive mathematics, truth coincides with the notion of provability. The most important com 2.1.3 Theory Set Constructive of Analysis and Comparison Note that the clauses for the that Note We will here give a variant of BHK-semantics tailored for the language of set-theory. We s We set-theory. of language the for tailored BHK-semantics of variant a give here will We proves all constructions all of the mind. bethecould collection of natural all numbers or more abstract notions such asthe universe of sets all or Such object”. an “being and object” “each about talk we can then Only discourse. of domain

of member 푓 relation. 퐵 ( 푥∈퐴 휙 퐴. ∈ ∀푥 퐴 . Intuitively, the expression 푥∈퐴 휙 퐴. ∈ ∃푥 푎 equal, one needs to prove that for each for that prove to needs one equal, 1 푥 휙 ∀푥. 푥 휙 ∃푥. ) 휓, → 휙 휓, ∨ 휙 휓, ∧ 휙 퐴, ∈ 푎

퐴 ∈ 푎 = ¬휙, ⊥, BHK-interpretation 퐵 ( (

푓 푥 푥

( ( )

( )

ℕ a member a 푥 푥 , 푎 ,

푎 ) ) in the just definedthewouldsense,we just in likesay thatto belongsit to set the 2 is a method giving instructions for its own evaluation and of which we can prove that i that prove can we which of and evaluation own its for instructions giving method a is , 퐵 , )

. If this is the case, we writ 퐴 of functions of if if if if if if never if if if if 푥 푎 Löf’s set theory .

is is an element of the set 푝 푝 푝 푝 푝 푝 푝 푝 푝 isatriple isamethod transforming aproof of isapair isamethod giving for objecteach proves isamethod transforming proofs of isapair isapair pair a is 퐵 ∈ 푏 ∃푥 휙 ∃푥 → ⊥ 휙 10 〈 〈 〈 〈 ( :퐴→퐵 → 퐴 푓: ,푞 푥, 푞 푛, 푟 푞, in such a way that equalities are respected, i.e. if if i.e. respected, are equalities that way a such in ,푞 푥, 푥 〈 ,푞 푟 푞, 푎, 10 )

and especially and – 〉 〉 〉 〉 stands stands for a , where , where 퐴

, where , where , where , where

, where where , this is why we included the word “canonical” in (i) and (ii). . This distinction between canonical and non- and canonical between distinction This . 〉 , where . To give an element an give To .

퐴 e and and write [ 푞 푥 푛 31 푥 푓 퐴 ∈ 푎

is a isa (construction ofan) object and isa natural number and is a canonical member of member canonical a is pro ( ] 푎 , 푞 ) method computingcanonical a member of [ isa proof of for the element 29 ves ~ ∀푥 휙 ∀푥 , ], [ ], 푓 15 ( 휙 푎 퐴 ∈ 푎 37 the “Brouwer the ( and and ~ ) 푥 ], ], [ 푥 = ) make only sense, if we have specified a certainspecified a have we if sense, only make aproof 휙 퐵 퐴 ∈ 푎 30 if if either into proofs into of 푟 푔 푓 proves ], for adiscussion see [ ( 퐴 ∈ 푎 of 푎 퐴 ) into aproof into of and and . Clearly, this equality is an equivalencean is equality this Clearly, . 푏 퐵 푝 as usual. To prove two elements 퐴 and and ( means to give a way to construct forconstruct to way a give to means 푎 -Heyting- 푥 휓 퐵 is a canonical member of )

푞 with equalities with of 퐴 proves 푟 and isaproof of 휙 휓. (

푥 ) Kolmogorov” 푞

proves that that proves 휙 휙 푞 ( if if proves 푎 ) 4 ca 0 = 푛

] or] [ nonical elements is elements nonical ℕ 휙 = ( and write and 퐴 푎 and and 64 푎 and and 휙 ) 1

( ] 푎 -semantics of -semantics ℕ = 푥 ). reduces to reduces )

. 퐴 휓 = WS WS

a domain a 퐴 푎 , else.

ay that ay 퐵 , accord- 2 respec- 2019 10 then , ,푔∈ 푔 푓, pre- 10 /20 ∈ 푝 t

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. this this notion toemphasize differencethe to classical sets. writingsearly his in that Note ated more or less freely from the preceding ones and species constructing new mathematical objects: Choice sequences Choice objects: mathematical new constructing toneed give a constructionof the continuum. In the second act of intuitionism, acceptedhe two ways of recog Laterelements. however,he denumerablymanyonly toconstruct allows intuitionus as standpoint [ 〈 describes the actfirst of intuitionism as [ In constructions. mental by justified is objects mathematical of existence The formalism. la ratherthan intuitionupon based mathematicsbe shouldpoint, his in suggests, ofschoolintuitionistic the of fatherfoundingthe as seen be canBrouwer Luitzen 2.2 Constructivism Theoryand Set In In particular, Brouwer rejects Cantor’s set theoretic construction of the continuum. According of apair. Hence We can give a proof the logical truth logical the proof a give can We Examples exactly proofs of natural deduction. e prove We In the following examples, we will use the choice is not by chance: Howard proved in 1969 that programs of the simply-typed Here, Here, as usual, “ 푞 푏 ,푟 푞, . Altogether, proves Completely separating mathematics from separating languageCompletely mathematics f mathematical hence and of all quality, it passes into itempty quality, into the of all form passes of the common substratumtwoities. of all described as fallinglifeinto of a moment of whichthings, two the distinctdescribed apart one gives way to the other, but is retained by memory. If the twoity thus born is is retained the divestedis gives other,but Ifway the to by memory. twoity born thus the phenomena of language described by theoretical logic, language theoretical of phenomena by the intui- described that recognizing origin in the perceptionmovethe a in origin of of time. This perception of of time be move may tionistic anmathematicstionistic essentially havingis of languageless mind its activity the

〉 And it is this common substratum, this empty whichis form, is this it the this And common intuition of substratum, basic , where where , Brouwer’s set theory 푎 ∈ 푥 : ( 10 푏 = 푎 ∧ 푎 ∈ 푥 푞 휆푝 ], ], the proves and and 휓 ↔ 휙 휆푥 pr . pr . 푟 1 continuum is given as an intuitive notion. It is impossible to conceive “all” its points, ( proves 1 pr 휙 ( ” is an abbreviation for 푥 and and 1 ( ) ) 푝 proves 푏 ∈ 푥 → ) ) 푏 = 푎 푟 ( proves 푥 , he referred to species as “Mengen” (germ. “set”); only later he changedhe later only “set”); (germ.“Mengen” as species to referred he , ) ( pr

( then , 휓 ∧ 휙 where , 2 ( 푝 휓 ( ) 휓 ∧ 휙 . Thus, Thus, . ) ) proves pr 휙 → 휆 1 -calculus informally to make our argumentation clearer. This mathematics. 푏 = 푎 ( ) 푟 . 휙 → 휓 ∧ 휓 → 휙 )( 휙 → pr ~ 푥 ( 푏 = 푎 ∧ 푎 ∈ 푥 1 )

16 ( proves me as follows: Suppose, follows: as 푝 ) ~ proves n etninl qaiy i.e. equality, extensional ans 푏 ∈ 푥 → 푎 ∈ 푥 – infinite sequences whose elements are cre- are elements whose sequences infinite – . If 휙 properties of objects previously acquired. ) , where , 푏 ∈ 푥 → = 푝 〈 ,푟 푞, pr . . Thus, Thus, . 푝 〉 1 is a proof of proves extracts the first component first the extracts pr thought. As the namethe thought.As nguage or any kind ofkindany ornguage 휓 ∧ 휙 휆 1 -calculus correspond ( 푟 ∀푥 )( 푏 = 푎 ∧ 푎 ∈ 푥 푥 ( , then , 푏 ∈ 푥 ↔ 푎 ∈ 푥 )( 푞 ) 8 proves to ], Brouwer ], WS WS 푝 nized thenized is a pair a is his early 2019 ro , i.e. m ∈ 푥 /20 ) .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. For example, the spread all naturalof numbers, denoted order of order of a species thus can We Russel’s. as such paradoxes block numbers 퐵 straightforward to define union, intersection {( object object A 2.2.1 Theory Set Constructive of Analysis and Comparison

laws laws called a called spread sequences spread). In[ spread). from functions of case special the for notions these reserves Brouwer (although We can think of a spread as tree where each node is labelled by a mathematical object. mathematical a by labelled is node each where tree as spread a of think can We This This is the reason, Brouwer developed the notion of spread. maythis seem, it does not permit asatisfying construction of the continuum.

can be defined as a species given by a method constructing from each element of element each from constructing method a by given species a as defined be can ,훼 푛, species (ii) (i) (B) (A) Λ

(

푎 푛

and and

푀 aia bet matical object: The complementaryThe spreadlaw Species and Spreads The spreadThe law ing restrictions:ing is an element of the spread the of element an is ) choice sequencechoice iii. iv. ) is a (constructive) property (constructive) a is ii. we can thus define the species 휔 ∈ 푛 : i. ( ( 훼 If the already constructed objects have order choicebers, sequences etc.) arespecies of order 0. rati numbers, natural of sequences numbers, (natural objects mathematical Concrete

Γ 푛 7 ( : 1 ], a theoryofconstructiblea ], ordinaland However cardinalpowerfuldeveloped. numbers is 푛 푛 , ) If If It It decides which sequences of length one are accepted. If cepted or not quence defined as is follows meaningful: ) 2 } 푛∈휔 ( ( ( … , . Note that many set-theoretic notions can be developed within this framework. It is It framework. this within developed be can notions set-theoretic many that Note . 푛 푛 푛 1 1 1 푛 , 푛 , 푛 , ) my e ersne a seis f order of species as represented be may , where each initial segment is accepted. The correspondingaccepted.sequence The segment initialis each where , 2 2 2 (

Λ 푛 , … , 푛 , … , 푛 , … , – 푛 decidesfiniteifsequencea of natural numbers accepted is or under not thefollow- Brouwer considers them to be perfectly justified mathematical objects. To eachTo objects. perfectlymathematicaljustified be considersthemto Brouwer 1 푛 , 2 푘 푘 푘 푛 , … , 푛 , ) ) i acpe, t eie fr eac for decides it accepted, is i acpe, hn hr i a aua number natural a is there then accepted, is 푘+1 푘 ) 푚 , is accepted, also 퐴 ) is accepted . 푃 Γ Note that the clause “of already defined objects” is essential to essential is objects” defined already “of clause the that Note assigns to any to assigns of already constructed objects. As usual, we write we usual, As objects. constructed already of [ 푀 ] of choice sequences of or ( the empty species. A function between two species 푛 construct species one step at a time, hence the notion of of notion the hence time, a at step one species construct ~ 1 푛 , 17 ( 2 푛 ( 푛 , … , 푛 1 ~ 푛 Λ 1 푛 , ( , then 푛 , -accepted sequence an alreadysequenceconstructedan-accepted mathe- 푛 2 휔 푘 2 푛 , … , 1 ) ) ) , has order, has A h 훼 ↦ 훼 ↦ 훼 ↦ ⋮ ⋮

푃 푚 spread 푘 applied to them has order ) i sm sequence some if , 1 2 푘 is accepted.

2 푀

nml te s pce o pairs of species as the namely , = 푀 . Note that for a choice sequence 1 . A recursive. A sequence of natural ( 푚 ,Γ Λ, sc ta te ucso se- successor the that such , ) is determined by the two species generated by a by generated species ( 푛 1 푛 , 퐴 an element of element an 2 푀 푛 , … , 1 + 푛 - sequences ( 훼 퐴 ∈ 푎 WS WS 푘 1 . onal num- onal 푚 , 훼 , 2 2019 ) … , if the if 퐴 i ac- is and are ) /20 is 퐵 훽

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. of functions: W Similarly, we assign tofinite binary choices of the binary spread instead intervals, as indicated: e hoyadCntutvs Constructivism Theoryand Set Finally, the continuum We can We can think of elements the of to check each initial segment of as as constructed by another spread it is in general undecidable whether or not Examples e • • • • can thus define the spread of the real unit interval unit real the of spread the define thus can

given by the correspondence complemen The sequences. successor some permit always will law this Clearly, Given Given an enumeration of the rational numbers themselves. cepted spread law: We accept all sequences of length 1. If 1. length of sequences all accept We law: spread of choices only step each in allows spread binary The finite sequence to itself, i.e. as law spread complementary The chosen. be to numbers natural every for step each in The species of all functionsall of species The If the spread law permits choicesno of natural numbers, we end up with the empty spread. : We give some important examples of spreads: 0,1 ℝ is given asthe spread of all equivalence classes of 2 1 훽 [ 0, 푆 ⇔ 훽 ≈ 훼 0 , . ( 휔 → 휔 2 1 ] 푛 asCauchy-sequences ofrational numbers. Wedefine equality on ( 1 푛 푛 , 1 푛 , … , 2 푛 , … , , denoted , | 훼 푘 ) 푛 푘 ) (푞 ↦ 훽 − |푞 ↦ ~ 푛 푘 ( 푛 휔 18 푛 푛 | 푞 − 휔 1 1 푞 < 푞 , … , 푛 ,

1 can be defined via a spread: The spread law permitslawspread The spread: a via definedbe can ~ 푞 , 푚 2 2 1 [ 푛 , … , 푛 < | 0,1 2 … , 푛 o l 휔. ∈ 푛 for all ( 푘 푛 ] ) 2 we define the following spread . To conclude, we discuss Brouwer’s notionBrouwer’s discuss we conclude, To . 0 . 1 푘+1 푘 1 or 푛 , … , 4 3 4 2 4 1 0, ) . , , , 4 1 4 4 4 3 4 2 . 1

. Again, finite sequences are mapped to mapped are sequences finite Again, . 푘 … … … ) … is accepted, then accepted, is

∈ 훽 [ 푆 ] [ under under 푀 ] , since we would need ( ≈ tary spread law is law spread tary 푛 . . 1 푛 , … , 푆 by by giving its WS WS 푘 signs each each signs 푚 , 2019 ) is ac- is [ /20 iff 푆 ]

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. istic function, then it is continuous in the sense that for thatsense the continuousin is it function,then istic 휙 Definition 2. Recall the definitions of continuity from analysis: of an of an intuitionistic function is formulated asthe axiom ℕ ∈ 푚 From this definition we immediately see the following continuity principle: If ( for each each for Clearly, this theorem is classically valid,not since function the Theorem2. Given Given a spread 2.2.2 Theory Set Constructive of Analysis and Comparison where we abbreviate we where If weputIf in general,in decide by afinite procedure whether continuous. cannotnot Note intuitionisticallyas we however,is that total,not speaking, this function is

Unfortunately, in defining functions ( style [ Brouwermathematicsdo to wishesone whenever formalism constructive any to added be may axiom

nal nal fashion and defined instead a function of the of the linear continuum” on associates function, the of definition” of “domain the form and […] continuum, unit the tions tions followingThe approachdefining of these is functions sequence an 훼 ( 1 훽 훼 , … , ( ) 퐖퐂퐍 , whenever

[ such that such for all 푀 59 Intuitionistic functionsIntuitionistic ] 푁 푀 ]). 훼 [ 휔 → ) 푀 ) -sequenc 11 . ] 푀 10 : Every function ℝ = and can be found in [ in found be can and -sequence based onthe law : A function 푀 ( 훼 , an in the in above uniformity principle, we can thus say 1 e 훼 … , 훼∈휔 ∈ ∀훼 훼 intuitionistic function 훼̅ 퐷 ∈ 푦 antrl number natural a 푛 푁 = [ ) 11 = ( :퐷→푊 → 퐷 푓: ]. ]. 훼 : 휔 ( | 1 푥∈휔 퐴 휔. ∈ ∃푥 . 훽 :ℝ→ℕ → ℝ 푓: 푦 − 푥 훼 , … , 1 훽 … , | 푛 푁 with with [ 50 2 < ) 푀 . Although clearly incompatible with classical mathematics, this mathematics, classical with incompatible clearly Although . )

iscontinuous. ]: A function A ]: ] – ( −푚 simply put 푓 → ,푥 훼, 푁 ( ,푊⊆ℝ ⊆ 푊 퐷, 휙 [ Φ 푥 0,1 [ 훼 →

푁 ) ∗ ) from from adantrl number natural a and wich correlates finite sequences of naturals such that = ] 훼∈휔 ∈ ∀훼 → ] | between two spreads, Brouwer did not stick to the inter- 푓 ~ ℝ → { 0 = 푥 ( 0, 0, 1, [ 푥 19 푀 ) is

푓 − ] to be “a law that, with each of certain point cores of max = 푁

~ to [ or moretheinspirit Brouwer’sof definitionfunc-of 푀 continuous if if

,훽∈ 훽 훼, ( 휔 휔 ] 푥≠0 ≠ 푥 푥=0 = 푥 푦 0 ≠ 푥 푛 .∀ 휔 ∈ ∀훾 휔. ∈ 푏 ∃푛, . , denoted → ) | 2 < [ 푁 ( [ 푁

푀 :ℝ→ℕ → ℝ 푓: . In textbooks,. In assuch [ ] is a law which corresponds to each each to corresponds which law a is −푛 훼 ] at 푁 , there exists some exists there . 훽 휙: 퐷 ∈ 푥 ) 휙 ). ). [ ( 푀 훼 with ) ] , if for each bsdo teiiil segment initial the on based 휔 → 휔 ( 휙: 훼̅ is a law 푛 [ 푀 훾̅ = ] 푛 휔 → 푁 ℕ ∈ 푛 66 퐴 → such that such 휙 ], ], definitionthe is an intuition- that computes ( there is some ,푏 훾, e point core point e WS WS ) ) , 휙 2019

( 훼 ) /20 푁 = - -

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. in Martin- in In layer). first the of elements(propositions,interplay their and layer)second the of (elements discourse of elements the between divide a is there sense, this In theory). set of example, (for ground logic, be it intuitionistic or classical) and secondly, the particular axioms of the respective t Axiomatized mathematical theories usually come Firstly, layers: two in adeductive system (some back- 2.3 Constructivism Theoryand Set like like Agda and Idris have been based on concepts featured in prog and Coq and NuPRL like assistants proof automatedvarious that surprise no of concepts from programming such as data structures, data types and programs. Hence, it will come as explained are functions and elements sets, as such objects mathematical intuition, this lowing To get tothe used ideas involved, let investigateus division the theorem for natural numbers, an object The task proach, Following the BHK-interpretation, this proposition corresponds to an algorithm an to corresponds proposition this BHK-interpretation, the Following

( numbers and this result,this namely, that a hypothesis of discontinuity of areal function leads to contradiction. We haveWe the following generalizedcontinuity principle: In [ In Theorem2. givesThis theus following result: ,푏 푎, 11 (i) (iii) (ii)

) of natural numbers and converting proofs of proofs converting and numbers natural of ], Brouwer, using his more general notion of real function, proved only a negative for negative a only proved function, real of notion general more his usingBrouwer, ],

Martin-

퐌퐋 of proving a theorem is identified with a special case of the mathematical activity of constructing – Löf’s we speak of this correspondence as If Φ Φ Φ Φ aimstobetter model the constructive activity ofa mathematician: 12 ( . ∗ ∗ 푚 < 푛 is not finally constant ( 훼 푝 : Every realfunction 훼 ) isaproof that Löf’s set theory 1 set u Φ sup = Π ∶ Φ 훼 , theory ( theory 2 then , 훼 , … , ( ( ,푏 푎, ∗ 푚 ( Φ 훼̅ ) 푛∈휔 푚∈휔 (훼 휔. ∈ ∃푚 휔. ∈ ∀푛 푎 ℕ ∈ 푏 ∀푎, ∗ . 푛 퐌퐋 ) ( ) ℕ × ℕ : 훼 , i.e. , 1 푏 < 푟 ), both layers come to the same basic notion: sets notion: basic same the to come layers both ), 훼 , 2 Φ 훼 , … , ℝ → ℝ )[ ( and and 훼

[ 퐺 → 0 > 푏 ) 0 is approximated by the segments theapproximated by is 푛 ( ) = 푎 푏 iscontinuous. Φ ≺ ) (Σ → 푞푏 ∗ 푚 ( propositions- ( 푞 = 푎 ∧ 푏 < ℕ 푟 ∈ 푟 ∃푞, 훼 ~ 푟 + 훽 = ( 1 ,푟 ℕ × ℕ 푟: 푞, 훼 , 20 . In symbols,. In we could write: 푚 2 0 > 푏 훼 , … , ~ → ( into triples into Φ 푚 as ) ( ) ( 훼 , i.e. i.e. , -types 퐌퐋 I ) ( ) 푎, 푛

Φ (see for(see example[ 푞푏 = (or ∗ ( ( 푟 + 훼 푞푏 Φ ( 1 Curry-Howard correspondence ( ,푟 푝 푟, 푞, 훼 , ) 푟 + 훽 Φ 2 퐺 × ) ∗ 훼 , … , ) ( ) 푛 훼̅ ] ) ) 0 푛 . , where where ,

( ) 푟 − 푏 . We say thatsay We . 푛 ) . In this “logic this In . i niiil emn of segment initial an is Φ 2 ) ramming languages ramming ], [ ], )] operating on pairs on operating 푞 푞 . . 33 and and

], ], [ 푟 39 Φ mulation of mulation are natural are ∗ WS WS ], [ computes - in terms in free” ap- free” contrast, BO 2019 heory ). ]). Fol- /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Note that in our example, proving the division theorem comes down to constructing a me a constructing to down comes theorem division the proving example, our in that Note and findingand that both computations converge to the same canonical element of ℕ × ℕ its quantification: existential to meaning constructive giving sum, disjoint (generalized) the ( oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison as central in first-order theories, this distinction is still present: Formulas play the r the present:Formulasplay still distinction istheories, this first-order incentral as From Frege tothe Principia the distinction ofpropositions and judgements has been vital: Although not 2.3.1 In a nutshell, the theory particular set to the metamathematical level. In In metamathematical level. the to

ones (there are rules for “ propositions- Martin- Following proofs. its ments of proofs of “ of proofs of thespecifiedinto codomain. Itgivesanatural interpretation for the universal quantifier. of) the division theorem with its proof. its with theorem division the of) “ Here, and “ and they standthey for statements can that bemade mantics. The only kind of judgements in these theories therefore is of the form “ (possibly)some via trueidentifiedas beenhave thatformulas arejudgements, In In set 퐴 퐴 퐴 퐴 퐬퐞퐭 푨 isaproblem (task) isan intention (expectation) isaproposition isaset 퐌퐋 with 1. 3. 2.

퐴 ∈ 푎 (we will discuss this important notion in Martin- are witnesses to the formula on the right: the on formula the to witnesses are , there are, there four kinds of judgements:

Π Propositionvsjudgement 퐴 set 퐴 퐴 ∈ 푎 퐵 = 퐴 ” denotes the product indexed by pairs of natural numbers, i.e. the set of functionof set the i.e.numbers, ofnatural pairs byindexedproduct denotes the ” 푘 elements) and gives rules for constructing new for rules gives and elements) ” can ” can have the following interpretations from as ( ( ( 퐴 0 > 푏 Π -sets 푎 퐴 is awell-formedset)

( is and ( ,푏 푎, anelement of ised f rpstosa-ye) Ipeetn ti ie, h asrin or assertions the idea, this Implementing propositions-as-types). of (instead ”; interpreting implications, “ implications, interpreting ”; 퐵 ) ℕ × ℕ : areequalsets) 퐌퐋 → ”, postulates the existence of certain ) [ "Π Löf’s early nomenclature, we may thus speak of this correspondence as correspondence this of speak thus may we nomenclature, early Löf’s … 퐴 ”, “ ] ) ) . Following the constructivist tradition, we have thus equated (the truth 퐌퐋 Σ (doing (doing the task) 푎 intentionthe (expectation) 푎 osition 푎 푎 푨 ∈ 풂 ” and so” and on). is a proof (construction) of the prop- the of (construction) proof a is isan element of the set i a ehd f ovn te problem the solving of method a is i a ehd f uflig (realizing) fulfilling of method a is we even go a step futher and futher step a go even we Φ

itself becomes a mathematical object and no longer restrictedlonger no and object mathematical a becomes itself 퐴 within

~ →

I the theory. the Theorems on other hand, the representing ( 21 ”, 푎, 퐴

as usually denotes function spaces, “ spaces, function denotes usually as 푞푏

~ Löf’s theory more rigorously in a moment) “ 푟 + [ 43 sets ) ]: are programs executing both both executing programs are 퐴 sets

퐴 and their canonical elements from givenfromelementscanonicaltheir and

, like identify ℕ (set of natural number) and the theorem with the set ofset the withtheorem the 퐴 퐴 퐴 퐴 issolvable isfulfillable (realizable) istrue isnonempty ℕ external 휙 . istru ole of propositionsofole formalism or se- formalismor e”. 퐺 0 Σ

s from s 푎 ( mber ” stands for stands ” 푏 and and WS WS ) index set index is the set isthe 2019 푞푏 Φ 퐴 set 퐴 ℕ × ℕ judge- of a of 푟 + /20 ℕ 푘 ” –

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. these assumptions the expression “ expression the assumptions these The rules of rules The from existing ones. We consider the following example of the Cartesian product of rules The 2.3.2 Constructivism Theoryand Set Here, the expressions introductionThe rule declares canonicalhow elements of 4. as“ ally, judgement 3. should 3. judgement ally, Example 퐵 Remark: × are obvious andwe will avoid writing down any trivialsuch assumptions thein future. should be read as follows: “Provided follows: as read be should × × to thesame result. 퐴 퐌퐋 “Provided when executed, yields a canonical element of the set set the of element canonicala yieldsexecuted, when are equal if, when executed,when if, equal are . -elimination -formation -introduction follows our intuitive definition quite closely. An element 4. 퐵 × 퐴 ∈ 푐

푎 and and The rules The of 퐴 ∈ 푏 = 푎 In formulating this rule, we should have written down assumptionsdown shouldwritten have we rule, formulating this In : We define the set 퐴 ∈ 푥 퐵 ∈ 푏 퐴 ∈ 푎 퐴 set 퐵 set 퐵 set 퐴 푏 × 퐌퐋 are programs yielding equalcanonical elements of -formation explain how to construct the set the construct to how explain -formation E ( ,푏 푎, postulate the existence of certain sets and describe how new sets may be constructed be may sets new how describe and sets certain of existence the postulate ( and and ( ,푑 푐, 퐴 × 퐵 set 퐵 × 퐴 푎 and ) ) 퐌퐋 퐵 × 퐴 ∈ 퐵 ∈ 푦 퐶 ∈ 푑 푏

( ( areequal elements o ( ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 푑 ,푦 푥, be read as “ as read be 푐 ( , the objects ( ) ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 퐵 × 퐴 ,푦 푥, ) 퐶 ∈ 푎 )

and and

퐶 ∈ vatefloigrls viathe following rules: ( ,푦 푥, 퐵 × 퐴 ( 푏 ) ,푦 푥, 푎 yield equal canonical elements of the set the ofelements canonicalequal yield ) 퐴 ∈ 푥 푑 is a program yielding a canonical element of element canonical a yieldingprogram a is

) 1 ) ( ” does not make any sense. However, assumptions like these like assumptions However, sense. any make not does ”

,푦 푥, and and and f ) the set and and 푐 ~ 1 퐵 ∈ 푦 푐 = 22 푑 푑 1 ( 2 2 퐴 퐴 ~ ,푦 푥, ( as a result. Two arbitraryelements Tworesult. a as ) ) , the object object the , 퐵 × 퐴 ∈ ,푦 푥, ) ( 퐵 × 퐴 ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 ) 퐵 × 퐴 푑 = 퐵 ∈ 푑 = 푏 퐴 ∈ 푐 = 푎 are equal elements of E 푎 ( of a set 2 푐 퐷 = 퐵 퐶 = 퐴 ( ( 1 look and like when they are equal. ,푏 푎, and when two such constructions leadconstructions such two when and ,푦 푥, 푑 , 퐴 푑 ). ). 1 퐷 × 퐶 = 퐵 × 퐴 ) ) ) ( ,푦 푥, = E = 퐶 ∈ 퐴 ) is a method (or program) which, ( ) ,푑 푐, 푑 ( ( is an element of element an is ,푦 푥, 푐 1 2 ( ) ,푦 푥, 푑 , ) 퐵 × 퐴 ∈ 2

퐴 set 퐴 ) ) ( 퐴 퐵 × 퐴 ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 퐶 ∈ 푑 = as results. Thus, actu- Thus, results. as 퐶 ( and and ,푦 푥, ( 2

푐 퐴 ( of two sets ) ,푦 푥, ” and and ” ) .” 퐵 set 퐵

) , respectively. ,푏 푎, 퐶 퐶 ∈ ( ) judgement ,푦 푥, WS WS

of the setthe of – ( without ,푦 푥, ) 2019 .” and .” 퐴 ) and

/20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ter be a pair We can now give our first “proposition logic to intuitionistic logic, it will turn out that there are other axioms of axioms other are there that out turn will it logic, intuitionistic to logic restriction these to victim fall to first Finally, the ruleof Theory Set Constructive of Analysis and Comparison The idea behind the theories behindthe idea The 2.4 set of proofs of set We can now, for example, derive the logical rule if if we substitute ∧ ∧ ( choice ( write × input an on follows as works tuitively, in elements of behavior the explain rules elimination The ∧ thus translate the in. Clearly, in. ,푏 푎, -elimination -introduction -formation -equality 5.

) of 퐴 prop 퐴 퐈퐙퐅 퐙퐅퐂 퐵 × 퐴 ( ,푏 푎, and ) 퐙퐅퐂 instead of instead – 퐵 × 퐴 ) the Hilbert- . Itthen returns of proofs is constructively not acceptable, so we try to carefully restrict it. Of course, Of it. restrict carefully to try we acceptable,so constructivelynot is 퐴 for 퐂퐙퐅 × . We therefore We . × -rules into rules familiar form logic: -equality declares how the 퐶

in in the rule of 퐴 set 퐴 푎 of style theory based on classical logic that most today’s mathematics is encoded and and 퐴 퐈퐙퐅 푑 and 푑 퐵 ∈ 푏 퐴 ∈ 푎 ( ( and and ,푏 푎, 퐴 true 퐴 define 푏 E of ∧ ) ( – 퐂퐙퐅 ) -elimination. We will continue discussing the rules of ( 퐴 ∧ 퐵 true 퐶 true 퐶 true 퐵 ∧ 퐴 however, even after restricting the original classical backgroundclassical original the restricting after even however, -as- ,푏 푎,

퐶 ∈ to indicate that there is some is there that indicate to 퐵 퐵 × 퐴 ≡ 퐵 ∧ 퐴 퐵 × 퐴 ∈ 푐 퐴 prop 퐵 prop 퐵 prop 퐴 퐴 true 퐵 true 퐵 true 퐴 . It is therefore natural to identify the proposition is simple: We start with the Zermelo-Fraenkel set theoryZermelo-Fraenkelwithset the start with simple:We is set” ) ( 푑 , ,푏 푎, ) -interpretation. Namely, a canonical proof of 퐴 ∧ 퐵 prop 퐵 ∧ 퐴 퐴 ∧ 퐵 true 퐵 ∧ 퐴 퐴 ∧ 퐵 true 퐵 ∧ 퐴 ) (,푏 퐶( ∈ 푏) 푑(푎, = E ~ . -operator works canonical on elements. 퐴 true 퐴 퐶 true 퐶 : It executes the method the executes It : 23 ( 퐴 true, 퐵 true 퐵 true, 퐴 ~ . When interpreting proposition as sets, we often we sets, as proposition interpreting When . (

( ,푦 푥, ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 퐵 × 퐴 ) ( 퐶 ∈ ,푏 푎,

. It introduces the introduces It . )

( ) ) ( ,푦 푥, ) 퐴 ∈ 푎 ) )

퐙퐅퐂 푐 to find a canonical element canonical a find to (some proof of proof (some that will still get us get still will that E -operator which, i which, -operator 퐵 ∧ 퐴 퐵 ∧ 퐴 퐌퐋 퐋퐄퐌 퐴 WS WS ). We can We ). in in chap- with its should 2019 will be will 퐋퐄퐌 /20 n-

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ioms if ioms if we instead defined equality viathe axioms for equality like It means that two sets are equal if they contain the same elements. Usually, the background logic con The The Extensionality retic axioms of our theory: relational symbol being “ being symbol relational The theory 2.4.1 Constructivism Theoryand Set This means, that unlike in unlike that means, This Zermelo-Fraenkel-style set theory th with up end we axioms, these all replaced Having theory. initial the restore to back ones equivalent (classically) by axioms these replace will We back. The The Pair It turns out that this seemingly innocent axiom already implies some unwanted instances of shortly, when discussion axiom the of pair. the the is that we are able to define notions by the tensionality such an element does not only exist but is also unique also is but exist only not does element an such tensionality pair. The singleton The pair. restrict our theory and end up with the Constructive Zermelo-Fraenkel-style set theory set Zermelo-Fraenkel-style Constructive the with up end and theory our restrict further We axioms. impredicative for true especially is This view. of point constructive the form atic It says that for each two sets two each for that says It start with a discussionbrief of larly. The larly. called a called ( The The Union 푥 1 푦 , axiom of extensionality axiom of pair axiom of union 1

) Zermelo-Fraenkel settheory ∧ 푓 ∈ relation (Kuratowski) ordered pair of of pairordered(Kuratowski) 퐙퐅퐂 ( 푥 is based on a classical Hilbert-style logic with equality, no function symbols and the only 2 . We often write often We . or 푦 , allows to us form unions of sets, i.e. 2 { pairing axiom ) 푥 푦 → 푓 ∈ } may be defined as defined be may ( 푏 = 푎 ∧ 푎 ∈ 푥 reads ∈ ”. 퐌퐋 We expect axiom systems like this to be known and focus on the set-theo- the on focus and known be to this like systems axiom expect We 1 푥 , we have a global equality on all sets. Another important application of application important Another sets. all on equality global a have we , 푦 = 퐙퐅퐂 and and ∀푥, 푦 푦 ∀푥, reads 푎푅푏 2 퐈퐙퐅 . ∀푥, 푦 푦 ∀푥, . Wewrite 푦 푥 there is a set containing exactly containing set a is there [ instead of instead . However, some remaining axioms of ∃푧∀푤 and ) ∀푧 ↔ 푦 = 푥 퐙퐅퐂 푏 ∈ 푥 → [ ∀푧 {

푦 ,푥 푥, ∈ ( (

is definedas is 푦 ∈ 푧 ↔ 푥 ∈ 푧 -relation by ↔ 푧 ∈ 푤 푓 } . Finite sets of more than two elements are defined simi- defined areelements two than more of sets Finite . ( and and ~ 푥 ( ,푏 푎, ) ( 24 for the thus for the unique 푦 ∈ 푧 ↔ 푥 ∈ 푧 ( ) 푏 = 푎 ∧ 푏 ∈ 푥 ~ ( 푅 ∈ 푦 = 푤 ∨ 푥 = 푤 〈 . A relation A . ) ,푦 푥, 푦 = 푥 → 〉 = ) .

{ { ) 푥 ] 푎 ∈ 푥 → – . – } )

∈ this way we can always put always can we way this , )] this justifies writing justifies this 푥 푓 { -relation 푦 ,푦 푥, and and is called a called is .

with with } } 퐈퐙퐅 . Wecould replace these ax- 푦 . A set of orderedpairs of set A . as elements. Now, by ex- by Now, elements. as ( ,푦 푥, may still seem problem- – we will see an example ) function 푓 ∈ . . e Intuitionistic e 퐂퐙퐅 iff iff { ,푦 푥, 퐋퐄퐌 WS WS . We first We . 푥 } 1 for the for 2019 . 푥 = tains 퐋퐄퐌 푅 2 /20 is ∧

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 푎 define their intersection as Given Given a set Separationschema sufficenot for internal induction or recursion on the natural numbers. Hence, we can set oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison Again, by extensionality, we may write may we extensionality, by Again,

The The of a function: For a relation a For function: a of It is clear that this set must be unique by extensionality. We will refer to it as Here,

simplest case we aregiven sets two Before formulating the axiom, we introduce the notation Infinity union, union, pair and extensionality (this can be seen by writing of and and set of set of natural numbers as follows: the setexistencethe axiom Extensionality justifies writing justifies Extensionality The The that that we can prove topossess the property 퐴 and in and our example, axiom of empty set 푏 contains exactly the elements of elements the exactly contains 퐴 is acollectionof is sets 푎 and and a formula postulates that there isaset containing these numbers (and these numbers only): postulates the existence of the empty set, ∃푥 ∃푥 [ 푏 ∪ 푎 ( ( 푛∈푥 푠 푥. ∈ ∀푛 = 푏 ∩ 푎 푥 = 푥 휙 푅 푢 such such that instead ofinstead . The union . and and ∅ ∀퐴 ∃푢 ∀푥 ∀푥 ∀퐴 ∃푢 ) for the witness of this formula. This axiom is often also formulated asformulated also often is axiom This formula. this of witness the for and the and empty set constructed asa corollary. { ( 푥푅푦 :푥∈푏 ∈ 푥 푎: ∈ 푥 푎 푛 ∀푎∃푥∀푦 and and ) 푠 = 2 ⋮ 푠 = 3 푠 = 1 ∅ = 0 푥 ∈ 푎 note that, because of because that, note 푎 and and { ⋃ is not free in

:휙 푎: ∈ 푥 [ 푏 ) 푈 ∃푤 ↔ 푢 ∈ 푥 { . By wemay pairing, form 푛∈푥. ∈ ∀푛 ∧ ( ( ( ,푏 푎, of all these ofall 휙

( ∃푥∀푦¬ 1 2 0 휙 ∧ 푎 ∈ 푦 ↔ 푥 ∈ 푦 푏 : ) ) ) . Again, extensionality justifies writing justifies extensionality Again, . } } = = ∪ ∅ = ~ . Another example is the definition of range and domain . ( { { 25 푥 ∅, ∅, ( ) 푥 ∈ 푦 } { ( { ~ for { ∅ ∅ 푚∈푥 푠 = 푛 푥. ∈ ∃푚 ∨ ∅ = 푛 ( ∅ 휙 } } 퐴 ∈ 푤 ∧ 푤 ∈ 푥 푢 } } , , we can form the (unique) set of all elements of 푠

{ shouldcontainelements all = ( ) 푦 ∅, 푠 푥 . . As an example, given two sets two given example, an As . (

{ ) 푥 { ∅ ∅ ) ∪ 푥 = ( } } ,푦 푥, = (

}} 푦 )

⋃ ) ) { . =

{ 푥, 푥 ) } { { { ] . This is justified by the axioms of { ,푥 푥, . ,푏 푎,

푥 } , } } { } , now the union sets the , now ofthe ( ,푦 푥, 휔 ). ). Now we can construct the 푚 . Note that this axiom does ) } )] } , we have we , .

푥 ⋃ of each of 퐴 for the union the for ,푦∈ 푦 푥, 푎 WS WS and and 푢 2019 . In theIn . ⋃ ⋃ 푏 we /20 푅 푎 .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. e hoyadCntutvs Constructivism Theoryand Set erset axiom it must be contained in contained be must it axiom erset Given Given a formula Replacement Later this in chapter, we will discuss the fact that axiomthis isconstructively not unproblematic.

This This axiom means intuitively, that given a function with domain viewpoint. For example, for every formula taining alltaining these We write We e a ue elcmn t gv aohr eiiin f oan n rne f rlto: Since relation: a of range and domain 푥. ∃! of 푅. definition another give to replacement use can We within the theory. This is usually done as follows: Let follows:as done usually isThis theory. the within We often write 퐴 example, replacement allows us, given any set any given us, allows replacement example, there isa setcontaining all subsets of In other words: there are no infinite no are there words: other In easy thisbut reasoning fails with an intuitionistic background logic. Hence, already thefinite (classically and This axiomThis says that every setmust have an Foundation { We define the subset-relation as Powerset 푦 ∈ 퐵: ∃푥 ∈ 퐴 퐴 ∈ ∃푥 퐵: ∈ 푦 ∗ = ) set ) { { ( 푥 푦 = 푧 ∃푦. } 풫 퐴 ∈ 푥 : 풫 ( ( 1 퐴 ) ) turns out tobe equivalent to the set of all formulas and hence to be highly non-trivial. ( for ( :퐴→퐵 → 퐴 푓: } ,푦 푥, ,푦 푥, . Indeed, we can apply replacement since replacement apply can we Indeed, . 푦 휙 s. Using separation, we can form , such that, such 푃 ) ) . It will turn out that that this axiom is not at all unproblematic from a constructivea unproblematicall from at not is axiomthis that that outturn will It . 푅 ∈ ) w obtain we , tomean that } ∀퐴 and similarly and range dom [( 푥∈퐴 !푦 휙 푦. ∃! 퐴. ∈ ∀푥 퐵 is not isnot free in ( ( 푅 푅 ∀푧 ↔ 푦 ⊆ 푥 ) ) 퐵 = = ∀퐴 cnann al f these of all containing 풫 { { 퐴 푓 ∈ 퐴 푃 ∀푥. ∃푃. ∀퐴. [ ( ∈ 푥 ∈ 푦 ∃푚 → 퐴 isafunction with { : -chains. This axiom allows us to prove properties by induction by properties prove to us allows axiom This -chains. range ∅ } ) ⋃ ⋃ ⋃ ⋃ ( 휙 . Of course, classically, course, Of . ,푦 푥, ( , we can form by separation the set 휙 ∈ 푦 ∈ 푧 → 푥 ∈ 푧 ( 푅 -minial element: , wehave ( ) ) ∅ = 퐴 ∩ 푚 ∧ 퐴 ∈ 푚 ~ ) . 퐴 ( 푦∈ ∃푦 : 푅 푥∈ ∃푥 : 푅 퐵 푥∈퐴 푦∈퐵 휙 퐵. ∈ ∃푦 퐴. ∈ ∀푥 ∃퐵. → , to “replace” each “replace” to , 푃 ∈ 푥 ↔ 퐴 ⊆ 푥 26 { :∃ = 푦 퐴 ∈ ∃푥 퐵: ∈ 푦 ~ 휙 ( 푥 dom = 퐴 ⋃ ⋃ ⋃ ⋃ ) ) . The be any property such that we can provethe can we that such propertyany be 푥∈퐴∃ = 푦 푦 ∃! 퐴 ∈ ∀푥 푥 B sprto, e a form can we separation, By . ) Powerset-axiom . 푅 . 푅 퐴 .

, we can form its range ( ) { 푓 ( ( ] ∈ 푥 . ,푦 푥, ,푦 푥, )

퐴 ∈ 푥 and and { 푥 { ) ) } ∅ ( } 푅 ∈ 푅 ∈ } ,푦 푥, 퐴 = range = 퐵 휙 : by { 푥 } } } )

} . ∗ ]

{ . We obtain some obtain We . ∅ = . states that, given a set . { 푥

∈ 푥 } , thus forming the set the forming thus , or { ( ∅ 푓 { } ∈ 푥 ) 휙 : . 퐵 } { . . As a simple By the Pow- ∅ } WS WS dom 휙 : } 2019 퐵 = ( con- 푅 푧∈ ∀푧 { ) /20 ∅ 퐴 = } , ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. example is called a weak counterexample: We say that a formula remove simply to sufficeUnfortunatel not does it however explicitly. axioms the all down writing skip we so discussion, informal less) or (more wa we now, For one. intuitionistic an with logic background classical our replace to is step oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison We start restrictingstart We 2.4.2 we take we union, union, empty setand infinity and arestricted version of the separation schema: choice Let Choice induction step Proof is ⓒ action,in let showus that set no can contain itself: the of example simple a As logic. intuitionistic on based systems in proofs inductive w if axiom thisdifferent formulationof a find must we that showalready lines These fying least a reasoning)(classicalis there then sets, all forhold not does it if cause saying that for each collection each for that saying It says that for all collection all for that says It of each see for example [ 퐙퐅 that it is constructively inacceptable in the next section).Therefore, one often likes to con each each of the sets 푎 Lemma 퐅 − 퐙퐅퐂 = , i.e. 푠푖푛푔 : For each set

( 휙 퐀퐂 F . By minimality, all itselements satisfy iff 푠 ∩ 푎 ( 퐁퐒퐓 ro . Alternatively,it is easily seen that the axiom ofchoice thehas equivalent formulation 푥 )

2.13 ) 휙 m reads: ma that mean { 퐀퐂 implies implies unwanted instances of

푎 – 퐙퐅퐂 } basic set theory set basic : No set can contain itself. 푎( ∀푎 . Kurt Gödel and Paul Cohen showed that the is independent from independent is choice of axiom the that showed Cohen Paul and Gödel Kurt . ∀푆 푎 ∅ = . 40 to to [( 퐀퐂 푎 ( ]. , i.e. , the set 퐙퐅퐂 ( 푥∈푎 휙 푎. ∈ ∀푥 푠∈푆 ∅ ≠ 푠 푆. ∈ ∀푠 is often considered problematic, even from a classical point of view. (We will show 퐈퐙퐅 푥 to obtain a theory compatible with intuitionistic logic. As noted before, firstthe theory compatible intuitionistica obtainwithlogic. to noted As ¬ cnan ol oe lmn, i.e. element, one only contains

( – 푎 ∈ 푎 counterexamplesweak { 푆 푎 ∀퐴 ( of nonempty sets nonempty of } 푥 must must have an element disjoint from itself, but the only possible candidate 퐴 – ) [ ) of nonempty sets nonempty of ) ( over the constructively unproblematic axioms extensionality, pairing, extensionality, axioms unproblematic constructively the over . ) 푎∈퐴 ∅ ≠ 푎 퐴. ∈ ∀푎 휙 → ∧ ( ∀푠 ( 푎 1 ) 푠 , ) . Then 2 퐋퐄퐌 .푠 푆. ∈ 퐋퐄퐌 휙 ) ~ and and by the inductive step, so must 푠 over some simple set theory. As the background theory 푓 → 퐴 ∃푓: → , there is a choice set choice a is there , 휙 1 27 from the background logic as we may observe what observe may we as logic background the from 푎 must must forhold all sets (if it holds for one): atleast Be- 푠 ∩ there is a choice function choice a is there ~ 2 ∅ = 휙 ∃푦 in in the language of set theory is a ⋃ ) [ ) ( 푦 ∈ 푥 퐶 푠∈푆 푠푖푛푔 푆. ∈ ∀푠 ∃퐶. → 푓 . 퐴 ( ) 푎 ∀푧 ∧ ) 퐶 푎 ∈ containing exactly one element one exactly containing ( 푦 = 푧 ↔ 푥 ∈ 푧 ] ,

∈ -least element -least 푓 picking one element of element one picking ( 푠 ∩ 퐶 푎 , contradiction. e want to conductto want e foundation axiom foundation ) ] ) . sider the theory

] . The The . weak counter- nt to have a have to nt 푎 WS WS not satis- not xo of axiom 2019 퐙퐅 /20 ∎ y ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Let us show that this assumption is justified: If If justified: is assumption this that show us Let i.e. Foundation,let that that there is some set “ sume, that for all all for that sume, stead of Ⓜ Constructivism Theoryand Set Ⓜ

dation, there is some Proof As a simple example of set-induction, let us give an alternative proof in proof alternative an give us let set-induction, of example simple a As or weaker the version 퐋퐄

common divisor of the theories the of divisor common that that set containsno itself 2. (Lemma Let Foundationas weak counterexample where where appear in their bounded forms only, i.e only, forms bounded their in appear is impossible and thus lematic in a constructive setting. We will therefore replace it with the more suitable wi proofs induction conduct to axiom Foundation the using that remarkedalready have We 푆 Taking thisTaking axiom isalso justified bythe thatfact axiomsboth are classically equivalent: “ ← , which means → 퐌 ”: Suppose, there is some set some is there Suppose, ”: Proposition 2.15 Proposition 2.14 푦∈퐴 휙 퐴. ∈ ∀푦 ”: Let ”: 퐋퐄 ′ : Let : . Assuming the full separation axiom, iteven implies 퐌 푎 is not free in in free not is 퐴 ′ be the law of excluded middle for bounded formulas. We will show that foundation implies foundation that show will We formulas. bounded for middle excluded of law the be 휙 . 휙 be any (bounded) set andformulaany be be any formula and assume and formula any be ( 푦 ) 퐴 ∈ 푚 . This immediately. This leads to the contradiction ¬휙 푎 ∈ 푦 . Altogether, : : Let 퐒 퐄 onain↔Set-induction ↔ Foundation ⊢ 퐋퐄퐌 + 퐁퐒퐓 ∈ 퐴 ′ 휙 퐋퐄 be such be that -minimal element such such that 휙

휙 and ( 퐋퐄 퐌 ( 푎 푦 ) ′ 퐌 ) , which is . By Set-induction, . If If . 휙 ′ be the bethe oflaw excluded middle for bounded formulas. Wethen have is a is 퐒 eaain⊢Fudto 퐋퐄퐌. → Foundation ⊢ Separation + 퐁퐒퐓 푥 ∈ 푎 퐙퐅퐂 (∀푎 ¬휙 ∨ 휙 푥 ¬휙 without without restricted formula restricted 퐴 ∩ 푚 , 13 ∀푎∃푥∀푦 ( ( 푦∈푎 휙 푎. ∈ ∀푦 , then there is some is there then , 퐈퐙퐅 퐴 퐋퐄퐌 ): )

. . We want to assume that this and and 푥∈푎 ∈ ∀푥 푆 ∈ 푠 ′ ∀푎 but only but for restrictedformulas. ∅ = ∈ ( ( -minimal element. Let element. -minimal 퐒 onain→ Foundation ⊢ 퐁퐒퐓 휙 ∧ 푎 ∈ 푦 ↔ 푥 ∈ 푦 퐂퐙퐅 푦∈푎 휙 푎. ∈ ∀푦 . If ( ∀푎 ∀푎 . Then,allfor . = 푆 푦 or ~ ) 0 = 푠 ) . The unwanted instances of instances unwanted The . 퐴 ( 28 푦∈푏 ∈ ∃푦 푥 ∉ 푎 { 휙 → is not not is ∈ 푥 (or ~ , then ( ( 푦 { ) 푎 bounded formula) bounded 0,1 ) , i.e. . 푥 ∩ 푎 ∈ 푦 ) ) ∈ 푎 휙 ∀푎. → ) 퐁퐒퐓 퐋퐄퐌 휙 → } mnml then -minimal, 휙 : 퐴 ∩ 푚 ∈ 푦 휙 ( ∅ = 푥 . If 휙 ∧ 0 = 푥

( will play the role of an easy to see least see to easy an of role the play will ( : 퐴 ( 푦 푎 ) ) 1 = 푠 ) . ) 퐋퐄 , hence , . Towards a contradiction, suppose, contradiction, a Towards . , .

휙 ( 푎 퐌 ( 퐴 ) 푡 : , then ′ . is )

, 휙(푦) )

be the formula “ formula the be , which means that quantifiers that means which , 1 = 푥 ∨ 퐁퐒퐓 + Set induction Set + 퐁퐒퐓 ∈ ¬휙 -minimal with this property, 퐴 . Wethus. can use = 푆 ∩ 1 = ∅ ′ ( = 퐋퐄퐌 푦 ) { } . This shows that shows This . :¬휙 퐴: ∈ 푏 . Since are usually full full usually are axiom of set-induction 푆 ∈ 1 { 푥 ∉ 푡 0 ( } 퐴 푆 ∩ of the fact the of WS WS ll be prob- be ll ) , by Foun-by , } ” and as- and ” 퐴 ∩ 푚 ∅ ≠ , i.e. 2019 푥 ∈ 푎 퐋퐄퐌 By . ∉ 0 in- /20 ∎ ∎ :

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. containing themselves, Ⓜ Theory Set Constructive of Analysis and Comparison The The notion of predicativity first emerged in the beginning of the 20 2.4.3 Russel’s paradox Russel in an attempt to analyze the newly found paradoxes in Cantor’s naïve set theory. Most istic logic, we stop at this point and define and point this at stop we logic, istic Diaconescu showed that the axiomthat of choice provides us with another weakcounterexample [ Axiomof choice as weak counterexample

Although there are other versions of choice (not necessarily equivalent to theoryThe Proof Lemma2.16: set induction and separation replaced with the following stronger axiom, called Proof question chapter in 6 semanticusing tools. elaborate weak counterexamples, not following directly from one of the axioms axioms the of one from directly following not counterexamples, weak elaborate oth with up come can one not or whether moment the at clear not is it Also, chapter. another beyond go axiomnot doesthisthat Showing Let Theorem 2.17 • • • 푓:

: Suppose, : Let

{ ,푏 푎, From 푏 = 2 = 푎 푓 푓 푓 ( ( ( 푎 푏 푎 휙 } ) ) ) be any beany (bounded) formula and set 2 → 퐈퐙퐅 1 = 1 = 0 = No set can contain itself. 퐈퐙퐅

beachoice function. one Then of the following cases hold: , then ∀푦 ∈ 푎 ¬ 푎 ∈ ∀푦 and , then , so ehv: : We have: to to [ = 푎 58 ¬휙 푓 ] ] 퐂퐙퐅 푏 ∈ 1 푎 ∈ 0 ( arises when one tries to form, according to Cantor’s notion, the set { 푏 2: ∈ 푥 = 푅 . ) ( ∀퐴

0 = 푦 ∈ 푦 – , we have , we have Predicativity [( { . Then . :푥∉푥 ∉ 푥 푥: ( 푥∈퐴 푦 휙 ∃푦. 퐴. ∈ ∀푥 휙 ∧ 0 = 푥 ) . Assume, 푓 퐒 eaain⊢ Separation + 퐁퐒퐓 휙 휙 } ( . Weimmediately arrive at absurd the equivalence 푎 . . ) ) 푓(푏) ≠ 1 = 푥 ∨ 푎 ∈ 푎 ( ,푦 푥, 푅 ∈ 푅 ⇔ 푅 ∉ 푅. 푅. ∉ 푅 ⇔ 푅 ∈ 푅 퐈퐙퐅 퐙퐅 ) = 푆 , and, as and, , to be all axioms of axioms all be to , then by hypothesis, ) ~ , requires some deeper results and will be postponedbe to will deeperresultssome requires and , } 퐒 ⊢ 퐁퐒퐓 퐵 푥∈퐴 푦∈퐵 휙 퐵. ∈ ∃푦 퐴. ∈ ∀푥 ∃퐵. → ad푏= and 푏 29 { ,푏 푎, ~ } 푓 , where 퐀퐂 퐀퐂 is a function, a is { :푥=0∨ 0 = 푥 2: ∈ 푥 퐋퐄퐌. → →

퐋퐄 퐌 th ′

퐙퐅 ,

century in writings of Poincaré and ¬

푏 ≠ 푎 but for foundation replaced with replaced foundation for but ( 푎 = 푎 ( ( ,푦 푥, 퐀퐂 휙 ∧ 1 = 푥 . If . ) ) compatible with intuition- ) ) ] , contradiction. 휙 .

, then , the collection schema – we will answer this answer will we ) } .

푎 ∈ 0 푅 of all sets not and WS WS famously, er, more er, 푏 ∈ 1 2019 21 : , so, ] /20 : ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. most notably Weyl’s Frankel-style set theory? Myhill writes in [ of kind own his formulating when predicativity about care constructivist a should why So coherent and foundation. of axiomatic set theory as developed by Zermelo, Skolem and Fraenkel, which gave mathematics a solid – Russel’s theory of ramified types was an attempt to give a foundation t a member of. vicious circle principle (VCP) in order topick out definitions as in (ii) on hold to convinced Russel, diagnosis, Poincaré’s to response In lb Poincaré [ inition inition is impredicative because Definitions like these are called called are these like Definitions Constructivism Theoryand Set explanation of predicativity and its seehistory, [ especially problematic in light of this paradigm are the axioms of Separation and Powerset. For a detailed that axioms Two sets. building at attempt bottom-up as seen be can theoryset constructive Thus, u hsnme a utbe eie ntels etne [ sentence. last the in defined been just has number this But definablefrom differentall number diagonalization, real(Define, Richard’sby paradoxa from analysis: Given a bounded subset bounded a Given analysis: from kno we that bound lower greatest of one the is definitionimpredicative another paradoxes, the sides [ set, a Other mentionable such paradoxes were Cantor’s paradox (in modern terms: the class of car an attempt that can be regarded today as failed [ “… ( “ erwise (on the(on erwise constructive come being viewsets them, thatonly define into aswe to explain what it is to satisfy the definingis explainwhat it to to theset;condition satisfy definingof that condi- that tion must only refertion to were sets which or havemight been defined previously, oth- ,푆 푦, Whatever contains Whateverbe apossible not contains anmust apparentvariable vari- value of that (ii) (i) because in orderin isbecause explainwhat it to an to element be we have of set, a certain

)

15 ∧ ∀푥 ∀푥 ∧ ], [ ], 51 in eachin an object 3 ], ], [ ( ] Burali-Forti paradox (the class of ordinals is not a set, [ set, a not is ordinals of class (the paradox Burali-Forti ]

lb 52 ( and were not there "all along") a vicious circle might result.might circle not "all were and there a vicious along") ,푆 푥, ] found] theythat arise because case infinite collections are regarded as “actual” or “completed”. ) 푂 Das Kontinuum 푦 ≤ 푥 → is defined by referring to a totality of objects containing ) , where where , 푦 predicative isdefined in reference tothe setof lower bounds [ 72 lb ], the discussion about predicativity abated with the development able ( 푆 ,푆 푥, of 48 , and , ) ] ℝ ” 푠∈푆푥≤푠 ≤ 푥 푆 ∈ ∀푠 ≡ (Russel, in [ in (Russel, , we say that that say we , ~ impredicative 23 49 30 ]. ]. ]. ]. Despite various other contributions to the question, ~ 57 푦 says that says

if they contain such a “vicious circle”. “vicious a such contain they if is its greatest lower bound, lower greatest its is 56 ]) ] Analyzing these kinds of paradoxes, of kinds these Analyzing ] to the actual infinite, formulated the formulated infinite, actual the to o mathematics respecting the VCP 12 푥 is a lower bound oflowerbound a is ], [ ], 17 푂 ] , and ), König’s paradox König’s ),

– asetthat ”

real numbers.real glb = 푦 dinals is not 푆 WS WS . This def- This . Zermelo- 푦 isitself 2019 [ ( 38 seem 푆 ) ] or ] Be- /20 iff w

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Again, our concerns with the powerset axiom are similar to what we had before: Given a set a Givenbefore: hadwe what to similar areaxiom powerset the withourconcerns Again, ImpredicativityPowersetof power set power 휙 for any restricted formula For the converse implication, let we have Ⓜ Theory Set Constructive of Analysis and Comparison sets of functions from one set to another. This is formulated in the exponent the in formulated is This another. to set one from functions of sets mathem Bishop-style sufficientfor is it numbers,ofnatural set given the that, notes Myhill Ⓜ Proof

Proof is not free in exponentiation check for check 휙 bounded quantification only et tee s function a is there ment, set a Given separation: with concerns our precise make us Let ImpredicativitySeparationof according to this axiom the set the axiomthis to according for any any for separation are equivalent the in classical context: stricted separation, form ( ( “Power set seems especially“Power set nonconstructive impredicativ and ,퐵 푛, 푦 other axioms:it the other involve,does as putting togetherothers do,not takingor apart Proposition 2.19 Proposition 2.18 sets that alreadyhas sets one but constructed ratherselecting out the of alltotality of ) : Let : : Given separation and a formula formula a and separation Given : 푥 ∈ ∅ ↔ ) . Myhill writes bounded 휙 ↔ 푥 ∈ ∅ 푥 휙 풫 to be in in be to be any formula. Because of Because formula. any be ( 휙 퐶 푦 states that for sets two ) . and and we may assume that sets those that standthose sets relation the in togivenof inclusion a se . Accordingto. separation, bounded formset the wemay formula . ByProposition 2.18,this shows Separation. : 퐶 ( 퐒 퐄 Separation ⊢ 퐋퐄퐌 + 퐁퐒퐓 퐁퐒퐓 is in 휓 [ { ): ): The axiom of separation is equivalent to the scheme ( 48 :∅∈푓 ∈ ∅ 퐴: ∈ 푦 휙 휙 ,퐶 푥, 푓 . Note that in this case in order to check whether ]: , where , dfnd on defined – ) this iscalled . To avoid this problem, we restrict this axiom scheme to formulas with formulas to scheme axiom this restrict we problem, this avoid To . = 퐶 퐴 be a set and { ∀푎∃푥∀푦 푥 푥 ∈ 퐵: ∀푦 휓 ∀푦 퐵: ∈ 푥 푎 is not free in in free not is ( and and 푦 휙 퐋퐄퐌 ⊆ 푥 ) } 퐴 , let , = with 푏 the ( { , we may define may we , , the set, the { 휙 ∧ 푎 ∈ 푦 ↔ 푥 ∈ 푦 ∅ = 푥

:휙 퐴: ∈ 푦

} 휙 axiom scheme restricedof separation . Then ~ ( ( 푓 ,푦 푥, 푦 { ( 31 ∈ 푦 ) 푦 휙 a formula. By assumption, there is a set ) ) 푏 . Let us show, that both bounded and unbounded and bounded both that show, us Let . } ~ ( 푥 = 푎 . Notice, that one of the instances of instances the of one thatNotice, . 푥 { 푦 offunctions 푦 ∅ ) is uniquely determined by } } 푦 휙 : . ad thus and } , where , = 푥 ( 푦 퐵 ) { and a formula a and ) ∅ ,

} :푎→푏 → 푎 푓: if if 푦 ∀푦 ∈ 퐴 퐴 ∈ ∀푦 is not free in free not is 휙 = 퐵 and and 퐵 ∈ 푛 isasettoo. e compared with with thecompared e ( { iation axiom: The axiom: iation ∅ = 푥 휙 :∀ 풫 ∈ ∀푥 휔: ∈ 푛 ∃푥 ∃푥 ( 푦 : 휓 ) we will have to verify ( ( 푓 ∈ ∅ ↔ 푥 ∈ ∅ ↔ 휙 if if 푦 ,푦 푥, 휙 ; hence, by replace- ¬휙 . Then . t.” ) , we can define can we ,

. In either caseeither In . 휓 atics tohave atics ( 푥 ( we have to have we 푦 퐶 휙 ↔ 푥 ∈ ∅ 푦 WS WS 퐶 ) ) such such that ) ) , where , form itsform , 휙 . B re- By . axiom of axiom 2019 ( ,푥 푛, /20 ∎ ∎ ) 푥 } .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. lection scheme and the collection schema strengthened tothe of axioms all Ⓜ Constructivism Theoryand Set “ Proof Proposition2.22: Clearly, the powerset of inverse relationinverse All other axioms seem acceptable form the constructive viewpoint. We thus set the axioms of axioms the set thus We constructiveviewpoint. the formacceptable seem axioms other All theoryThe Definition 2.20 Definition discuss in this thesis. We will make this moreaxiom comprehensible giving following the definition: [ example for (see predicativity of notion the of tation justificationThe for taking axiom this instead of Exponentiation isgiven by aproof-theoretical interpre- In

is asetby The The Powerset. This axiom is called set ← 퐂퐙퐅 ”: Let Proposition 2.21 퐶 axiom of fullness of subsets of subsets of : “

however, we require a generalization of Exponentiation to hold, which itself is a weakening of weakening a is itself which hold, to Exponentiation of generalization a require we however, → [ ∀푢 ∃퐶 푏 ∀푎, 푥∈푎 푦∈푏 휙 푏. ∈ ∃푦 푎. ∈ ∀푥 ”: Just use 퐶 푅 be 푢 퐂퐙퐅 퐈퐙퐅 푏 × 푎 ⊆ 푎 -full and suppose, that for each ∀푥 ∈ 푎. ∃푦 ∈ 푏 . 푥푅푦 → ∃푑 ∈ 퐶 퐶 ∈ ∃푑 → 푥푅푦 . 푏 ∈ ∃푦 푎. ∈ ∀푥 : For two sets two For :

푅 but separation replaced by bounded separation, powerset replaced by the subset col- subset the by replaced powerset separation, bounded by replaced separation but − 퐒 oest⊢Fullness ⊢ Powerset + 퐁퐒퐓

is full isfull between 푏 휙 : is is the assertion that for each pair of sets 퐒 ustcleto Fullness ↔ Subset-collection ⊢ 퐁퐒퐓 and bounded and separation. Moreover, it is full; hence, there is ( ,푦 푅 푦, 푥, 푎 -full 푏 ( ∃퐵 ∃퐵 ,푦 푢 푦, 푥, is ) iff for each full relation full each for iff 푎 ( 푥푅푦 ≡ -full: 푎 푥∈퐴 푦∈퐵 휙 퐵. ∈ ∃푦 퐴. ∈ ∀푥 푥∈푎 푦∈푑 푥푅 푑. ∈ ∃푦 푎. ∈ ∀푥 the subset collection schema and and ) 푑∈퐶 ∈ ∃푑 → 푑 푥푅 . and and 푏 , we call a relation a call we , 푢 ∀퐴[ 휙 ∧ 푏 ∈ 푦 ∧ 푎 ∈ 푥 ↔ 푦 푎 ( , i.e. , i.e. ( 푥∈푎 푦∈푑 휙 푑. ∈ ∃푦 푎. ∈ ∀푥 푥∈퐴 푦 휙 ∃푦. 퐴. ∈ ∀푥 . ( 푥∈푎 푦∈푑 푅 푦∈푑 푥∈푎 푥푅푦 푎. ∈ ∃푥 푑. ∈ ∀푦 ∧ 푥푅푦 푑. ∈ ∃푦 푎. ∈ ∀푥 ( 푢 ~ ,푦 푥, 푢 , 푦∈푑 푥∈푎 푥푅 푎. ∈ ∃푥 푑. ∈ ∀푦 ∧ 푦 푥∈푎 푦∈푏 휙 푏. ∈ ∃푦 푎. ∈ ∀푥 32 ) 푦∈퐵 푥∈퐴 휙 퐴. ∈ ∃푥 퐵. ∈ ∀푦 ∧ ~ 푅 24 between

( : ], [ ], 푅 ,푦 푥, between 61 ( 푎 ) ( ,푦 푢 푦, 푥, ], [ ], ) and and strong collection schema ,푦 푢 푦, 푥, → 60 푎

( ]), which we do not have the space tospacethe have not do we which ]), and and ) 푏 ,푦 푢 푦, 푥, ) 푎 there is a 푦∈푑 푥∈푎 휙 푎. ∈ ∃푥 푑. ∈ ∀푦 ∧

and and 푢 푏 ( 푦. ,푦 푥, , there is a is there , )

. Then 푏

) full ] , 푎

iff -full set 퐶 ∈ 푑 ∀푥 ∈ 푎 ∃푦 ∈ 푏 푥푅푦 푏 ∈ ∃푦 푎 ∈ ∀푥

퐷 ∈ 푑 ( ) with 퐶 ,푦 푢 푦, 푥, .

. such that the that such ) WS WS )] 퐂퐙퐅 ,

2019 to be to . A A . /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

formulas all for Clearly, constructivist analysis of ters we will give a meaning-preserving interpretation of interpretationmeaning-preserving a give will we ters to beconstructively very well justified. 휙 where , 퐵 i ntfe in free not is 퐂 퐙퐅 does not end with checking all the axioms. In one of the later chap- 휙 Aan e tl ed o hw that show to need still we Again, . ~ 33 ~ 퐂퐙퐅 into Martin- into Löf’s theory Löf’s 퐙퐅

rvs hs axiom. this proves 퐌퐋

which seems which WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. (DET Inference rules section3.2. Finally, 3.5 in weclear relationship the between in concepts common some give we 3.4, and 3.3 in theories the between differences discussing Before theories in more detail. We start with spelling out the whole axiomatization of the theories in theoriesthe axiomatizationof whole the outspelling with start We detail. more in theories

Axioms of identity predicate logic) as well as the all axioms theoriesand we would like to consider. Axioms 3.1.1 For reasons of clarity, we write down all the axioms and inference rules of logical system 3.1 motivated having After 3 theories set constructive Axiomatic (UG) (EI

) (HPL14) (HPL13) (HPL12) (HPL11) (HPL8) (HPL6) (HPL4) (HPL10) (HPL9) (HPL7) (HPL5) (HPL3) (HPL2) (HPL1)

∃푥 휙 ∃푥 Axiomatic constructive set theories theories set constructive Axiomatic (ID5) (ID3) (ID4) (ID2) (ID1)

) 휙

Setting the stage 휙→∀푥 휓 휙→∀푥 휙,휙→휓 ( 휙→휓 Heyting’s predicatelogic 푐 ( ) 푥 휓 →휓 )

→휓

(

푐

( ) where where 푥 ) 푢∈푎 휙 푎. ∈ ∃푢 푢∈푎 휙 푎. ∈ ∀푢 ( 휙 ∀푥 휙 ∀푥 ( → 휙 ( → 휙 ( → 휓 ( 휓→ (휓 → 휙 ( ( 휙→ (휙 푥 = 푦 → 푦 = 푥 푥 = 푥 → 휙 where where 휓 ∨ 휙 휓 ∧ 휙 휓 → 휙 휓 ∧ 휙 푥 ∈ 푧 ∧ 푦 = 푥 푧 = 푦 ∧ 푦 = 푥 푧 ∈ 푦 ∧ 푦 = 푥 ( 푐 ) ( ( ( ( ( → ∃푥 휙 ∃푥 → 휓 ∨ 휙 휙→휓 → ¬휙 휙 → 휓 푥 휓 ∨ 휙 (

푐 ) ) ) 휒 → 휓 is free for ) ) 푐 → 휙 → 휓 → 휙 → ( → is free isfree for ( ( 푢 푢 퐈퐙퐅 ( ( ) ) ) ) 휒 ∧ 휙 ( )

( ( 휒 → 휙 ( ) ∃푢 ↔

∀푢 ↔ 푥 ¬휓 → 휙 푐 ) ( → ) anf anf ) ) )

) )

, where , where 푦 ∈ 푧 → 푧 ∈ 푥 → 푧 = 푥 → ) ) 푥 [ – 퐂퐙퐅 [

) ( 휙 ∧ 푎 ∈ 푢 휙 → 푎 ∈ 푢 in

휓 → 휙 푥 퐈퐙퐅 ( → in in WS WS ) in section 2.4, in this chapter, we will discuss these axiomatic set axiomatic these discuss will we chapter, this in 2.4, section in 휙 ¬휙) → 퐇퐏퐋 and and 푐 푐 (

and occurs and free in neither 휙 is free for is free for

휒 → 휓 2019 ) and occurs and free in neither →

퐂퐙퐅 ( 푢 ( /20

( 푢 )] ) 휒 → 휙 )]

휒) →

~ 푥 푥 in in in in 34 ) ) )

휙 휙 ~ . . 퐙퐅퐂 –

퐈퐙퐅 , 휙 퐈퐙퐅 nor 휙 nor and and and 휓 . 휓 퐂퐙퐅 . 퐂퐙퐅 .

퐇퐏퐋

(Heyting’s section 0 section . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 3.1.2 Theory Set Constructive of Analysis and Comparison Definition 3.1 Definition

Fraenkel set theory

( eml-rekl e theory set Zermelo-Fraenkel (9’) (9) (8’’) (8) (7’’) (7’) (7) (6’’) (6) (5) (4) (3) (2) (1)

extensionality. We denote the (unique) witness of this formula by We write of the of the form ality. for all for all formulas Set induction Foundation Subset collection Powerset Strong collection schema Collection schema Replacement schema Bounded separation schema Separation schema Infinity Empty set Union Pair Extensionality for any formulafor any where we define the relation the define we where for all for all formulas ation. We write Here refer tothe witness of this formula by where where where where for all for all formulas

Set axiomsSet [ 푥∈푎 푦∈푏 휙 푏. ∈ ∃푦 푎. ∈ ∀푥 : ∀푥, 푦 ∃푧 ∃푧 푦 ∀푥, 푠 : ( ≔ 푦 ∩ 푥 휙 : Let : 푢 bounded ∀퐴 ∃푢 ∀푥 ∀푥 ∀퐴 ∃푢 : isany formula. ) ∃휔 : ⋃퐴 { ) and) : ∪ 푢 ≔ ∀퐴 ∃푃 ∀푥 ∀푥 ∀퐴 ∃푃 ,푦 푥, ∃푥∀푦 ¬ ∃푥∀푦 푥∈퐴 ∈ ∀푥 : 퐋퐄퐌

[( for the union of ∀퐴 [ : } 푤 ↔ 푧 ∈ 푤 ∀푤. ∀푛 ∈ 휔 휔 ∈ ∀푛 : for the pair obtained from (∀푎 { 퐁퐒퐓 [ 푦 ∀푥, 휙 휙 휙 :푧∈푥∧푧∈푦 ∈ 푧 ∧ 푥 ∈ 푧 푦: ∪ 푥 ∈ 푧 휙

∃푚 → ∅ ≠ 퐴 formula { ( [ , where , where , where , where , where , where : , where 푢 law of excluded middle excluded of law ∀푎, 푏 ∃퐶 ∀푢 ∃퐶 푏 ∀푎, ∃푤 ↔ 푢 ∈ 푥 or ( ( : } : 푥 ∈ 푦 푦∈푎 휙 푎. ∈ ∀푦 ( ∀퐴 . This exists by union and pair and is well-defined by extensionality. We will ), ( ∀푎∃푥∀푦 ( basic set theory [ 푃 ∈ 푥 ↔ 퐴 ⊆ 푥 ,푦 푢 푦, 푥, : 푥∈퐵 ∈ ∃푥 ( ∀푧 ∃퐵 (∀푥 ∈ 퐴 ∃푦 ∈ 퐵 휙 퐵 ∈ ∃푦 퐴 ∈ (∀푥 ∃퐵 퐈퐙퐅 ∀퐴 [( 푠 ( ( ∀푥 ∈ 퐴 ∃푦 휙 ∃푦 퐴 ∈ ∀푥 푛 ) 푦 ∈ 푧 ↔ 푥 ∈ 푧 퐵 퐵 퐵 [( : 푥 휙 , ) ( ∀퐴[ is free not in is free not in ) is free not in is not free isnot in ( ∀푥 ∈ 퐴 ∃! 푦 휙 푦 ∃! 퐴 ∈ ∀푥 , where nutoitc eml-rekl e theory set Zermelo-Fraenkel intuitionistic 휔 ∈ 푦 = 푤 ∨ 푥 = 푤 . ( 푑∈퐶 ∈ ∃푑 → ( ( : ∀푧 ↔ 푦 ⊆ 푥 퐴 휙 ∧ 푎 ∈ 푦 ↔ 푥 ∈ 푦 ∅ = 퐴 ∩ 푚 ∧ 퐴 ∈ 푚 ( 푦

∀푎∃푥∀푦 ( 퐴 ∈ 푤 ∧ 푤 ∈ 푥 and and ) ∀푥 ∈ 퐴 ∃푦 휙 ∃푦 퐴 ∈ ∀푥 ) ) ) 휙 → ∧ (∀푛 ∈ 휔 (푛 = ∅ ∨ ∃푚 ∈ 휔 휔 ∈ ∃푚 ∨ ∅ = (푛 휔 ∈ (∀푛 ∧ ) by) 푥 ) 푦 ∪ 푥 , is not free in ( ,푦 푥, ( ( 휔 ( } 푎 ) 푥∈푎 푦∈푑 휙 푑. ∈ ∃푦 푎. ∈ ∀푥 ) 휙 ∧ 푎 ∈ 푦 ↔ 푥 ∈ 푦 . This . This set is formed by(bounded) separation.

(again, this isjustified by extensionality). ) ~ be the axiom schema axiom the be ( 휙 푦 = 푥 → 휙 휙 휙 ( for ) ) → ∀푎 휙 ∀푎 → ) ,푦 푥, 푦 ∈ 푧 → 푥 ∈ 푧 푥 ) . . . . Wewrite ) ( 35 and and ] ( → ∃퐵 ∀푥 ∈ 퐴 ∃푦 ∈ 퐵 휙 퐵 ∈ ∃푦 퐴 ∈ ∀푥 ∃퐵 → ,푦 푥, , ,푦 푥, ) ) ⋃ ] ) ~ , { ) ( → ∃퐵 ∀푥 ∈ 퐴 ∃푦 ∈ 퐵 휙 퐵 ∈ ∃푦 퐴 ∈ ∀푥 ∃퐵 → ,푦 푥, ) 푦 푦 ) ∧ ∀푦 ∈ 퐵 ∃푥 ∈ 퐴 휙 퐴 ∈ ∃푥 퐵 ∈ ∀푦 ∧ ] ) , ] . This is. This well-defined byextensionality. ) 휙 ( , → ) ∅ } 푎 . This means all quantifiers are bounded, i.e. , . Both definitions are justified by extension- .

) { ,

:휙 푎: ∈ 푦 ) ( . We write We . ,푦 푢 푦, 푥, ( 푦 ) ) , ( ¬휙 ∨ 휙 ( ) 푥 푠 = 푛 푦∈푑 푥∈푎 휙 푎. ∈ ∃푥 푑. ∈ ∀푦 ∧ ) } ( for ( ,푦 푥, 풫 ,푦 푥, ), ), ( ( . We define the theories the define We . 푚 ( 퐴 푥 퐂퐙퐅 ) ,푦 푥, ) ] ) . This is well-defined is . This by ) , ] for the powerset-oper- the for )

) ) ( ] ] osrcie Zermelo- constructive , , ( ,푦 푢 푦, 푥, WS WS ) 2019 )] ,

/20 퐙퐅

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. It iseasily seen that

Definition 3.2 Definition 3.2 theories set constructive Axiomatic or of our thought our of or perceptionour of objects distinct definite, of whole a into togethergathering “a as set a defined Cantor 3.2.1 axiomatic set theory is the notion of a class: A A class: a ofnotion the istheory setaxiomatic

theories just defined: as containing all sets satisfying us Before we go on and convey the expression that dealingwe are with some type of higher order logic, let notion, as each class is given by a formula and in fact it fact in and formula a by given is class each as notion, and and guistic abbreviation for abbreviation guistic ample, for a set f e ter.Hwvr wa e en y hs epesos is expressions these by mean we what However, theory. set of ∀푥 on classes existing strictly on the metalevel the on strictly existing classes on Sometimes we like to think be pedantic for a moment and insist on the fact that a class is actually a actually is class a that fact the on insist and moment a for pedantic be ( • • • • • • • • 휙

퐵

퐴

Basic concepts in , w ( 푥 We say that a relation a that say We We say that arelation Unioning and Bounded Separation. ( dom range { ie sets Given Classes ment and bounded Separation to show to Separation bounded and ment o ay set any For instead of instead Given two sets A set We define the Pairs, unions and empty the set have already been defined above. × 퐴 ,푦 푥, ) e often use terms like “ 휙 → ) ( { 푅 ( 푅 푏 {푥, = 푓 | } 퐵 is called : 푑 ) 퐵 ∈ 푏 : ( The following definition are justified within justified aredefinition following The ) 푎 퐵 ⊆ 푥 , we can define the formula dom = ( ) { 푅 ,푦 푥, – ) ,푦 푥, 푅 and and

and . 퐁퐒퐓 which are called elements of the set.”, the of elements called are which define , } ordered pair of ) 퐴 is aset and finally } 푅 ∈ } relation and 푓 ∈ ( ∃푥 ∀푦 ∃푥 isthe weakestof the four theories. 푅 푑 ∀푥 ) w dnt by denote we , . We write We . about whether a given “class is a set” or not. How can that be after 푑 ∩ ( , then , 퐵 휙 푅 푓 퐁퐒퐓 = dom 퐂퐙퐅 = = 퐂퐙퐅 , we define its 퐈퐙퐅 = = 퐈퐙퐅 퐴 iff is aset ofordered pairs. Instead of ( 퐙퐅 is a is a is . Wewill use notationthis mainly in the case when 휙 ( 퐁퐒퐓 휙 ∀푥 ∈ 퐴 휓 퐴 ∈ ∀푥 푥 퐴 – 퐴 ) (

= = ( 퐈퐙퐅 { ( 푦 relation between 푥 푅 휙 ↔ 푥 function 푥 WS WS and ) ) } ( ( ( ( )

, and and 1 1 1 1 푥 ∈ 푦 ↔ , i.e. = :퐴→퐵 → 퐴 푓: { ) ) ) ) ,푦 푥, 2019 퐵 -(5)+(6 -(5)+ () 6 +(7 (6) -(5)+ () 6 7 8 9 +퐋퐄퐌 (9) + (8) + (7) + (6) -(5)+ { 푦 ( = 퐵 × 퐴 푥: ∃푦 ∃푦 푥: ( by by 푥 푥 푅 퐂퐙퐅 } = 퐴 iff ) ) | Cartesian product /20 ∈ ) 푑 ”, “ ? The statement “ statement The ? resp. ( 휙

( ⋃

) h rsrcin of restriction the ( [( and say that say and ,푦 푥,

6 { . Classes that cannot are not sets are called called are sets not are cannot that Classes . { ( ,푦 푥, class 푎 ∃푥 ∈ 퐴 휓 퐴 ∈ ∃푥 ′′ ′′ :휙 푥: 푅 ,푦 푥, 푦 ~ } +(7) )+ ) ⋃ ) and

) 퐴 +(7 퐵 × 1 푎 ∈ 푦 ≡ 36 = ) ) 푆

퐴

and 퐴 by the by Union axiom. ∧ 푓 ∈ 푅 ∈ ( 〈 is a set for each for set a is is given by a formulaa by given is 푥 ′′ ,푦 푥, ,푦∈ 푦 푥, ~ ′ 8 +(9 (8) + ) )

)+ 퐵 } ]} ( . iff

〉 푥 and is ( ( . Then the class { = ) ,푦 푥, 8

푓 ”

= 퐵 × 퐴 ⋃ ⋃ dom ′′ nothing more nothing [ , is a function from function a is 퐁퐒퐓 14 푅 ) { “ 2 퐴 푥 range +(9 퐵 ⊆ 퐴

) to

]. The modern pendent of this notion in notion this of pendent modern The ]. } 푅 is a set” a is ( 푦 → 푓 ∈ ,

푅 . So, we justify we So, . { and are thus meaningful for all the set the all for meaningful thus are and 푑 ,푦 푥, ∀푥 ) ′ ′ ( by ) ) {( ,푦 푥, ( 퐴 ⊆ ”, 푅 ( } ,푏 푎, 퐴 ∈ 푎 휙 ) }

푅 “ . ) 퐴 and = 1 퐵 = 퐴 | 푅 ∈ ) ( 푑 can be thought of as a metalin- a as of thought be can 푦 = ,푏∈퐵 ∈ 푏 퐴, ∈ 푎 : 푥 { 풫 than a formula. For classes For formula. a than = :∃푥 푦: . Use them again, to show to again, them Use .

) ( 휙 we usually write range 휓 → 푎 2 {( 퐴 . We usually write usually We . ) ” strictly metamathematical strictly ,푦 푥, ( [( = 푥 퐴 mimicking the language dom 푅 ( ) to to ,푦 푥, 푥 is afunction. ( and can be thoughtofbe can and ) { 푅 :푦⊆푎 ⊆ 푦 푦: ) :푥∈푑 ∈ 푥 푅: ∈ ) 퐵 ) ( ) , 푅 iff 퐵 ⊆ } 푅 ∈ ∃푥 : First use Replace- ) and dom ( . ]} 휙 proper } Nt ta if that Note : 퐴 is a set in range ( } ( 푥푅푦 we insisted 푥 Nt that Note . 푓 ) ) 푓 휓 ∧ . For ex- For . 퐴 = . ( ( 푥 푅 ) ( ) and 푥 푦 = us- = 푆 퐙퐅 ) ) 퐴 ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 퐴 ≺) (퐴, set of transitive sets is called an an called is sets transitive of set Definition 3.3 allows us to talkabout ordering of sets in the post-countable setting. notionThe ofordinals is central axiomaticin settheory. Itgeneralizes the notion ofnatural number and 3.2.2 Theory Set Constructive of Analysis and Comparison

and 퐴 ≺ (퐴, transitive and satisfies the trichotomy law question (it turns out that “powerclasses” such Proposition3.4 ( results thatresults resemble properties of som provewill numbers.naturalWeordinalsandbetween similarities many are there noted, As Note that paradoxes such as Russel’s cannot arise, when talking about classes, since there is a 푓 Proof objects the like ofclass all classes. form cannot we hence classes, never sets, only contain classes Proper classes: and sets between archy ( 푎 1 7) 6) 5) 4) 3) 2) 2) 1) 7) 6) 5) 1) 3) 4) 퐈퐙퐅 ) 퐴 :

a ) ≺ Ordinals ments of ordinals, all elements of Suppose, This follows by 1). Let This is clear. ⊆ 훼 Let Th 1 + 훼 elements All of an ordinal areordinals. 핆ℕ All the natural numbers are ordinals. 휔 Let Let ⋃ 훽 ∩ 훼 and 퐵 by the powerset-axiom, but the question whether or not this holds true in true holds this not or whether question the but powerset-axiom, the by well-order is anordinal. 핆ℕ ∈ 푈 푓 is trivial. is not aset. ( 휔 ∈ 푛 핆ℕ ∈ 훼 ∈ 푥 ⋃ 푎 is anordinal for each is anordinal for 2 퐴 ≺ (퐴, 푈 ) : A set ⋃ . . This showsthatThis . . Then ∃푥 ∃푥 , for each set of ordinals 퐁퐒퐓 푈. iff it is well-founded, i.e. each subset has a has subset each i.e. well-founded, is it iff and 퐵 Ti en,teei oe ordinal some is there means, This ( ) 핆ℕ = 푥 푇 ) : smrhc f tee s bjcie function bijective a is there iff isomorphic is called 훼 ∈ 훽 휔 ⊆ 1 + 푛 ∈ 푛 ) . By definition, . By 3),. By ,훽∈핆ℕ ∈ 훽 훼, transitive ⋃ ordinal 핆ℕ ∈ 훼 푈 휔 is transitive. But because of 1), and the fact that all its elements are e elementsare itsallfact that andthe 1),transitive. ofbecause is But = 훼 . . For ordinal an

. The class of all ordinals is referred to as to referred is ordinals all of class The . ⋃ ⋃ 푈 iff every element is a subset of . . 푈 핆ℕ ∈ 푥 푎 푎 ≺ 푏 ∨ 푏 = 푎 ∨ 푏 ≺ 푎 퐴 ∈ 푏 ∀푎, 훽 aretransitive too. is asubset of 풫 ~ . But then ( 푎 37 ) are never sets in 훼 푈 ∈ 훼 ~ , wedefine 훼 훼 ∈ 훼 , hence each element of such that such ≺ is a contradiction is a to Lemma2. -minimal element and element -minimal 푠 ≔ 1 + 훼 훼 ∈ 푥 퐂퐙퐅 :퐴→퐵 → 퐴 푓: 푇 , i.e. . We call two well-ordered sets ). . But .

( 훼 푇 ⊆ 푎 → 푇 ∈ 푎 ) ∪ 훼 = 훼 sc that such is transitive, hence transitive, is 훽 핆ℕ must must be transitive. 퐂퐙퐅 . As usual, we call we usual, As . { 훼 } ≺ . is a nontrivial a is is irreflexive, is . A transitive 푎 WS WS 16 1 strict hi ≺ . 퐴 2019 e basice 푎 2 ⊆ 푥 /20 er- le- ⇔ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. “ Let us adopt the more usual notations Proof Proposition 3.8 Proposition 3.5

Proof Ⓜ 퐼 In this section we justify the fact that ordinals can be seen as generalization of the natu 3.2.3 theories set constructive Axiomatic is problematic in 휔 without the second conjunct, thus Lemma3.7 on What makes the natural numbers different to the ordinals in the intuitionistic setti Proof show show We therefore obtain our usual principle of natural induction: Let Suppose,

inductive hypothesis, the the Proposition itmust be equal is called 푚<푛 < ∀푚 as the least inductive set. However, this requires unbounded Separation or the Powerset axiom which 휔 Corollary 3.6 • • • • •

istrichotomous: : By natural induction on : We use natural induction on : Weshow by

푛∈휔 휙 휔. ∈ ∀푛 푛 < 푚 푚 = 푛 푚 < 푛 푠 푠 Set recursionSet and the natural numbers ( ( ” tosignify “ 푚 푚 푦∈푎 휙 푎. ∈ ∀푦 inductive o o ) ) ( 퐁퐒퐓 + Set induction Set + 퐁퐒퐓

푛 = 푛 ∈ , then , then , then by the last lemma, 푚 = 1 + 푛 푚 < 1 + 푛 : ( , then ( , then 푛 퐒 e nuto nutv 퐼 ⊆ 휔 → inductive 퐼 ⊢ induction Set + 퐁퐒퐓 퐁퐒퐓 + Set induction Set + 퐁퐒퐓 ( 퐂퐙퐅 퐁퐒퐓 + Set induction Set + 퐁퐒퐓 ) iff ( ∈ 1 + 푛 < 푚 1 + 푛 < 푚 휙 → 푦 -Induction on the formula ) . Thus, our formulation of the infinity axiom states “there isaleast inductive set”: 푚∈푛 ∈ ∀푚 푠 퐼 ∈ ∅ and and 푠 퐼 ∈ 푎 ( ( ( 푚 푚 푠 , , or ( ) ) 푛 and 휔 ∈ 푎 . By inductiveness of ∪ 푛 ∈ ∪ 푛 ∈ 푛 = ) ) . , ”. Using the lemma, we can show the trichotomy law forthe relation “ and and 푛 . Let 푠 → 퐼 ∈ 푥 ): ): – to . If {

푛 ∀푛 ∈ 휔 ∀푚 ∈ 푛 푛 ∈ ∀푚 휔 ∈ ∀푛 퐈퐙퐅 stating “there is an inductive set”. One then usually proceeds to define the whole of 휙 } 푛 WS WS ∅ = 푎 휙 푠 = 휔 ∈ 푚 ( ): ): . Let and and { , natural induction schema): For formula any 0 ( 푛 1 + 푚 = 푛 푚 :푛<푚∨푛=푚∨푚<푛 < 푚 ∨ 푚 = 푛 ∨ 푚 < 푛 휔: ∈ 푛 ∀푚, ) 0 2019 } . In other words, ( ) 푛 푠 = 푛∈휔 휙 휔. ∈ ∀푛 ( 푛 ∈ 푚 and and we are done. If there is some 퐂퐙퐅 ) 푥 . By inductive hypothesis, we have one of the following cases: . /20 ) ( 퐼 ∈ 푛

∀푛 ∈ 휔 휙 휔 ∈ ∀푛

) 휙 . , then by inductive hypothesis one of the two holds: . Usually, in ~ 휔 for 퐼, ( ( 푥

. Thus, 푠 = 푎 38 푠 ) ( 푠 = 푛 퐼 ∈ 푥 → 휔 ∈ 푥 ≡ 푚 ( ~ 푛 ) ( ) 푛 ( 푠 ∨ 푛 ∈ = 퐼 푚 ) 휙 → ( . . ) 푚 퐼 ∈ 퐙퐅 { ) :휙 휔: ∈ 푛 ( and and 푠 ( and and . ( 푚 푛 휙 ) ) be any property and suppose, we can 푛 < 푚 ) 푛 =

퐈퐙퐅 : ( 푛 ) , the infinity axiom is formulated ) . . } for is an inductive subset of 휔 ∈ 푛 푛 ∈ 푚 with ng ng is that the ordering . Also, we often write 휙 , we have the rule ral numbers. A set 푠 = 푎 ( 푛 ) , then by 휔 < . By ”: ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ⓒ nal nal an an Proof ¬ ht w ordinals two that law trichotomy the show to remains it so axiom, Foundation the exactly is well-foundedness Proof rable with at least one other ordinal. and thatand ⓒ ordered sets All All make sure the class class the sure make we would have work in work in in ordinals of applications important the of Many ordinals. ofdifferences and similaritiessome discussbriefly will we section this In 3.3 Theory Set Constructive of Analysis and Comparison The axiom of Foundation says that every set has an an has set every that says Foundation of axiom The 3.3.1

Corollary 3.9 Altogether, We will often write often will We eiiin 3. Definition union: separation pairing, infinity, of axioms the using numbers, natural of sequences finite of there is a numeration of the elements of finite iff there is a natural number natural a is there iff finite {( ⋃ ( 푛∈휔 ( Proposition 3. Lemma 3.11 훾 ∈ 휂 ,푎 0, ∈ 훽 ∈ 훾 퐶 ∩ 훼 ∈ 훽

-minimal element. : Let : Irreflexivity and transitivity also hold in the intuitionistic setting and will be shown in Lemm

휔 퐙퐅 0 ) 푛 Applications ofFoundation ) … , 퐈퐙퐅 . 훽 must by minimality of minimality by must ¬ ∧ 훾 ≠ 휂 ∧

퐶 ∈ 훼 vs is ( , but (the class of) ordinals fail to be well-orders in 푚 < 1 + 푛 ,푎 1, − 푛

– ∈ , which must be 퐈퐙퐅 10 ( arole that lostwill be in ( -least among-least ordinalsthe 퐁퐒퐓 + Set induction Set + 퐁퐒퐓 . Either 퐒 퐄 Foundation + 퐋퐄퐌 + 퐁퐒퐓 W dfn fr each for define We : 훽 ∈ 훼 12 or the problem with trichotomy 휂 ( 푛−1 and and 퐒 퐄 Foundation + 퐋퐄퐌 + 퐁퐒퐓 핆ℕ ,,,… 0,1,2, ( 휂 ∈ 훾 , which , which is impossible by incomparability. Thus, or ) 훼 is well-ordered by well-ordered is ) isalready 푎 : 훾 푚 = 1 + 푛 ae noprbe f te ae cutrxml t te rcooy a, i.e. law, trichotomy the to counterexample a are they iff incomparable are ) 푖 instead of instead . Towards a contradiction, suppose that there is such an ordinal an such is there that supposecontradiction, a Towards . 휔 ∈ ∈ -minimal in } 훽 W dfn te e of set the define We . ): Equality): on be comparable to comparable be 휔 ∈ 푛 ∈ or By -minimal, or 휔 ∈ 푛 ,푠 0, 1 + 푛 < 푚 – 퐈퐙퐅 Lemma 3.11, we may assume that 푠 Ordinals in 훼 and a bijective function bijective a and ): ): 0 ( isincomparable with. 푠 , . 0 핆ℕ ∈ 퐶 te e of set the , 1 ) . Furthermore, the ordinals in ordinals the Furthermore, . too. 푠 , 푠 , … , ): ): iswell-founded, i.e.each nonempty subclass ( ( ~ 푠 ℕ ∈ 핆ℕ, 휔 ( . 푛−1 ∅ ≠ 퐶 ∩ 훼 0 isdecidable. 39 )) 훼 … , ofthe elements of 풁푭 ~ ) ∈ , thus , is a well-order and so is . We will refer to a set a to refer will We . -minimal element. Together with Together element. -minimal eune o ntrl ubr o length of numbers natural of sequences

iie eune o ntrl numbers natural of sequences finite . By Foundation, there is an 퐙퐅 훼 ∈ 훾 퐈퐙퐅 , like induction and recursion pertain to pertain recursion and induction like , :푛→푆 → 푛 푓: , . 훼 = 훾 훼 ⊆ 훽 푆 or , or tobe more precise: setA is . In In . 훼 퐙퐅 퐙퐅 . is 훾 ∈ 훼 and and 퐁퐒퐓 form paradigms for well- for paradigms form ∈ 푆 ( -least -least with this property as ,∈ 훼, . In the last two cases, two last the In . , we can define the set the define can we , 퐈퐙퐅 (Kuratowski) ) for each ordinal by the example ofexample the by ∈ , collection and collection , -minimal ordi- 퐋퐄퐌 핆ℕ ⊆ 퐶 훼

WS WS as incompa- 푛 this will this : We say We : as

finite 2019 휔 a 3.16 휔 <푛 has 푛 iff /20 훼 ∎ ∎ = = . ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. counterexample: The The problem with well-ordering ordinals is the trichotomy on Proof

Axiomatic constructive set theories theories set constructive Axiomatic Ⓜ Lemma 3.3.2 sameThe argumentation applies to: and and ⓒ

훼 Lemma3.14 Theorem3. We can now prove the following fundamental theorem of We know from above, that Proof

morphism property and our assumption on ments, so it remains to show that it is transitive itself: Let itself: transitive is it that show to remains it so ments, hs, e a asm that assume may we phism, Note that given given by 푓 is isomorphic to a (unique) ordinal (unique) a to isomorphic is Now, suppose, thatsuppose, Now, =퐺 = If If Proof ( ). 퐴 = 퐺 Corollary 3. 휂

Proposition 3.17 ) , we thathave 훾 = 훽 : Irreflexivity isLemma and2.16 transitivity is guaranteed bydefinition. : We first show that for all for that show first We : 푓 ∈

: Uniqueness follows from Lemma 3.14. We denote by denote We3.14. Lemma from Uniquenessfollows : Ordinals inOrdinals

, then then , 3. ( 푎 훼 16 푓 , we would have the impossible Note that ) isin fact an isomorphism between 15 . ( : Let 퐁퐒퐓 + Set Induction Set + 퐁퐒퐓 : Every well-ordered set 13 ( ,≺ 퐴, ( 푓: 퐒 퐄 Foundation + 퐋퐄퐌 + 퐁퐒퐓 ( 퐒 eaain⊢ Separation + 퐁퐒퐓 ) ( : Trichotomy implies forms of ,≺ ↓, 푎 퐈퐙퐅 is isomorphic to isomorphic is 푓 ,∈ 훼, ↓ 푎 is not the identity and let andidentity the not is

) is well-ordered by ) → isisomorphic to 푓 ( ( ,∈ 훽, 훾 ) 푓 = 훿 consists of ordinals only, hence 퐒 ⊢ 퐁퐒퐓 – 훼 ∈ 훾 ) ): ):

be an bean isomorphism. Then 퐈퐙퐅 ( WS WS 푓 ↔ 훾 ∈ 훿 ℕ ∈ 핆ℕ, 휉 ( 훼 ∈ 훾 푎 and and 휌 ⋃ , and form the function the form and ( ) 2019 푓 [ [ ,≺ 퐴, 푎∈퐴 , 훼 핆ℕ ∈ 훽 ∀훼, 훼 핆ℕ ∈ 훽 ∀훼, ( 푓 = 휖 훾 ) 퐂퐙퐅 훼 ∈ 훽 . By minimality, itmust be comparable to is irreflexive and transitive and so is ) 휉 ): ): /20 ≺ ) ran is an ordinal. It is clear, that clear, is It ordinal. an is 푎 isisomorphic to a unique ordinal 훾 as well. Let ∈ . If If .

( ,

-least elements are unique in 훿 ( ( ( . Thus, 휂 ,≺ 퐺, ~ 푓 ) 훼 ∈ 훾 퐴 ≠ 퐺 ) ) 푓 ∈ and and 퐋퐄퐌 40 핆ℕ ∈ ( ( ) 훼 ∈ 훽 ∨ 훽 = 훼 ∨ 훽 ∈ 훼 훼 ∈ 훽 ∨ 훽 = 훼 ∨ 훽 ∈ 훼 and ( be the least ordinal such thatordinalsuch least the be ~ 훾 , let , 훽 ∈ 훾 : ) 훼 ∈ 휌 ∈ 휂 , acontradiction. 푓 ∈ 훿 ↔ 퐺 ran 퐙퐅 be the set of elements 퐴 ∈ 푎 푓 ∈ 훿 , i.e. 푓 푎 ↓ = ↓ 푎 (details can found be in [ ( isthe identity mapping. 푓 핆ℕ 푓 ( ) 푎 푓 and and be the least element not in not element least the be 훽 ⊆ 훼 ( B transitivity, By . ( , since it would provide us with a weak ) ( 훾 훾 { 훾 ) 휉 = ) :푥≺푎 ≺ 푥 퐴: ∈ 푥 ) .

and let and must be must equal to ran . Altogether 푎 with domain with ( 푓 푓 ) ) ) ( ] ] 핆ℕ 훾 핆ℕ ∈ 훿 ∈ 휖 → 퐋퐄퐌. → ) consists of transitive ele- transitive of consists (and thus in each ordinal 훼 퐋퐄 } ( the initial segment ofsegment initial the . ,∈ 훼, 퐴 ∈ 푎 . Since . . 퐌 훽 = 훼 훼 ∈ 휂 푓 ′ 훽 )

. (

for each ordinal . In the. In cases 퐺 훾 such such that 훾 40 ) by replacement. by after all. ad hence and , contradiction. 훾 ≠ ] 푓 ): is an isomor- an is

. By the iso-theBy . 퐺 . Since Since . ( ,≺ ↓, 푎 훾 ∈ 훽 = 휖 ↓ 푎 훼 ∎ ∎ ∎ ∎ 퐴 ) .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. We show this by ordinal induction that that induction ordinal by this show We mains and that for each for all The functionsThe Ⓜ dinals. A similar proof can be found [ in rec via classes and (class-)functionsdefining of applications important the discuss We 3.3.3 Theory Set Constructive of Analysis and Comparison 퐺 To verify that this works we have to show that the local approximations local the that show to have we works this that verify To Proof:

domain o te eod at let part, second the For Proof App chotomy we have have we chotomy shown so far. So, by Replacement and Union, we may define 1 + 훽 For the uniqueness of uniqueness the For Lemma2.16. In the second casewe would have and and Finally, we can set 훼 ∈ 훽 Example universe of all sets ( ,푠 훼, hoe 3.18 Theorem 2. 1. ( 퐹 푑 : Let :

for all all for . But then But . ) ′ First, let us give a such 훼 푑 ∩ 푑 ∈ 훽 are identical. tobethe unique Recursive on definitions the ordinals 훼∈핆ℕ 퐹 ∈ ∀훼 ∀훼 ∃! 푦 휓 푦 ∃! ∀훼 ℎ , 푑 ∩ 푑 ( 훼 휙 퐈퐙퐅 ) be any (bounded) formula and let and formula(bounded) any be . Again, we do this by Set induction: Suppose, we have defined such funct such defined have we Suppose, induction: Set by this do we Again, . 훼 ∈ 훽 ′ or ℎ and and App can can be thought aslocal approximations to ′ ( 퐹 : 훼 ∩ 퐙퐅 ,푦 푥, ( ( ( Fixing the domain lets us assume that these functions are unique by what we have we what by unique are functions these that assume us lets domain the Fixing 훼 훼 ℎ App ( 퐁퐒퐓 + Replacement + Set Induction Set + Replacement + 퐁퐒퐓 1 ∈ 훽 ∨ 훽 = 1 ∨ 훽 ∈ 1 ): As an important example we can define the the define can we exampleimportant an As ): ) ) ,ℎ 푑, 훼 ) . Then , so 퐺 = 퐺 = ℎ = 퐹 ( 휓 훼 ) 푦 푑 ∩ 푑 , suppose, we are given two such functions such two given are we suppose, , ̃ ( 휓 , there is some ( safnto dom ∧ ℎ is a function ≡ such that such ( ,푦 훼, 핆ℕ ∈ 훼 ,퐹 훼, ∪ defines afunction ,퐹 훼, ℎ { ( 훼 ℎ (훼, 퐺 ) 휓 훼 ′ App | ℎ ∩ ℎ , | 훼 ) 훼 ≡ . Let us abbreviate ) ) and 퐺 = [ ( 퐺 = W ne t dfn a function a define to need We . 훼 ∈ 핆ℕ ∧ ∃푑, ℎ ℎ ∃푑, ∧ 핆ℕ ∈ 훼 ,ℎ 푑, ̃ 휙 | ′ ( 훼 ) 퐹 ,ℎ 훼, ( ( . ) ) ,푠 푦 푠, 훼, ,퐹 훼, is uniquely thisby property. } App ∧ ℎ and and 40 Te is cs i ipsil, for impossible, is case first The . ℎ | defined taking a value on 훼 ( ]. ′ ) 훼 | 훼 ) ) 퐺 = 퐹 . Then there isa formula 푑 ) ( ℎ = , where 훼 푑 퐹 = = 훽 ′ ~ ( 휙 1 + 훼 = ( ℎ , ℎ ,ℎ 훼, [ ′ and in the third App ) ( 41 ′ ′ ( ) 훼 { = 푑 ⊆ 핆ℕ ∧ ∀훼 ∈ 푑 ℎ 푑 ∈ ∀훼 ∧ 핆ℕ ⊆ 푑 = 훼 :휙 1: ∈ 푥 ) App → ′ ) | for all all for ~ 퐹 ( 훼 . We may thus infer by ordinal induction that induction ordinal by infer thus may We . : Assume, ): ,ℎ 푑, ( ) and conclude and 훼 ℎ = ) is the unique ) 퐹 ( ℎ ∧ 푑 ∈ 훼 ∧ } . Our candidate for 푑 ∩ 푑 ′ 푑 ∩ 푑 ∈ 훼 ( ℎ 1 ⊆ ̃ 훼 = ) . So . Clearly, . ′ ⋃ ℎ ∩ ℎ , von Neuman universe Neuman von

∀훼 ∈ 핆ℕ ∀푠 ∃! 푦 휙 푦 ∃! 핆ℕ ∀푠 ∈ ∀훼 ¬휙 훽∈훼 훼 ℎ ℎ ∩ ℎ . For the first part, we need to show 퐹 훼 . 휓 ′ App and and . Suppose, we have we Suppose, . dfnd on defined ( ℎ such that such 푦 훼 ′ 훽 ) ( with ) and and ′ . 훼

ℎ ( indeed forms a function with 훽 푦 = ) ℎ 퐹 agree on their common do- common their on agree 1 ⊆ 훽 ∈ 1 is an ordinal; hence, by tri- by hence,ordinal; an is 훼 퐺 = ′ and say they agree on all on agree they say and 푑 , 푑 ] 휓 ̃ ] 휓 훼 ( . 훼 =

( ,푦 훼, ) istherefore ,ℎ 훼, . 푑 . Clearly, ) ( 훼 | . ,푠 푦 푠, 훼, wud contradict would 훼 with

ursion on the or- the on ursion ) ion ion 푉 .

, also called called also , ℎ ℎ ) 훽 ad define and ( 푑 ∈ 훼 훽 WS WS with App ) ℎ = 2019 ( 훼 ℎ ̃ 푑 and ′ 푑 , 훽 ( /20 the ̃ 훽 ∎ ∎ = ) 퐹 ) .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

Axiomatic constructive set theories theories set constructive Axiomatic given set. Myhill notes [ notes Myhill set. given all construct to procedure any describe not does it that sense, the in non-constructive and section2.4.3In we have discussedthat the axiom ofPower setmay becriticized for being impredicative 3.4 Bishop’s necessary for asolid foundation of mathematics may therefore berebutted bythorough investigationof of mulation

the secondthe ofname obtained is this way starting from the empty set. It turns out that this class is the cla construction. up” “bottom a us gives construction simple This Let us Let us consider the first few stagesof the This This definition is justified by the form the Collection. In the case of We can now set 0 = 훼 Usually,whengivinginductive definitions ordinalsthe on in = 푉 Proof setting setting as the distinction between successor and limit ordinals requires Theorem 3. ment of ment Replacementthe axiom. We now have

⋃ : By set induction. Suppose, , 퐈퐙퐅 훼∈핆ℕ 훼 훼 book. is a successor ordinal (i.e. of the form the of (i.e. ordinal successor a is 1 + 훽 = 훼

vs 푉 19 훼 퐂퐙퐅 . Although more readable, this definition like this would make sense in the intuitionisti the in sense make would this like definition this readable, more Although .

( 퐂퐙퐅 퐈퐙퐅 , is enough to do mathematics Bishop-style. The argument, that the Powerset axiom is axiom Powerset the that argument, The Bishop-style. mathematics do to enough is , 푉 훼 푉 훼 or 퐹 = ). The definition of definition The ). or how to live without the powerset operation 푉 퐹 = : 퐙퐅 ( 훼 ( 48 ): ): 퐙퐅 훼 ) and observe and ) 푥 푉 ∈ 푥 ∀푥. ], that the weaker Exponentiation axiom, thus replacing Powerset in the for- the in Powerset replacing thus axiom, Exponentiation weaker the that ], , we may assume that the 퐺 = Theorem ( – ,퐹 훼, 푦∈푥 ∃훼 푥. ∈ ∀푦

퐈퐙퐅 WS WS , or written out, | 훼 and and 3.18 as follows: Let ) 푉 푉 푉 2019 푉 = 1 2 0 would thus look like look thus would 푉 = = ∅ = 퐂퐙퐅 푦∈range 훼 /20 푦 : { { 푉 = 푉 ∅, ∅, 훼 ℕ 푉 ∈ 푥 핆ℕ. ∈

⋃ = { { 푉 ⊆ 푥 1 + 훽 = 훼 ~ ∅ ∅ ( 훽∈훼 훼∈핆ℕ ⋃ } } 퐹 ⋃ } 42 , |

{ 훼 푥 훼∈핆.푥∈푉 ∈ 푥 핆ℕ. ∈ ∃훼 ∀푥. { ) ⋃ 풫 ∅ 훼 풫 ~ 푍 푉 푦 } ( ( 훼 } and so are 푉 푦 , 훼 . 훽

{ ) 푦 ) and and ) ∅, 휙 ) . In the case of = , (

∈ { ,푠 푦 푠, 훼, ∅ 훽∈훼 ⋃ -minimal to satisfy the uniqueness require- 퐙퐅 } 푉 ∈ 푥 훼 }} 푉 푉 , the definitionthedividedis , casesthe into a limit ordinal ( ordinal limit a 0 풫

is then the class of all sets that can be can that sets all of class the then is ) ∅ = ( be the formula 퐹 ⋃ ( 훼 푍+1 훽 , . 퐋퐄퐌 푉 ) ) 훽+1 . 퐈퐙퐅 = . 훽∈훼 ⋃ 풫 = , form a set 풫 ( 훼 푉 ( ss of all sets justifying 훽 = 푦 cannot be written in written be cannot 푉 훽 ) , ) 푉 .

⋃ 휆 푍 푦∈range = of such ⋃ subsets of a a of subsets 훼∈휆 ( 푠 푉 ) 훼 풫 훼 and ( 푦 푦 by ) ∎

c .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Powerset axiom from the Exponentiation axiom. By axiom. Exponentiation the from axiom Powerset of assumption weak seemingly the that out, turns It ing restricteding separation. which ofcourse stands for the set the 푓 axiom. But then Proposition3.22 Proposition3.23: Ⓜ We start with showing that theuse axiom ofFullness the in following discussion. th we convenience, of sake the For equivalent. fact in are Fullness and collection Subset of oms shown (Proposition have we Remember,thatExponentiation. thanstronger isturn in We start with formally showing that the Powerset-axiom is indeed stronger than Subset collectio 3.4.1 Theory Set Constructive of Analysis and Comparison Proposition 3.21

푎 ∈ 푥 structures may thus be conducted in a recursive fashion. It turns out that out turns It fashion. recursive a in conducted be thus may structures contain Proof define Proof Proof class One specific application of the Powerset to mathematics in general and set theory in particular is that the the hierarchythe being sets to ( 푥 퐂퐙퐅 Proposition 3.20 ) : Let : : One direction is clear. For the other, let : The direction from left toright is clear. For the other direction, let =

. Then . 푉 . of all sets forms a hierarchy of sets of hierarchy a forms sets all of = 푥 = 퐶 { Powerset,Subset Collection, Exponentiation ∅ ∈ 푦 : 퐶 { be an be { ∈ 푦 푓 { { ∅ :∅∈푓 ∈ ∅ 퐴: ∈ 푥 ′ is a full relation betweenrelation full a is } 푧 ∈ 푥 : { = 푑 : : ∅ 퐒 xoetain+ Exponentiation + 퐁퐒퐓 퐒 ules⊢Exponentiation ⊢ Fullness + 퐁퐒퐓 푎 } 퐁퐒퐓 ⊢ Powerset ↔ Exponentiation + "풫({∅}) is a is set" "풫({∅}) + Exponentiation ↔ Powerset ⊢ 퐁퐒퐓 휙 : -full set of subsets of subsets of set -full : { 퐒 퐋퐄퐌 ⊢ 퐁퐒퐓 } 푓 } for

′ with with ( – 푥 ( most ofthe recursive characterof ) 푥 퐴 ∈ 푥 푎 ∈ 푥 : 푧 ∀푏. ∃푧. ) 퐋퐄 푦 } 풫 ∈ 푓 : not not free in 퐌 . Then ′ ′ } may be reduced to the statement that all sets either contain ( ∀푥 ↔ = 푧 ∈ 푏 ↔ 푎 ⊆ 푏 ( { {( ∅ 풫 ∈ 푓 ,푓 푥, } ( ) 푎 푥 ∉ ∅ ∨ 푥 ∈ ∅ 휙 퐴 푏 × 푎 and and ( 퐋퐄 } . If . If 푥 . Clearly, . ) ( – 퐌 퐴 ) { definitions and proofs of properties on that and similar and that on properties of proofs and definitions 푥 ∈ ∅ ∅ 푎 ∈ 푥 : . For . 푏 × 푎 be any set and define by exponentiation and replacement ′ } ~ Powerset ⊢ ) ) 퐴 . 43

and and , then , then , hence there is some thereis hence , :푎→푏 → 푎 푓: } 풫 ⊆ 퐶 ) ~ 풫 "풫(푎) is a set" is "풫(푎) 푓 = . ( = 푧 { ∅ 휙 . Thus, } , if , if ( . ) { define 퐴 푉 to be a set is enough to to deduce the full the deduce to to enough is set a be to :∅∈푓 ∈ ∅ 퐴: ∈ 푥 ) may bepreserved when passing from 푥 ∉ ∅ . For the other inclusion, let inclusion, other the For . 푏 푎 푓 can can be identified as a subset of , then we mean the formula the mean we ′ 푏 × 푎 → 푎 : . 휙 ( 퐶 ∈ 푑 푥 be restricted any formula. We ¬휙 ) – } apart from the stages the from apart . This shows. This . according to the fullnessthe to according by 푓 ′ ( 2.21) that the axi-the that 2.21) 푥 ) = 풫 = 퐶 퐴 ⊆ 푧 푧 풫 = 푧 ∃푧. ( ,푓 푥, ∅ WS WS or do not n, which ( ( and let and 퐴 erefore 2019 푥 ) ) 퐶 . 푉 ) ( 훼 us- for 퐈퐙퐅 푎 /20 of ∎ ∎ ∎ ) ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. result from [ from result Let this Intuitively, the following:the The GeneralThe Uniformity Principle ( 풫 and by and reasonable todoubt that the where axiom, Powerset the of instances constant. From fact the that 풫 theory set Zermelo-Fraenkel in axiom set Power the about section the in this Proposition 3. Proposition the full Powerset axiom from the Exponentiation axiom. By axiom.Exponentiation the from axiom Powerset full the instances where the resulting power-set is classically finite. Intuitionistically finite. classically is power-set resulting the where instances Altogether,

Ⓜ theories set constructive Axiomatic Proof By This is a very strong result: It turns out, that in restricting in that out, turns It result: strong very a is This By bounded separation GUP Proof ( ( 푎 1 Theorem 3.24 퐋퐄 • • ) ) . Wewill argue inside the lattertheory for the rest ofthe proof.

isequivalent tothe set of all formulas of the underlying language oftheory. set , which of standscourse for : Towards a contradiction, suppose that : By It turns out, that the seemingly weak assumption of assumption weak seemingly the that out, turns It By : 퐌 For For For For GUP ′ 푦 , be fixed. 푥 ∉ ∅ ∨ 푥 ∈ ∅ = 푥 ∅ = 푥 , GUP ∅ = 푎 71 22 ], one concludes that concludes one ], { says that every mapping from the universe of all sets into a given set given a into sets all of universe the from mappingevery that says ∅ , , it suffices to show that show to suffices it , : For } , acontradiction. = 푦 , = 푦 ⋃ ∅ ≠ 푎 . In the. In case,first , ⋃ {

:∅∈∅ ∈ ∅ 푎: ∈ 푤 { 풫 푤∈푎 ∈ ∅ 푎: ∈ {푤 :∅∈푥 ∈ ∅ 푎: ∈ 푤 ( 휔 , 퐂퐙퐅 풫 ) ∀푎 is consecutively justifiable.The peculiar side isthat we also lose innocent ( ∅ isconsistent with a weaker version ofthis axiom case (the – ( ) 푧 ∀푏. ∃푧.

푥 푦∈푎 휙 푎. ∈ ∃푦 ∀푥. is not aset isnot in 퐈퐙퐅 GUP WS WS 푦∈풫 ∈ ∀푦 푥 푦∈풫 ∈ ∃푦 ∀푥. } { GUP ∅ and and } ∅ = 퐙 ∃.푧=풫 = 푧 ∃푧. → ∅ ≠ 푎 ⊬ 퐂퐙퐅 is asettoo, for every set } 2019 ) isthe) following schema: 푎 = } ( = 푥 푧 ∈ 푏 ↔ 푎 ⊆ 푏

. 풫 is consistent with consistent is 퐂퐙퐅 ( /20 ( 푎 { . 풫 { ) 퐂퐙퐅 ∅

∅ 푥 = 푦 ∃푥. .

( -operator is applied to infinite sets. For example, it seems it example, For sets. infinite to applied is -operator ( } } 푎 ,푦 푥, ) , in the second, in the case, ~ is a set. In fact, we show we fact, In set. a is 퐂퐙퐅 ) proves that = 푦 . 44 ) : 푦∈푎 푥 휙 ∀푥. 푎. ∈ ∃푦 → ~ { { ) :∅∈푥 ∈ ∅ 푎: ∈ 푤 :∅∈푥 ∈ ∅ 푎: ∈ 푤 . 퐈퐙퐅 퐂퐙퐅 "풫(푎) is a set" is "풫(푎) 풫 풫 ( to ( 푥 푎 . Given this result, one can easily show easily can one result, this Given . 푎 ( . Hence, ) { ) . ∅ 퐂퐙퐅 is a set for

} } ( } ∅ = 푥 . , ,푦 푥, )

to be a set is enough to to deduce to to enough is set a be to one not only loses the unwanted the loses only not one ) ) . . 풫

we mean the formula the mean we ( { ∅ ≠ 푎 ∅ 퐙퐅퐂 – } and we have discussed have we and ) = 2.4.1 . Then so does { ∅, 푎 { – ∅ , must in fact be fact in must , already the set the already } } . Let . 휔 = 푎 ⊆ 푥 ) and a and ) 푧 = 푧 ∃푧. 퐙 + 퐂퐙퐅 { ∅

∎ ∎ } .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Ⓜ nition of this (and other important classes) in sets thedefining in that Note 3.4.2 Theory Set Constructive of Analysis and Comparison Ⓜ set separateprovesome introducedefinea lemma,notation:existence-partlet us the inwe We Before for Definition 3.25 conclusion under

closed class rule rule infer some with logic a considercould example,we For accidental:not terminologyis familiar The We call aclass the conclusion of conclusion the be inferred from it is theit is deductive closure of where thewhere Furthermore, we thehave following inductionprinciple: Lemma3. 3.26 (inductiveTheorem)definitionTheorem 푋 Φ and and a class (or set)

: This rule can be represented as a class of pairs of class a asrepresented be can rule This : Inductive Inductive definitions 퐼 푎 퐼 s satisfy 27 . Itcan be written as 푌

: An ( Φ

퐒 Collection + 퐁퐒퐓 푀 훷 푥 . For any class - by under under closed inductive definition Φ in in one step. We will be interested in the smallest 푀 iff Φ , . Then, for any set any for Then, . Γ 푀 Φ 푉 ( with respect with to 훼 푎(푏∈푎 푥∈퐼 ∈ ∀푥 푎. ∈ (∀푏 ∀푎 푌 we used the Powerset-axiom in a decisive way. In order to give a defi- a give to order In way. decisive a in Powerset-axiom the used we 푀 푀 푌 ) ): There): is aclass Γ , wedefine the class of ∈푋 푋 푌 ⊆ Φ = ( = 푌 is a class of ordered pairs. If . { ) 푎: { 푎: ∃푌 ∈ 푋 푋 ∈ ∃푌 푎: = 〈 ,푎 푋, { 퐼 퐽 :∃ 푋⊆푌∧ 푌 ⊆ (푋 ∃푋 푎: 푎 푎 퐂퐙퐅 푥∈퐼 휙 퐼. ∈ ∀푥 Γ = 〉 Γ = Φ 푀 = 퐼 푀 ∈ ~ . of formulas, of 푏 as well, aswell, we to need consider inductive definitions. Φ Φ : For any inductive definition inductive any For : 휙 . 〈 45 퐽 푎∈푉 ⋃ ,푎 푌, ( such that such ( } ( , ⋃ 푏∈푎 ⋃ 푏∈푎

푥 ~ 퐼 〉 ) ( 〈 푎 푥 ,푎 푋, 푀 ∈ 푥∈퐼 ∈ ∀푥 → 퐽 , 퐼

) 푏 Φ 푏 ). 〈 ). -conclusions from premises ,푎 푋, 〉 } , where ,

= Γ 〉 Φ 푌∈푋 ⋃ Φ ∈ ( 푎 푀 휙 . 〈 푀 ) ,푎 푥, 푋 } is the class of all formulas that canthat formulas all of class the is ( . 푌 푥

is a finite set of premises and premises of set finite a is ) . 〉

) Φ ∈

Φ -closed class: In our example, , we call Φ , there is a smallesta is there , 푥 a 푌 , premise WS WS and 2019 ence 푎 푎 /20 Φ is a

a -

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. which shows that

Axiomatic constructive set theories theories set constructive Axiomatic Ⓜ Φ sion ission trivial). Suppose, To show minimality,let show To that this does the job. Indeed,let job. the does this that Proof of the theorem the of Proof 퐼 Proof h idcin rnil i ntig oe u st nuto: Assume, induction: set but more nothing is principle induction The 퐽 퐺 Let Let We giveanother formulation ofTheorem 3.26 in terms of ordinals: Let that that By collection, there is aset For other the inclusion, let 푎 푦 isgood. As -closed -closed class 휙 . . So, bycollection, there is aset Theorem 3.28 (inductive definition on ordinals) on definition (inductive 3.28 Theorem = 퐽 = 퐺 퐽 ∈ 푎 Γ ∈ 푎 ( : We call a set 푥 ) ⋃ ) { . By setinduction, 푋 〈 { Φ ,푎 푋, . Then Then . 퐺: 퐺 is good 퐺 is 퐺: (퐽 ∈푋 〉 〈 } ,푎 푋, 퐽 ) . Furthermore, ∪ . Thus, 〈 ⋃ ,푎 푋, ⊆ 푋 〉 : Let the Let : 퐺 퐺 ∈ 푍 ofordered pairs . Then . } 〉 . The lemma. The in our new notation reads ⋃ 퐺 ∈ , wehave 퐽 푋 푦∈푌 퐼 퐼 푥 ′ be another be Γ ⊆ for some good set good some for 푍 Γ ∈ 푎 퐼 ⊆ 푎 푥∈퐽 ∈ ∀푥 ∀푎. 퐽 퐼 such that such ⋃ 푎 푦 Φ s be as in the lemma. Of course, we set we course, Of lemma. the in as be s and and hence 푍 ′ (퐽 for all for all is good and because of because and good is Φ – ∈푋 푦∈푌 퐺∈푍 sgo n 퐺 ∈ good andis 푦 퐺 푍. ∈ ∃퐺 푌. ∈ ∀푦 〈 퐽 ∈ 푎 (퐽

푌 퐈퐙퐅 ,푎 푋, , such that, such ) WS WS ∈푋 푦∈푌 퐺 sgo n 퐺 ∈ good is and푦 퐺 ∃퐺. 푌. ∈ ∀푦 . 퐼 and and 푎 ) 〉 푋 Φ 푋 ∈ 푥 휙 . 푋 good . Then 2019 〈 Φ ∈ . Thus -closed class. We show We class. -closed Γ = ,푎 푋, ( 푥∈푋 푦∈푌 퐼 ∈ 푥 푌. ∈ ∃푦 푋. ∈ ∀푥 Γ ∈ 푎 퐂퐙퐅 푥 iff ) /20 and and Φ . But . But by monotonicity of 〉 and thus and ( Γ ∈ 푎 → 퐺 ∈

〈

,푎 푌, = 퐼 Γ Φ 푥∈푋 ⋃ 퐺 Φ ~ ( 퐼 ⊆ 푋 , which means which , ( ⋃ 〉 퐽 퐼 훼∈핆ℕ 46 ∈푋 푦∈푌 ⋃ 푦 Φ ∈ Γ ⊆ ) : For any inductive definition inductive any For : ) . Then for each each for Then . ~ 푥∈퐽 휙 퐽. ∈ ∀푥 퐼 퐽 ⊆ for some 퐼 푦 훼 ) 〈 Φ , ,푎 푌, Φ

푋 퐼 = ( ( . 퐺 퐼 ) 〉 ∈푋 푦 푌 퐽 Φ ∈ , 퐼 ⊆ (

푋 퐼 ⊆ ) 퐼 푥 Γ ∈ 푎 , 푋

Γ = ) 퐽 ⊆ 푌 ′ . ∈푋 and and 퐼 ⊆ . .

∈푋 Φ .

Γ Φ 푋 ∈ 푥 Φ ′ (퐽 퐼 . ∈푋 by set induction (the other inclu- other (the induction set by (퐺

푎 , 푎(푏∈푎 푦∈퐽 ∈ ∀푦 푎. ∈ (∀푏 ∀푎 퐺 ⊆ 푌 ∈푋 , i.e. 퐽 = ∈푋 ) there is some is there . Weshow this asfollows: ) 푎 . Since Since . and and 푦∈푌∃ 퐽 ∈ 푦 푋 ∈ ∃푥 푌 ∈ ∀푦 ∈푋 also = 퐼 Φ 퐺 Γ ∈ 푎 , there is a smallest a is there , ∈푋 ⋃ 푎∈푉 퐽 ⊆ 푦 푏 Φ such that that such 휙 . ( 퐼 ∈푋 퐺 푎 ( and claim and ∈푋 i follows it 푦 ) ) 푥∈ ∀푥 → 푥 ; hence ; , so ∈ 푥 ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. choose doublechoose induction over single two inductions. Example written as written Theory Set Constructive of Analysis and Comparison 퐂퐙퐅 theories the between relationship looming the establish to like would completeness,we of sake the For 3.5 collection and strong collection strong and collection Proof and it is easily entangled into two separate inductions for inductions separate two into entangled easily is it and 퐽 ⊆ 푋 The inductionThe principle follows easily. ℕ푥∈퐼 ∈ 핆ℕ 푥 푉 In later chapters, we will often prove statements by statements prove often will we chapters, later In nlso w so that show we inclusion thewhere weaker theories. Furthermore, we thehave following induction principle scheme 훼 = ,

: Effectively, all we need to show is that the smallest the that is show to need we all Effectively, : 퐈퐙퐅 ⋃ 훾 Relation between the theories , where , where 훽∈훼 and and 훼 : Let : . By collection, there is some is there collection, By . 풫 퐼 훼(∀ 푥∈퐼 ∈ ∀푥 훼 ∈ ((∀훽 ∀훼 ⋃ 훼 ( 훼∈핆ℕ s satisfy 푉 퐙퐅 Φ 훽 = 훾 ) be the class of pairs of class the be . First, we need some results about the interplay between the axioms of replacement, of axioms the between interplay the about results some need we First, . for all for all 퐼 훼 ⋃ . So, let So, . 1 + 퐵 푉 훼 훼 = 퐼 . Suppose, this holds true for all ′ . By definition, is 훽∈훼 ⋃ 훼∈핆 ∀ .∀ 퐼 ∈ ∀푥 훼. ∈ (∀훽 핆ℕ ∈ ∀훼 퐼 ′ Φ 훽 = 퐽 휙 coe. Let -closed. 훽 ⋃ – ( = the only axioms that are strengthened when passing from passing when strengthened are that axioms only the 푥 훼∈핆ℕ ) 훽∈훼 ⋃ 푥∈퐼 ∈ ∀푥 → ) 〈 ,푎 푋, 퐵 퐼 Γ 훼 Φ containing all these all containing 〉 , we show that indeed that show we , Γ ∈ 푎 , where where , 푥∈퐼 휙 퐼. ∈ ∀푥 ( 퐼 훾∈훽 ⋃ 〈 훼 ,푎 푋, 푥∈퐼 휙 퐼. ∈ ∀푥 Γ = Φ 훼 퐽 ~ ( 휓 훾 〉 퐼 Φ ) 푋 Φ ∈ 훾 47 ( ( ) ( 푥 is any set and and set any is = 푥 훽 ) 퐼 = 훽∈훼 ⋃ double recursion double ) 휙 . 훼( ∀훼 ) ~ , where where , 훽∈훼 ⋃ 휓 ∧ ( 푥 ( 훾+1 퐼 푥 ) 훽 훼 ∈ 훽 ( 풫 ) ( ). 푥 . 푥∈퐼 ∈ ∀푥 → 휙 ) Φ

( and and 훾∈훽 훼 ⋃ -closed class -closed 푥∈퐼 ∈ ∀푥 푋 s. Hence, by Proposition 3.4 it holds thatholds it 3.4 Proposition by Hence, s. , then i a ust of subset a is 퐼 = 퐼 퐽 훾 푋 ⊆ 푎 휓 ) . However, it will be convenient to convenient be will it However, . for an inductive class inductive an for 훼 훼 = 휙 . 휓 ′ . Obviously, . 훽∈훼 ⋃ ( ( . We set We . 푥 푥 ) ) ) ) 퐼 풫 from Theorem 3.26 can be can 3.26 Theorem from

푥∈퐼 ∈ ∀푥 → ( 푉 훽 퐽 W have We . ) 푉 . 훼

= 퐼 ′ 훼 ⋃ 퐼 ⊆ 휙 훽∈훼 ( . For the other the For . 푥 ) 퐼 퐼 푥∈푋∃ ∈ ∃훼 푋 ∈ ∀푥 ) 훽 . This is the is This .

WS WS and show and 퐙퐅 2019 to the to /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. and theand equations in in ⓒ ples this of chapter the and previous one). = 훼

Proof We canWe now gather together our results from the previous sections toestablish: 3.5.2 Ⓜ Proof Ⓜ 3.5.1 theories set constructive Axiomatic Ⓜ Ⓜ And we may assume that assume may we And ⓒ tain a set a tain Proof tion 2.22 and Proposition 2.21. Propositionand 2.22 tion show in Corollary 4.27, that tion.

퐙 퐄 = 퐋퐄퐌 + 퐈퐙퐅 퐙 퐄 = 퐋퐄퐌 + 퐂퐙퐅 퐂퐙 퐙 ⊊ 퐈퐙퐅 Theorem 3.31 Proposition 3.30 ⓒ ⋃ : : Suppose, : Suppose, 퐅

Proposition 3.29 Ⓜ , which shows that the inclusion isstrict. 퐵 Proofs ofProofs inclusion Replacement,Collection, Strong Collection isan ordinal and 퐙 퐈퐙퐅 ⊊ 퐂퐙퐅 퐙퐅 퐵 such as required in the strong collection schema, set schema, collection strong the in required as such : 퐙퐅 푥∈퐴 푦 휙 ∃푦. 퐴. ∈ ∀푥 푥∈퐴 푦 휙 ∃푦. 퐴. ∈ ∀푥 퐙퐅 proves Collection by Proposition 3.29 and Set induction by Proposition 2.15. We will We 2.15. Proposition by induction Set and 3.29 Proposition by Collection proves 퐙퐅 : We stricthave the inclusions : Foundation follows Propositionfrom 2.15. : Separation followsfrom Proposition 2.19 and setPower fromProposition 3.23. : : 퐈퐙퐅 퐙 Strong collection ⊢ 퐈퐙퐅 : 퐙퐅 proves Strong CollectionProposition by 3.30and Subset Collection by Proposi- 푉 Collection ⊢ 훼 훼 푥 퐋퐄퐌 ( ( = is the smallest such such smallest the is ,푦 푥, ,푦 푥, By ⋃ – is indeed not derivable in ) ) 푥∈퐴

. By collection, we get a set . Then 퐈퐙퐅 Theorem 3.24, the Powerset axiom does not hold in its full generality full its in hold not does axiom Powerset the 3.24, Theorem 푥∈퐴 ∃훼 퐴. ∈ ∀푥 WS WS 퐙 퐄 퐙 퐄 = 퐋퐄퐌 + 퐈퐙퐅 = 퐋퐄퐌 + 퐂퐙퐅 푉 and and 훼

푥 2019 and therefore and 푥∈퐴 푦∈푉 ∈ ∃푦 퐴. ∈ ∀푥 퐂퐙퐅 /20

퐙 퐙 ⊊ 퐈퐙퐅 ⊊ 퐂퐙퐅 푥

ℕ 푦∈푉 ∈ ∃푦 핆ℕ. ∈ ~ 훽 48 . By replacement, form the set the formreplacement, By . ~ 훼 휙 . 퐙퐅 ( ,푦 푥, 훼 퐈퐙퐅

푥 퐵 휙 . ) ′ (and hence so are all weak counterexam- such such that . (

퐙퐅 ,푦 푥, = 퐵 .

) .

{ :∃ .휙 퐴. ∈ ∃푥 퐵: ∈ 푦 푥∈퐴 푦∈퐵 ∈ ∃푦 퐴. ∈ ∀푥 퐵 of all such all of ( ,푦 푥, ′ 휙 . ) ( } ,푦 푥, by separa- by 훼 ) . To ob- 푥 . Then . ∎ ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. containing as constants all natural numbers and standard operators such as such operators standard and numbers natural all constants as containing Proof (left and right projections), logical operators etc. We will write We will write will We style notation. lam a use Turing-machinesor lambda-calculus and recursivemean functions, the actually Theorem 4.1 (Recursion) this is not problematic. Let recall us somebasic results: number further discussion we will hence, when referring to we will not clearly distinguish between a term and its Gödel-number and write assume that there is aGödel-numbering ofthese terms. We write write can becan given in terms of theseone of concepts (this assumption asis known eral idea of computability pretty well, it is reasonable to believe that all effectively computable functions of functions. Additionally, three asall concepts make minimal assumptions seem and tomodel gen- the [ functions.in shown recursive and lambda-calculus the Turing-machines, are computability of els reader is familiar with the basic concepts and will therefore be relatively brief. The most promine th that assume We theory. computability from concepts and notion some recall will we section this In 4.1 that showto able be will a seen be can principles) Church’s or Markov’s as tive set theory to possess. Many of these properties arise naturally from the BHK-semantic In this chapter we will discuss some metamathematical properties that we expect an axiomatic construc- 4 Theory Set Constructive of Analysis and Comparison The The expression

from computability theory needed later on. recalli section short a with chapter the start will We computability. effective on based use use the concept of realizability

Metamathematical properties of constructive axiomatic set set theories axiomatic constructive of properties Metamathematical : Let 휙 Some Some aspects of computability theory ( 휏 푛 ) 휏 . For terms For . if if 36 fix 휏 ] and [ and ] ≡ converges to converges 퐾푙 ↓ 휏 휆푧 (after Kleene) for the structure being able of forming lambda-terms and generously and lambda-terms forming of able being structure the for Kleene) (after . means means that ( 67 휆푦 ,휃 휏, ], that the three models turn out to be equivalent, i.e. they describe the same classsame the describe they i.e.equivalent, be to out turnmodels three the that ], 푧 . ( , we write we , : There isa term 퐂퐙퐅 푦푦 푛 ) )( and and and and 휏 – 휆푦 converges (to any term) and for a formula which itself can be seen as a specification of the BHK-interpretation (see 푧 . 퐈퐙퐅 휙 휃 ≃ 휏 ( ( 푦푦 푛 indeed satisfy some of these properties. For this purpose, we willwe purpose, thisFor properties.these of somesatisfy indeed ) holds. As usually, the set of all terms is countable, and we may we and countable, is terms all of set the usually, As holds. ) ) if the terms converge to the same term (if they do converge). do they (if term same the to converge terms the if . Weverify the property: 퐾푙 휏 fix 휏 ⊨ (the (the fix ~ 휎 ≃ 휎 fixed-point combinator s starting point of a subbranch of constructivism.We of subbranch a of point starting s 49 programs ~ ( 휏 fix 휎 , algorithms ) 푛 ≃ 휏 .

{ 푒 if if the term } for the term for the with number or ) such that for all terms effectively computable functions 휙 Church-Turing-thesis ( 푥 퐬 휏 ) 푒푓 on natural numbers, we , converges to the natural 퐤 instead of , 퐢 , , 퐩 ng some notions some ng (pairing) s, others (such bda-calculus- t a been has It { 푒 WS WS } 휎 푓 푒 nt nt mod- , ). In). the . Often, , where 2019 퐥 and and /20 e 퐫 ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof Lemma4.3 (Double recursion): Proof Definition 4.4 Definition Theorem4.2 (Fixed-point): axiomatic constructive of properties Metamathematical The The disjunction property is the property that whenever the theory 4.2.1 “ let discussion, following the In possess. to theory In this section we will discuss some metamathematical properties that we expect proper con 4.2 Set Furthermore, we assume that one of one of bility bility may be formalized within by

∈ ”, 휇푚 휏

1 : Given : Let some constant some

Metamathematical properties 퐥푖 ≡ 휏 . 휙 Disjunctionproperty the the least number or ≡ 휎 and and 휓 휎 : 1 휆푥 : The lest number operator number lest The : and and 휏 2 휏 . 퐫푖 ≡ ( 푥 휎 휔 ) 2 denoting the set of natural numbers and constants and numbers natural of set the denoting and and . Wecheck the first property: Using the property, fixed-point we have , apply Fixed-pointthe theorem to find afixed point 휏 1 푚 휏 ≡ 푖 푖≃퐥 ≃ 퐥푖 ≡ All terms such that such 휏 ≡ 푖 퐓 decides equality on fix For terms 퐓 휎 퐾푙 . fix If If . Indeed, recursion by the theorem, ( WS WS 퐩휎 휎 ≃ 휎 휏 ⊨ 휓 ∨ 휙 ⊢ 퐓 휎 ≃ ( 휎 ≃ ≃ 휏 휏 ()≃0 ≃ 휏(푚) 1 fix ( ( 푥 ( 1 푖 퐫푖 퐥푖, 휎 휆푦 ) ( 퐩휎 휎 ≃ with ( 휇

휏 휎 ( 휏 휎 . fix is known from recursive functions. For a term a For functions. recursive from known is 1 휆푦 1 fix ( ) 1 ( 휎 ( ,푦 푥, 휎 푥 퐫푥 퐥푥, ( 휎 . 휎 푦푦 ) , then , then 2 휏 . ~ 푥 ) . ( 1

( 푖 퐫푖 퐥푖, free have afixed-point ) ) 휏 , 푦푦 ≡ and and )( 퐓 50 set 2 ) be a theory over a language containing the relationthe containing language a over theory a be 휎푖 ) 휔 휆푦 ) 휎 )( 휙 ⊢ 퐓 ) theories theories ~ 2 휏 ∧ in in the usual way and that some form of computa- ) ≡ ( 휎 . 휆푦 휎 푥 퐫푥 퐥푥, 휎 ≃ 2 2 ( ( ( 휎 . 푦푦 휆푥 휎 ≃ ,푦 푥, or 1 ( 휏 . ) ( 푦푦 ) ) 푖 퐫푖 퐥푖, ) 2 . 2019

(

휓 ⊢ 퐓 there are ( ) 푥 휏 ) ) 1 ) ) ) 휏 , 퐓 /20

휏 ≃ 푖 휎 ≡ proves 2 . )

.

1 ( ( 푖 휏 휏 푖 ) 1 1 , this means, this ,,,… 0,1,2, .

and and 휏 , 푖 휓 ∨ 휙

of 2 )

.

휏 2 , it must actually prove denoting its elements its denoting such that such 퐾푙 휏 ⊨ 휏 structive set ( , we define we , 푖 ) 푖 ≃ . ∎ ∎ ∎ .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 퐅 ¬ ⊢ 퐙퐅퐂 For the classical set theory erty). number and h cnium hypothesis continuum the possess the disjunction property (indeed, we will show, that both by the formula 푥 A theory A 4.2.2 Theory Set Constructive of Analysis and Comparison 3 The 4.2.3 if

Surprisingly, the existence property does not hold neither for neither hold not does property existence the Surprisingly, numerical existence property. ∀푥 ⊢ 퐓 푛 then it also proves with aconstruction of a witness there has to be a naturalnumbera be to has there ( the in formulated colors, many countably have you where case the is generalization A same color!” be the formula the be seem unsatisfactory. However, the theory the However, unsatisfactory. seem proof a BHK-interpretation, the Following structive theory.structiveexample,For BHK-interpretation:considertheproof A But both But We will discuss this interpretationchapter in 5. interpretationof Note that the disjunction property is actually a special case of the numerical existence property: 4.7. 4.7. germ, indecomposable indecomposable germ, UR free such that such free of you have two colors, two have you ): If whenever If ): 휃 Unzerlegbarkeits-rule

, given by the numerical existence property. In particular, this shows, that [ Existence property numericaland existence property Unzerlegbarkeits-rulevariants and 휓 퐂퐇 퐂퐙퐅 ( 퐓 푥 has the has . )

푞 ¬휓 ∨ and and proves 휃 ( 퐓 ⊢ ∃! 푥 푥 ∃! ⊢ 퐓 . The 휙 → 0 = 푛 퐂퐙퐅 ( 퐈퐙퐅 푥∃ .휓 휔. ∈ ∃푦 ∀푥 ⊢ 퐓 푛∈휔 휃 휔. ∈ ∃푛 푥 existence property existence ) in Martin- in ] have the numerical existence property existence numerical the have numerical existence property 휙 , then if if “ ( [ the only way only the 휃 UzR 0 = 푛 퐙퐅퐂 ( ) 푥 퐂퐇 ( 퐓 ⊢ ∀푥 휓 ∀푥 ⊢ 퐓 ∨ ) 푛

) ( ) 휙 ∧ , this property does not hold: Take any statement independent of Te, by Then, . 휓 → 0 ≠ 푛 . Now one can decide whether states that the universe of sets is unzerlegbar is sets of universe the that states Löf’stheory,type todoeswitnessesextract whichallow proofs. from us and and 푦 푛 along along with aproof of ( such that such 푥 ( ) if whenever if ,푦 푥, 휓 ] ( . This means that the witness of the formula the of witness the that means This 푥 , else. Hence, it is reasonable to expect for a constructive theory to to ) ) , color all the sets the all color ∨ ∀푥 ¬휓 ∀푥 ∨ then ) . If . 퐋퐄퐌 퐂퐙퐅 푝 of 휙 ⊢ 퐓 퐓 퐓 ⊢ ∃푦 ∈ 휔. ∀푥 휓 ∀푥 휔. ∈ ∃푦 ⊢ 퐓 may still be defended by the fact that there is a natural a is there that fact the by defended be still may has the numerical existence property and proves and property existence numerical the has , ~ is a weakening of this: Whenever 휓 ∨ 휙 퐅 ⊢ 퐙퐅퐂 ( 퐓 ⊢ ∃푥 휙 ∃푥 ⊢ 퐓 푥 51 ( ) 푛 . In the In words of McCarty in [ ) ~ must indeed be a pair a be indeed must . Again, both properties can be expected of a con- ofa beexpected properties can both Again, . 퐂퐇 휙 ( ( in 푦 ¬ ∨ 푥 ) the [...] universe [...] the ) . , then there is a formula a is there then , – 퐓 퐂퐇 which we will show in sections 4.6 and 4.6 sections in show will we which proves 퐂퐙퐅 ( ,푦 푥, 퐂퐙퐅 bt aosy neither famously but , nor for nor ) and and . This is indeed a generalization: Ifgeneralization: a indeed is This 휙 푝 or 퐈퐙퐅 of 퐈퐙퐅 휓 3 ∃푥 휙 ∃푥 have the disjunction prop- 〈 by a property: Whenever property: a by ,푞 푛, by inspecting the witness is 퐙퐅퐂 ( [ (

to 〉 푥∈휔 휙 휔. ∈ ∃푥 ⊢ 퐓 ( 44 25 , where , make everything theeverything make 푥 휙 does not possess the ) ], this], rulesays that can be constructedbe can ], [ ], should provide should us 휃 63 ( 푥 Uniformity-rule ]), which may which ]), 퐅 ⊢ 퐙퐅퐂 ) 푛 with exactly with is a natural a is WS WS ( 퐙퐅퐂 Let 푥 퐂퐇 2019 ) 휓 ∨ 휙 , then , like 휃 nor ( /20 푛 )

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. This reasoningThis shows a clear incompatibility of Church’s rule This isThis the rulefirst we discussed, that holds in (trivially) the classical setting. then a program is never terminating, then it does terminate: If Church’s rule Church’s 4.2.4 axiomatic constructive of properties Metamathematical 4 Note all that rules so far (Unzerlegbarkeit, Church, Markov) came in the form 4.2.6 Markov’s rule 4.2.5 fetvl cmual fntos whenever functions: computable effectively or optbe Hne tee s o a t efciey opt wtess o te formula the for witnesses compute { effectively to way no is there Hence, computable. For example, we can formulate the Unzerlegbarkeits-principle as We can associate toevery such rule a theory a For example, the Halting-problem inspires the following function in true. not clearly is this classically but case, intuitionistic the in assumption reasonable a as seems This with Due to Due .휃 휔. course, course, ∀푥 ¬휓 ∀푥 Clearly, This counterexample does not work i work not does counterexample This 0,1 ∅ ≠ 푥 } ( 푓 . ,푦 푥, 푛∈휔 휙 휔. ∈ ∃푛 ⊢ 퐓 푥∈휔 휙 휔. ∈ ∀푥 ⊢ 퐓

( ( Church’s RulesPrinciplesvs. rule Markov’s 푥 퐋퐄퐌 . . 퐅 ∀푥 ⊢ 퐙퐅퐂 퐱 ) 퐙퐅퐂 ) ) , where where , 퐓 , depending on 푧 = hs h Uiomt rl ad proves and rule Uniformity the has , this function is total in in total is function this , does not enjoy any of the two properties: Take for example the formula the example for Take properties: two the of any enjoy not does ( ( , (we may, (we assume that CR MR , do not confuse with Church-Turing-thesis) says that all total rules on rules total all that says Church-Turing-thesis) with confuse not do , [ 휃 ), central), to Russian Constructivism, says that if we can prove it that is impossible that ∅ ≠ 푥 ∨ ∅ = 푥 rule ( ( ( ,푦 푥, 푛 푥, ) . {

푒 ) } ≡ ( 푦 푥 . ( ) 휓 ) . (

푥 ] ) , but both 0 = 푦 ∧ 푓 n WS WS ( ,푚 푛, 퐂퐙퐅 퐱 퐙퐅퐂 principle codes pair the or ) ) . From Turing’s famous argument ( argument famous Turing’s From . If If ∨ = 푥 ∅ = 푥 ∀푥. 퐈퐙퐅 퐴 ⊢ 퐓 ( ¬휓 { 푥∈휔 푦∈휔 휙 휔. ∈ ∃푦 휔. ∈ ∀푥 ⊢ 퐓 ,otherwise. 0, 1, , of formthe 퐵. → 퐴 ⊢ 퐓 , as we cannot, in general, decide for each set each for decide general, in cannot, we as , ~ ( 푥 , then , then if ∀푥 52 ) set

{ and and 0 ≠ 푦 ∧ 푛 [ ( theories theories ~ 휓 } ,푦 푥, ( ( 푚 퐵 ⊢ 퐓 푥 푥 ∅ ≠ 푥 ∀푥.

) ) 푛∈휔 ∈ ∀푛 ⊢ 퐓 ) converges, ). ¬휓 ∨ )

. By Uniformity, we either have either we Uniformity, By . . 2019 with the axiom ( 푥 are in absurd /20 ) ( ] ,푦 푥, te i ms as prove also must it then , 퐙퐅퐂

(

휙 ( ) 푛 , : te tee s oe number some is there then ) ¬휙 ∨ 퐋 [ 퐄퐌 68 퐙퐅퐂 ( 푛 ]), this function is not is function this ]), . ) ) 4 . ∀ .¬휙 휔. ∈ ¬∀푛 ∧ 휓 ( 푥 휔 푥 , whether , ) are given by given are 퐱∈휔 푧∈ ∃푧 휔. ∈ ∀퐱 ∅ = 푥 ≡ ∀푥 휓 ∀푥 푥 푦∈ ∃푦 ∀푥. ( ∅ = 푥 푥 . Of . ( ) 푛 or or ) 푒 ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. atomic case and bounded quantification, we can rules transfer the the in BHK-semantics: abstract an be it (let discourse of domain our specifying further Not oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison elements set a imagine we 2.1.2, section in As case. atomic the motivate now us Let 4.3.1 If discussion this proof structure will interpre realizability proof, of notion precise Realizability is a semantical method inspired by the BHK-interpretation. While the latter lea 4.3

.∃ .휙 휔. ∈ ∃푦 휔. of proof every transform can we however, Metamathematically, computable. effectively is rule theory follows a rule, we must show how to transfersh To content: in significantly differ but description, in similar seem may formulations Both For example, as both both as example, For rules,ing but the converse is thenot case: Knowing about the Curry-Howard correspondence, this fact is perhaps not too bi corresponding rule. We can think of this situation as the fact that fact the as situation this of think can We rule. corresponding f they that however, show, will We principle. Church’s follow possibly cannot theories both principle, ing musthowever,give we Although with it, i.e. we can consistently extend also We denote the Unzerlegbarkeits principle, Uniformity principle, Church ciple as ciple 휙 ⊩ 푒

퐅 퐓 ⊈ 퐙퐅퐂

Realizability of The The proofstructure , wesaythat UzP 푥 휓 → 휙 ⊩ 푒 퐂퐙퐅 , but also by (coding of) proofs 휓 ∨ 휙 ⊩ 푒 휓 ∧ 휙 ⊩ 푒 , 푒 ⊩ ∀푥 휙 ∀푥 ⊩ 푒 푒 ⊩ ∃푥 휙 ∃푥 ⊩ 푒 ( ,푦 푥, UP ′ ¬휙 ⊩ 푒 ). and and

, ⊩⊥ 푒 ) CP into a procedure a into 퐈퐙퐅 and and

푒 퐂퐙퐅 realizes

cannot cannot follow Church’s principle, will show that both theories are are iff iff iff iff iff iff iff MP and 퐂퐙퐅 respectively. ∀푥 ⊢ 퐓 ∀푓 [ ( ⊥ 풱 ∈ ∃픞 풱 ∈ ∀픞 ∀푓 푉 ( 푒 휙

퐈퐙퐅 푒 ) ∗

[ [ or that ) 0

¬ ( 0 휙 ⊩ 푓 ∧ 휙 ⊩ are subtheories of subtheories are ( ∧ 0 = 푉 휙 ⊩ 푓 [ 푒 휓 [ [ ∗ computing for each for computing 휙 ⊩ 푒 휙 ⊩ 푒 and and proofs will be interpreted as effectively computable functions. ( 퐓 퐓 푥 ) 푒 -proofsof to a theory ) ( (

→ 푒 ) is a a is realizer for 푒 ¬휓 ∨ 푛 ] ) ) (

( 1 of the fact that ( 1 픞 픞 { 휓 ⊩ ) ) ts proofs as elements of a certain “proof structure”. In our In structure”. “proof certain a of elements as proofs ts 휙 ⊩ 푒 ] ] }

(

( 푥 ~ 푓

) ] ) ] 퐵 → 퐴 ∨ 53 → 휓 ⊩ 퐓 퐓 [ ( 퐙퐅퐂 ′ -proofs of ~ [ following this principle. Of course, 푒 ∀푥 휓 ∀푥

) ) 휙 ] 0 . Clearly,principlesthe . implytheir correspond-

. , where Church’s principle conflicts with conflictsprinciple Church’s where , ∧ 0 ≠ 휔 ∈ 푥 픞 ∈ 푥 ( 푥 ) ∨ ∀푥 ¬휓 ∀푥 ∨ 퐴 ( . Thus, what we can do is to some 푒 into into ) 퐂퐙퐅 1 휓 ⊩ and and 퐓 { 풱 푒 ( -proofs of 푥 ] } ’s principle for now), and leaving out the out leaving and now), for

( ) 푥 ] 퐈퐙퐅 . ) 픞

to be given not only by its by only not given be to 휔 ∈

do not “know” that each that “know” not do such that such g 퐵 . For the correspond- a surprise. and Markov 퐓 ′ ves open the 휙 identify compatible 퐙퐅퐂 ⊈ ( WS WS 푥, ow that a that ow ollow the ollow { ’s prin- 2019 푒 } (and 퐋퐄퐌 푥∈ ∀푥 ( 푥 the /20 ) ) . ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. axiomatic constructive of properties Metamathematical We will want to carry out these constructions inside constructions these out carry to want will We Lemma 4.5 do the this in following lemma: 푎 in [ in actually is Note this that inductivean definition Theorem justified and by structure3.28. This isstudied set at the sameat the time show inductively that verse Proof: ture, presented in [ in presented ture, ment ment We can now give the definitions of realizability for atomic formulas refer to truth inside Let simultaneous recursion and incorporatesextensionality the axiom: n suppose, and Γ Φ and and ( 픞 44 ⋃ Φ with all such pairssuch all with 훽∈훼 bethe inductive definition with ] and [ and ] 푥 푉 For each set or class or set each For 〈 itself must be of this shape. To follow this idea, one will thus have to define inductivelytothus havedefinethisthe follow one will idea,shape. To uni- ofthis be must itself ,푚 푎, ( 퐾푙 퐽 훽 ) ) : The classes 〉 픟 = 픞 ⊩ 푒 as: 픟 ∈ 픞 ⊩ 푒 . Let ∗ 54 푚 = ]. For our purposes, however, we will use a similar, but slightly more complicatedstruc more slightly but similar, a use will we however, purposes, our For ]. ( 푉 훽 푉 ∗ . The definition. The ofthe universe 훼 ) ∗ 〈 ′ 55 푉 푋, 퐂퐙퐅 ≔ 훼 푉 =

∗ ]. The reason for this is, that in order to verify the rules of section 4.2, we want to want we 4.2, section of rules the verify to order in that is, this for reason The ]. 〈 = = ,푚 푎, ⋃ 푉 푉 in in our definition of realizability. Define for a pair 훽 iff iff 푉 훼 〈 훽∈훼 ∗ ∗ 훽∈훼 훽∈훼 훼 ⋃ ⋃ ,푥 푛, for all all for ∗ = = and 〉〉 푋 퐽 〉 훼∈핆ℕ 훽∈훼 Γ 퐽 ⋃ , define , ∀,픡 ∀푓, °=픟 ∧ 픟° = 픞° ∃픡 ∧ 픟° ∈ 픞° 훽 Φ ∈ ⋃ 훽 . The problem, however, is that if we want sets to be foundational, the ele-foundational,the be to setswant we if that isproblem, however, The . Φ . Then,

푉 ( { 〈 ∗

훼 ∈ 훽 훾∈훽 ⋃ 푉 ,푚 푎, are definable in 훼 f 푎∈푋 ∈ 푎 iff 푉 ∗ [ WS WS (〈 . (

퐽 푋 퐾푙 훾 〉

,픡 푓,

′ ) 푉 ∈ 푎 : B Term .8 w have we 3.28, Theorem By . ( ) =

[ 푉 〉 〈( = 훼 { ∗ → 픞 ∈ :∃푚. 푎: ) 푒 { ′ ′ ) 〈 훽 ad푚⊆휔×푋 hr 푥∈푚. ∈ where ∀푥 푋, × 휔 ⊆ and 푚 0 푉 = ,픞 푛, 픡 , 푉 × 휔 ⊆ 푚 ∧ ~ 〉 훼 〉 { 〈 푉 푉 ∈ 픞 ∧ 휔 ∈ 푛 : ∧ 픟 ∈ ( . ,푚 푎, 54 퐂퐙퐅 푒 ∗ set looks asfollows: ) 0 theories theories 퐂퐙퐅 ~ } 〉 . ( ( 푓 푋 ∈ 푒 ) )

(definition inside (definition 1 훽 픟 ∈ 픡 ⊩ ∗ } 픡 = 픞 ⊩ . We use Theorem 3.28 to define the define to 3.28 Theorem use We . 푥∈푚. ∈ ∀푥 ∧ 2019 ( 퐾푙 ) /20 ) ∧ ] } 픟 ∈ 픞

.

(〈 = 퐽

( ,픡 푓, 푥 ∗ and and 〈 ⋃ ) ,푚 푎, 〉 푎 ∈ ° 훼∈핆ℕ → 픟 ∈ ( 퐈퐙퐅 푥 〉 픟 = 픞 ∗ the expressions ) } is less problematic). Weproblematic). less is 퐽 푎, ∈ ° , 훼

{ ad o each for and ( . This definition is by 푒 )

1 } ( 푓 ) 픞 ∈ 픡 ⊩ 〈 ,푚 푎, 훼 푉 훼 , ) ∗ and ] 퐽 〉

훼 = ° = - Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison We can now give the definition of realizability we will work with. As we want to prove 4.3.2

systems. What inside truth to refer so needs realizability of kind our 4.2, Section in discussed rules the follow The inductiveThe step follows from inductive definition theorem: Let Also, the definition of the clauses for clauses the of definition the Also, 〈 n i bt cases both in and and theand other inclusion is clear. ,픟 ,푒 0, 픟, 픞, = 푎 = 푎

〈 〈 ,픟 ,푒 0, 픟, 픞, ,픟 ,푒 1, 픟, 픞, Definition ofDefinition realizability 〉 퐽 ∈ and and ( 〉 〉 푉 , where 픞° = 픟° and ∀ 픟° = , where 픞° hr °∈픟° and ∃픡 ∈ , where 픞° 훼 we define is hence a variant of “realizability with truth” from ) ′ 픟 ∈ 픞 ⊩ 푒 = ( = = ⊇ = = 푉 = = = ,픟∈푉 ∈ 픟 픞, { { { { 훽∈훼 훽∈훼 훽∈훼 ⋃ ⋃ ⋃ :∃푚. 푎: :∃푚. 푎: 푎: :푎∈ 푎 푎: 훽∈훼 ⋃ 훼 ∗ . 〈

{ ,∅ 푎, { { 〈 퐽 〈 iff iff ,푚 푎, 〈 훽 ∗ ,푚 푎, ,푚 푎, ) , 훽∈훼 ⋃ 〈 〈 〉 ,푚 푎, ,푚 푎, ′ 〈 휔 ∈ 푒 ∈

,픟 ,푒 1, 픟, 픞, 〉 〉 푉 ∈ 푎 : 〉 ∈ 푎 : 푉 훽∈훼 ( ∈ 푎 : ⋃ 훽 〉 〉 〈 ,푎 푥, } ( ∈ ∈ 〈 Te smallest The . { [ ,푓 픡, 푉 = 〈 훽∈훼 ( 〈 ⋃ ,푚 푎, 훽 〉 픡, 〉 푉 훽∈훼 ⋃ beamember of 훾∈훽 ⋃ 픟 = 픞 ⊩ 푒 훽 퐽 ∈ 푉 × 휔 ⊆ 푚 ∧ 〉 ∗ 훼 ( { ) 푒 , ∈ 픞 픞 ∈ 〈

′ 〉 ) ,푚 푎, . 퐽 푉 ∈ 푎 : 퐽 푉 × 휔 ⊆ 푚 ∧ 0 훽 훾 〉 ) ) (〈 ∧ 픟 ∈ ′ ′ 〉 ,픟 1, 픟, 픡, } 푉 ∈ 푎 : × 휔 ⊆ 푚 ∧ ~

훽 and and 푉 × 휔 ⊆ 푚 ∧ 55 〈 Φ 훽 ,픡 0, 픡, 픞, ∗ { 훽 픟 ∈ 픞 ⊩ 푒 ( 푥∈푚. ∈ ∀푥 ∧ ~ coe class -closed 푒 푉 × 휔 ⊆ 푚 ∧ 훽 ∗ Φ ) 0 푥∈푚. ∈ ∀푥 ∧ iff } ( 훾∈훽 ⋃ 푓 푒 〉 ) 1 푥 ∈ 〉 퐽 훽 must be justified: Again, we invoke the invoke we Again, justified: be must 훾 ∗ 푥 ∈ ( 푥∈푚. ∈ ∀푥 ∧ 푥∈푚. ∈ ∀푥 ∧ 푥 ) ∗ ∀ ∧ ] ) ( 훽 , ∗ 퐽 푎 ∈ ° 푥

dfns h relations the defines 푥∈푚. ∈ ∀푥 ∧ ∗ 〈 ) ,푓 픡, 푎 ∈ ° }

〉 ( ( ∈ 픟 픟 ∈ 푥 푥 } ∗ ∗

[ ) ) 55 푎 ∈ ° 푎 ∈ ° (〈 ( 푥 ,픞 1, 픞, 픡, ]. ∗ ) 푎 ∈ ° } } }

{ ( that } 푒 ′ ) } 1

} 픟 = 픞 ⊩ 푒 퐂퐙퐅 푓 〉 WS WS 푥 ∈ and and 2019 ) those , , or 퐈퐙퐅 /20 iff ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. merely for cosmetical realiza reasons: In giving of realizers for definition the axioms of our in quantifiers of types different semantically as treated are of abounded existential statement We writeWe axiomatic constructive of properties Metamathematical Theorem4.6 (Completeness) bility: Owing to our reference to truth within the system, we can now easily prove the completen 4.4 Before proving the theorem, we need the following notion of standard representatives of sets o sets of representatives standard of notion following the need we theorem, the proving Before Let realizability reads as follows: implication, negation, unbounded existential and bounded universal quantification. For Proof of Theorem 4.6 Theorem of Proof We are now ready to give proof the of the completeness-theorem: The propertyThe Definition4.7 within ,픟 픡 픟, 픞, 휙 = 휙

A A simple Completeness Theorem 푉 ∗ range over range : 푉 ( 픞 ∗ 푥∈픞 휙 픞. ∈ ∀푥 ⊩ 푒 푥∈픞 휙 픞. ∈ ∃푥 ⊩ 푒 1 휙 ⊨ 픞 , … , ( : Wedefine for every set 푥̂ ) iff there is somethereis iff 푥 = ° 휓 → 휙 ⊩ 푒 푛 휓 ∨ 휙 ⊩ 푒 휓 ∧ 휙 ⊩ 푒 픟 = 픞 ⊩ 푒 픟 ∈ 픞 ⊩ 푒 푒 ⊩ ∀푥 휙 ∀푥 ⊩ 푒 푒 ⊩ ∃푥 휙 ∃푥 ⊩ 푒 ) : By induction on induction By : with all the free variables shown, we define ¬휙 ⊩ 푒 푉 isshown by simple recursion. ∗

⊩⊥ 푒 and and ( ( 푥 푥 ) )

,푓 푐 푓, 푒, : If iff iff iff iff iff iff iff iff iff iff iff iff 푉 over 휔 ∈ 푒 ∗ ⊥ ∃픡 ∀픡 ∃픠 휙° 픞°. ∈ ∀푥 ( ( [ ( 픡 ∀푓, ∧ 픟° = 픞° ∃픡 ∧ 픟° ∈ 픞° { 푥∈픟 휙 픟. ∈ ∃푥 WS WS ( ( °→휓° → 휙° ¬휙° 푒 휃 ⊨

푒 푒 ) [ ) [ 〈( ) [ 0 휃 휙 ⊩ 푒 휙 ⊩ 푒 0 1 휔 푥 , then such that such ∧ 휙 ⊩ 푒 . The base cases are clear by definition and so are the cases of cases the are so and definition by clear are cases base The . } ) ∧ ퟎ = its 픞 ∈ 픡 ⊩ 푓 ) . Wedefine recursion 푥̂ = ∀푓 ∧ 0 픠 , ) ( standard representative ( 〉 ( 푥 ∀푓 ∧ 픡 픡 ( 〈 퐙 휃 ⊢ 퐂퐙퐅 픞 ∈ ( [ ) [ 푥, ) 푥 ( ) ¬ 푒 〈( not somewhere not the in ] 푒 ] ) ~ )

( { ) [ 푒 1 ∗ ∀ ∧ 〈 휙 ⊩ 푓 1 (〈 휙 ⊩ 푒 [ ) ) ,푢̂0, ∧ 56 휓 ⊩ ( ] set 0 휙 ⊩ ,픡 푓, 휙 ⊩ 푓

픡 , ( 〈 ,픠 푓, 푒 theories theories 〉 ~ ° 〉 ) 푥 ∈ 푢 : .

〉 ] ) 1 픟 ∈ . Note thatNote unbounded . boundedand quantifiers 픞 ∈ 〉 ] ∨ 휙 ⊩

) 픞 ∈ [ ∗ ( → ∗ ∧ 푒 } → ∗ ( ) ( 〉

( 픠 2019 0 [{ . { ) 푒

{ 푒 ] ∧ ퟎ ≠ 푒 ) (

} 휙 1 } 푒 휓 ⊩ 푓

휙 ⊩ 푓 /20 푥̂ ° ) 퐂퐙퐅 픡 = 픞 ⊩ by 0 in in }

픟 ∈ 픡 ⊩ 푓 ( class , it will be crucial to find a witness 휙 푉 푒 ) ( ) ( ∗ ] 픠 1 byrecursion:

픞 ) ] 1 ° 휓 ⊩ ]

픞 , … , 푉 ∗ but rather but in the ) ]

∧ 푛 ° ) (〈 . The full definition of ,픡 푓, bility. This is not is This bility. 〉 픟 ∈ essrealiza- of ∗ → set

f f 픟 . 퐂퐙퐅

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. For Now for (2), assume (1) for (1) assume (2), for Now Proof By inductive the hypothesis, we conclude 〈( tails tails Lemma4.9 (closure): 4.5.1 Ⓜ nextThe sections we will with spend proving the soundness theorem: 4.5 Theory Set Constructive of Analysis and Comparison All we have to do is to give realizing terms for the underlying axioms and inference rules of rules inference and axioms underlying the for terms realizing give to is do to have we All that that Theorem 4.

픟 have have ular, for any set any for ular, equality equality thenand do the same for the axioms of For restricted existential quantification, suppos quantification, existential restricted For universal quantification. Suppose , then for all all for then , 푒 ) Theorem 4.8 (Soundness)4.8 Theorem (2) 5) (1) 1) 2) 3) 4) 0 푉 ∈ 픟

퐙 ⊢ 퐂퐙퐅 °=픟° = 픞° : By simultaneous induction. To prove the(1), inductive hypothesis is that for all 픡 ,

(

Soundness Theorem

푥∈픞 휙 픞. ∈ ∃푥 〉 ( 퐢 푉 ∈ 픟 퐢 퐢 퐢 퐢 Realizing equality Realizing ퟏ 풓 풔 풕 ퟎ 픞 ∈ 푉 ∈ 픞 훼 푒 ⊩ ∀푥 휙 ∀푥 ⊩ 푒 ∗ ⊩ 픞 = 픟 → 픟 = 픞 ⊩ 픞 = 픞 ⊩ ⊩ ⊩ , there is, , there by definition, some ( ∗ and and hence 10 ( 훼 ( ( 휃 ⊩ 푡 ∗ 픠 = 픟 ∧ 픟 = 픞 훼 ∧ 픞 ∈ 픠 ∧ 픟 = 픞 픠 ∈ 픟 ∧ 픟 = 픞 ∗ : There are 훽∈훼 ∀픠 (푉 훼. ∈ ∃훽 → 〈 ( 푉 ∧ ( ,픡 푓, 푒 푥 푥

) ) ) 1 , we obtain that obtain we , ∗ ( . ) 〉 푥 픟 = 픞 ⊨ 휙 ⊩ ° 픟 ∈ ) 푥∈픞.휙° 픞°. ∈ ∃푥 ≡ . Thus, for any for Thus, .

푉 ∈ 픟 ( ∗ , 픡 ) ) )

퐢 ) ( 풓 픟 ∈ 픠 → 픠 = 픞 → 픠 ∈ 픞 → . 푒푓 ) 퐢 , B definition, By 훼 푉 ∈ 픟 → 훼 ∗ 풔 ) ∗ . 퐢 , : For every theoremevery For : . Let . 1 푉 ∈ 픠 → 픟 ∈ 픠 ⊨ 풕 픞 ∈ 픡 ⊩ 퐢 ,

ퟎ ( 퐢 , 퐂퐙퐅 푉 ∈ 픞 푥 훼 ∗ ) ퟏ

∃픡 푉 ∈ 픞 . ( 휔 ∈ 푉 ∈ 픡 proves , which gives which , [ 훼 〈( 훼 ∈ 훽 ∗ and choose and 푒 such that such for all ∗ 푉 ∈ 픠 픡 훽 ) , ∗ 0 ° 휙 ⊩ 푒 훽 푉 ∧ 픡 , ∗ 픞 ∈ , s.t. ) ( 〉

훽 휙(푥̂) ∗ ~ e ∗ 픟 ∈ ° . 퐂퐙퐅 ad y nuto hypothesis, induction by and 픡 = 픠 ⊨ 푥∈픞 휙(푥) 픞. ∈ ∃푥 ⊩ 푒 ( 휃 픟 57 픞 ∗ of ∗ ) ) . 푉 × 휔 ⊆ ∧ . By induction hypothesis,induction By . 푉 ∈ 픡 ° 훼 ∈ 훽 ~ and thus and 퐂퐙퐅 ( 푒 ) ) 1 훽 , there exists a closed application term applicationclosed a exists there , ,픟 푉 ∈ 픠 픟, 픞, ∗ 푉 ∈ 픠 → such that such 픡 = 픠 ⊩ . Altogether, Altogether, . 훽 ∗ . Let 휙 훽 ° . This means, there is there means, This . ∗ ( ] 픟 ∈ 픠 ⊩ 푒 . . 푥

∗

, ) ∀픠 , showing the case for unbounded for case the showing , ( 픟 푉 ∗ ∗ 푉 × 휔 ⊆ . This . This means, 푉 ∈ 픠 → 픟 ∈ 픠 ⊨ 퐙 ⊢ 퐂퐙퐅 휙 ° 훽 ∗ ( . Also Also . 픡 ° 훼 ∈ 훽 ( ) 휙 Atgte we Altogether . 푉 ∈ 픡 ( 픞 훽 ∗ ) 픟 = 픞 ⊩ 푒 ) , ) . ° If If ∗ WS WS . In partic- In . such that such 퐇퐏퐋 = 픞 ⊩ 푒 2019 푡 , such , with en- /20 ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. axiomatic constructive of properties Metamathematical

corresponding axioms of identity hold in hold identity of axioms corresponding uh that such Moreover, for each formula Proof application terms 〈 Axioms 3 and 4 are shown by simultaneous induction on the structure of structure the on induction simultaneous by shown are 4 and 3 Axioms and thusand For 2,let Let For induction 3the hypothesis isthat for all By 퐢 Let ퟎ ,픟 푓, then then then then then 휎 ≃ Lemma 4.3 Lemma 휎 푉푉 ∈ 픢 〉 : Itgoes withoutsaying: throughout that theneedbother proofwe truth not aboutin 1 픞 ∈ and and 2 ( 퐢 풕 , then then , 퐢 퐢 퐢 , 훼 ⏟ (( ∃픦 (( ⊩ ℎ 퐢 풔 흓 ∗ ퟎ 퐢 휎 ퟎ 흓 ,

≡ 퐩 ℎ ℎ

픞 = 픞 ⊩ 2 ) 〈((( 〈

) ) (((( . ,픤 푔, bedefined as follows: 퐩 ≃ 0 0 휆푥

( (Double recursion) (Double 픣 = 픢 ∧ 픢 = 픡 ) )

( ℎ 1 0 퐢 퐩 . 〉

ℎ ((( 픢 ∈ 픤 ⊩ 푔 ( 흓 ) 픡 ∈ 휆푦

) 0 휏 ) 휎 휎 ( 푖 0 1 ) 0 퐫푥 . 1 ℎ 2

) 0 - 퐩푦퐢 . 퐩푓퐢 ≃ 푓 ( ( 휏 0 ) and and 푔

,푏 푎, ,푏 푎, )( 6 0 푔

) asfollows: ) ) 0

퐥푥 0 1 픦 , 흓 푔

) ) 푖 ) 〉

ℎ )( ) )

≡ ≡ . This term. This swaps entries of apair, which isall we need. ) 0 such that such 픟 ∈ 픣 ∈ 픤 ⊩

휆푦 휙 흓

휆푦 픣, ∈ 픦 ⊩ 휆푦 . As As . 퐢 ( 흓 and and 퐩푦퐢 . 푥 ( 퐩 . (휏 퐩 . ) ⊩ there exists 〈

( 흓 휆푥 WS WS ( on double recursion, there are there recursion, double on (((

휙 ∧ 픟 = 픞 2 ) 퐩푓퐢 ( 푎 . Let . 푦 퐢 ℎ 휏 휏 휏 휏 ퟎ ) 4 3 1 5 ) 흓 ( ( 0 푥 )≡퐫 ≡ 푦) (푥, 푥 )≡퐫 ≡ 푦) (푥, 휏 푥 )≡퐫 ≡ 푦) (푥, 푥 )≡휏 ≡ 푦) (푥, 휏 퐩휏 ⊩ ) 푏휏 ) 2 6 퐂퐙퐅 0 0 ( 푦 퐩 ≡ (푦) 1 ( 픟 , 푉 ∈ 픞 푔 푦 6 픠 ∈ 픟 ∧ 픟 = 픞 ( 푉 ∈ 픠 ( ( ) ) ,푦 푥, 〉 퐢 푦 ,픠 픞, 1 . For 1 we use the Fixed-Point Theorem to define to Theorem Fixed-Point the use we 1 For . 흓 퐥 ≡ 픞 ∈ ) 픦 = 픤 ⊩ ~ 훼 ) 휔 ∈ ) 1 훼 ∗ ). by induction hypothesis since may assume ∗ 휏 픠 , … , ( and and ad assume and 2 ( ( ( , 58 ( 5 퐫푦

set 퐥푦 퐫 퐥 ( 퐥푦 ( ( such that such for all 푦 ( ,푦 푥, 퐥푦 퐫푦 ) ) ) ) theories theories ~

( ( 푛 (

) ( ( 퐥 퐥 퐫 ) ) 푥 ( ℎ 퐩푓퐢 ) ( ) 푥 ( ) 퐫 ) ) 퐥 ) 퐫푦 ) ( (

픠. ∈ 픞 → ( ) 1 휙 →

퐥푦 퐫푦 흓 픣 = 픢 ⊩ ) 휆푥 ) ) ) ) 푥

1 2019 푥

( 푉 ∈ 픣 ) 푎 . ,픠 픟, ) 픟 = 픟 ⊩ 퐢 ) ) 흓

(

1 /20 퐩휏 픟 = 픟 ⊩ 훼 , where by the closure lemma (2), we 픠 , … , ∗ 퐢 . ,픟 픠 픟, 픞, 풕 4

and and ( ,푦 푥, w hv shown have we , 푛 ) 1 . )

for all all for 퐢 픠 , … , 휏 ퟎ 5 such that such 푉 ( ∗ ,푦 푥, . We first need to define to need first We . 푛 푉 ∈ ) 푉 ∈ 픟 ) , ) ∗

퐢 훽 풕 ∗ with 휎 ≃

( 퐂퐙퐅 퐢 흓 1 ) ( 0 , as all theall as , 퐢 픞 ∈ 픟 ⊩ 푓 훼 ∈ 훽 풕 퐢 , 퐢 흓 ퟎ to be to ) and If .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ( For teed automatically. Proof We can compute Theory Set Constructive of Analysis and Comparison Theorem4. 4.5.2 thus, we have shown Finally, for 5, the term

As With propertiesWith the of ment showsment does the Let job: The term The For the (4), inductive hypothesis is that for all Let vided by 푒 then then then iff iff ) 0 ((( 푉 ∈ 픠 HPL ∧ ퟎ = : Let us showaxiomssoundnessthe ofLet inferencesometruth us rules.and in for: Again,

ℎ SoundnessTheorem for ) 훼 0 8, we claim that claim we 8, ∗ ) 퐢 and and 퐢 ( ( ⊩ ℎ 퐢 풓 퐩 0 ퟎ 흓 ℎ ℎ ( 퐢 , 푔 ( 11 ((( 푒 is constructed by recursion on recursion by constructed is ) ) 퐩 퐢 풔 ) ) 0 0 ퟎ 퐢 , 1 1 : ( ( ℎ ( 픟 = 픞 ⊩ 픟 = 픞 ⊩ ℎ ℎ 픠 ∈ 픟 ∧ 픟 = 픞 If If 풕 휏 ≡ 휙 ⊩ 퐩휏 ) such that such 퐢 , ) 퐩 1 ⊩ ℎ 퐇퐏퐋 0 ퟎ ((( ) 4 (( 퐢 0 1

( ퟏ and or ) ( ,ℎ 푔, ℎ ℎ ( ,ℎ 푔, ≡ ( ) 퐢 퐢 ) 픞 ∈ 픠 ∧ 픟 = 픞 휙 ⊢

1 ퟎ ퟎ ( 퐢 1

휆푥 ) ퟎ ( ) 푒 픠 ∈ 픞 ⊩ ℎ 퐢 ) 1 ) 퐩 ퟏ 휏 , wehave that 0 ) and and ) 푥푦푧 퐝 휆푥휆푦휆푧. ≡ ℎ 5 . 0 퐢 . ) ( , then ( ( ℎ 픡 = 픞 ⊩ ) ퟎ ∧ ퟏ = ,ℎ 푔, 퐢 )

ퟎ

(퐩 and and and we mayassume 0 ( (( 휏 ≡ 푖 퐩 ) ( ( ℎ ) 푉 퐫 . As ℎ ) ∃픡 ∃픡 ( ( ( ∗ 픡 ∈ 픤 ⊩ 1 ) ℎ )

퐫푥 푒 퐇퐏퐋

5 0 ∀ ⊨ ) , let ) ) 1 ( (( 〈(( 1 1 ) ℎ ℎ ,ℎ 푔, ) ̅ ) ℎ ) 픠 ∈ 픟 ⊩ 휓 ⊩ 푒푓푔 wasarbitrary, 휙 ℎ (퐥

( 퐢 ) 〈(( 픠 ∈ 픞 ⊩ ퟏ 푦 1 ) . 퐢 ) (( ( for all 픟 ∈ 픠 ⊩ ℎ 1 풕 ) ( we have shown have we 퐥푥 . Assume, ℎ 1 ) 퐫푥 ℎ 퐝 ≃ ⊩ 0 ) ) ) ) 휙 ) 픡 , 1 ) 0 ( (

) . The inductive steps are easy and the base cases are pro- are cases base the and easy are steps inductive The . 푉 ∈ 픟 )( 퐩 ≃ 퐥 ) 픣 = 픢 ∧ 픢 = 픡 〉 훼 ∈ 훽 ( 0 (

0 푓 픡 , 퐫푥 ∧ 픠 ∈ 푧 ~ (( 〈 ( ,픤 푔, ( 〉 by inductive hypothesis ( 퐫푒 ℎ 퐫푥 ) . 훽 59 휏 픞 ∈ ) ∗ ) 2 for some and )≃퐢 ≃ )) ) 1 ) 〉 휒 → 휙 ⊩ 푓 ( (( )( ) ) 퐢 ℎ ~ 픣 ∈ such that such 0 ퟎ ( ℎ 푔 ) 퐥푥 픟. ∈ 픡 ⊩ ⊩ ) ) ( . Altogether, we have 푉 ∈ 픢 1 ) (퐢 퐫푒 ퟎ ) ) does the job. Assume job. the does ( 1 ( 퐢 ퟎ 픠 ∈ 픟 ∧ 픟 = 픞 픣. = 픡 → ) ퟎ 퐩 ( ) 픡 = 픟 ⊩ ( 휏 (( 훽 ( 훼 ∈ 훽 퐩휏 ∗ 6 퐥푒 , and and ℎ (

1 ℎ (( ) ) ( ퟎ. 1 ) ℎ ,ℎ 푔, ) )

1 ) 휎 ≃ ) 휒 → 휓 ⊩ 푔 , where by the closure lemma (1), lemma closure the by where , 1 ((( ) ) 1 휏 ℎ ) 5 픠 = 픡 ⊩ ) 1 ( 픠 ∈ 픞 → 0 ,ℎ 푔, ( ) 퐢 0 풕 퐢 , (( ) ퟎ ) ℎ 퐢 . Then ) . Then 풕 ) 퐢 ≃ ℎ 픣 ∈ 픤 ⊩ . 1 휎 ≃ 휓 ∨ 휙 ⊩ 푒 ) 0 ) 2 ) ퟎ (

ℎ, 퐢 . A similar argu- similar A . 풕 퐢 ,

ퟎ 퐂퐙퐅 ) . Then, either Then, . 픣 = 픡 ⊩ is guaran- is WS WS 2019 . /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. In the course of realizing the set axioms we will often be in the situation to construct a witn a construct to situationformula in the in be often will we axioms set the realizing of course the In 4.5.3 axiomatic constructive of properties Metamathematical We need to find a realizer a find to need We Pairing by If If Lemma4.12 By properties the of other direction is symmetric and truth in and weand easily seethat Proof For Let Extensionality such such creasing the set 푒푔 there is there

For 〈( Conversely, if For EI, assume EI, For Altogether, we have shown that ℎ ∀푧 ⊩ ℎ ( ) °= 픭° 푒푓 퐢 0 HPL 풓 HPL : For each For : 픠 ,

훽 beas thein last section and ) 〉 ad let and s Realizing Realizing theset axiomsof ⊩⊥ 푉 ∈ 픟 { 픞 ∈ 13, let 13, ° 픟° 픞°, ( , let 9, 픟 ∈ 푧 ↔ 픞 ∈ 푧 , and , and finally 푉 with : Let ∗ ∗ . Wewill therefore apply the following result: Given asubset } such that such ∀푢 ⊩ 푒 and and 푢∈픞 휙 픞. ∈ ∀푢 ⊩ 푒 〈 휓 → 휙 ⊩ 푒 퐴 ∀푥 ⊩ 푒 ,픠 푘, = 훼 ( 퐴 푐 ℎ does not change this. beasubset of 〉 ) 픭 1 퐝 퐴 ∈ ∗ ⋃ [ , 픲 = 픠 ⊩ = 휙 → 푎 ∈ 푢 퐩 ≡ 푒 ( 퐶 ) ¬ 휙 and and there is some is there 휙 ⊩ 푔 Then . { ( ( 〈 휙 ⊢ 푔 ,픞 0, 푥 , 푒 ) such that given that such ( ⊩ 푓 ( 퐩ퟎ 〈 휓 → . This implies. This 푢 〉 ,픡 푓, , ( ) 〈 픟 . This means, for all for means, This . 퐢 푉 × 휔 ⊆ 퐴 ) ,픟 0, ( 퐫 ( ) ( 퐩 for all 〉 푉 × 휔 푢 ) )( ¬휓 → 휙 ≡ 푒 . Altogether,. . This shows that for all for that shows This . 픞 ∈ ) ℎ 〉 퐩ퟎ WS WS ∀푤 휆푒휆 ] } 푒푓푔 and and . ByLemma 4. 휆푦 . As As . 퐢 퐂퐙퐅 ( 퐫 훽 .퐢 ℎ. ∗ ) °↔ 픭° ∈ 푤 휔 ∈ 푔 . Then 퐩 . such that such does the ofjob realizing the axiom. ≃ 퐴 퐂퐙퐅 훼 〈 ) 흓 ∗

,픠 푓, 푐 ( 퐩푓퐢 { ad assume and lo o each for Also, . 퐢 휆푥 (퐩 푓 = 푔 흓 ,픟∈푉 ∈ 픟 픞, ( 〉 isapparent. ( ( . 퐥푦 . 퐫푒 풓 ( { 퐫푒 퐩 푒푔 픞 ∈ 〈 퐫ℎ 픠°: ∃푘 ∃푘 픠°: ,퐴 퐴, (( 픞 ∈ 픡 ⊩ ~ ) ) ( , , if ,if , ( ) 퐩푥퐢 ℎ 휓 ⊩ , then °∨푤=픟° = 푤 ∨ 픞° = 푤 60 ( 〈 set ) 12, 12, 푐 푒 ,픠 푘, 1 , if , if ∗ 〈 〉 ( there is some is there )( 〈 풓 ,픠 푓, and any and theories theories 퐥ℎ ~ . We have seen that we can takecanthat seen havewe We . ,픠 푘, ) 푉 ∈ 픭 〉 ( ( 푒 )( , we have that have we , ) 푒 푒 ( 퐩푓퐢 푉 × 휔 ∈ 〉 ) 〉 ℎ ) ) 휆푥 휙 ⊩ 푔 픞 ∈ 푒 ) 0 0 ) 퐴 ∈ 0 풓 퐫푦 . ퟎ ≠ ퟎ = 〈 푉 ∈ 픞 ∗ ) ( ,픠 푘, , . Obviously, 픞 ∈ 픠 ⊩ ) 퐥ℎ } 푒푓 2019 푒 〈 . (

} ) ,퐵 퐴, ( 〉 훼 Thus, . 퐩푥퐢 ∗ ) ℎ 휙. ⊩ 퐴 ∈ 휙 ⊩ . Using collection, form the set the form collection, Using . ∗ ) ( ) , /20 0 휆푒휆푓 ) 풓 〉 휙 ⊩ 푒 픭 with . Thus, ) , 휙 ⊩ ,

with (

)

°∈퐴 ∈ 픠° 픠 .

( ) 푒 . 푒ℎ . If If . 푒푓 ( ( 픲 ( 퐴 ⊇ 퐵 ) 픞 ) 픭 ∈ 픟 ∧ 픭 ∈ 픞 ⊩ 푒 퐩푓퐢 0 픞 ∈ 픲 ⊩ ℎ 휓 ⊩ ) . 푉 ⊆ 퐴 푒 푐 ≃ 푓 and hence and ( 휓 → 퐩푓퐢 풓 ) , 푐 ) ( is in isin . Assume . 풓 realizes this axiom. 푒푔 ℎ ∗ ) , wedefine ) 0 휙 ⊩ ¬휓 ⊩ , then there is some is there then , ( 퐩푓퐢 푉 휆푥 ( ∗ 〈 픠 . 푥 . 풓 ,퐴 퐴, ) ) . Ti shows This . . We define We . to realize EI. to ∃ ⊩ 푔 픟 ∈ 픡 ⊩ 푐 〉 푉 ∈ ess to a to ess 퐶 푥휙 훼 of all of . The The . ∗ In- . , i.e. , ∎ ∎ 픭

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

Lemma4.13 With this lemma it is now easy to realize to easy now is itlemma this With Proof ally theally formula InvokingLemma 4.12, we define for Union (the (the base case Clearly, for all Let term andobviously To represent Infinity Putting things together, wehave that The representative of the empty set is just just is set empty the of representative The Empty set 3) 2) 3) 2) 1) 1) 픞 ∈ 픵 ∧ 픵 ∈ 픶 ⊩ ℎ : Let

Then, by 1),we have that Truth in 푒 푒 Again, 푒 푒 1 1 2 3 ( ( ( 퐢 푚 ⊩ 푘 푘 푛 풓 ) ) be the realizer bethe with ) : For all 푘 ⊩ 퐩푘퐢 ≡ 푦∈푛+1 + 푛 ∈ ∀푦 ⊩ 휔 0 푛 ⊆ in in (푘 푉 ∈ 픵 퐂퐙퐅 = 푛 ∈ ) 푉 ∘ , for all 〈 풓 ,∅ ∅, = 푛 ⊆ 푘 = ∗ is again easy. Let . This . This means 푘 ⊩ ∗ ¬ for all 1 + 휔 ∈ 푛 , wedefine for , ( °∈ 픶° 〉 푛 ∈ ∅ = 휆푥 푛 ≤ 푚 . ( ̂ 푛 < 푘 푛 ∈ 푦 , since clearly 퐩 . 〈 isimplicit). Then ,∅ ∅, ( ( there are realizers 퐥퐥 푛 . 〉 푒 퐢 푥 ) . 풓 푛 = 푦 ∨ ° ° Un Un 3 )( . The realizer we need is ) ∃픠 ∃픡 ( 픞 = 픞 ⊩ °∈ 픵° . 푛 퐫퐥 1 + 휔 ∈ 푚 ( ( [ [ ) 픞 픞 〈(( 푉 ∈ 픞 퐝 〈(( 푥 ≡ 〈(( ) ) be be the operator with ) = ° ∗ Un ℎ ℎ ∀푦 ⊩ 휆푥 ) = ℎ 푚 ) (푘 , for ) 1 + 푚 0 ( ) for all for all 1 ∗ 퐝퐢 . ⋃ { 픞 ) 0 ) ) , the set, the = ∅ ̂ 〈 0 ) ) 0 ∘ ,픶 ℎ, 픠 , ∃푤 ↔ ° 0 픡 , ( 풓 휔 ∈ 푛 = = 푛 ∈ 푘 = 〈 , 픠 , ∃푤 ( °, 픞° 휔 푛, 〉 〉 퐩푘퐢 〉 〉 〈 ~ ∧ 픵 ∈ 푠 = ∧ 픞 ∈ ∃ : ,∅ ∅, { 푉 ∈

∈ 푉 ∈ 픞 푒 ( 〈 픞 ∈ 푤 ∧ 푤 ∈ 푦 1 ,푘 푘, 풓 61 . 〈 Un 푒 , ) Un ( ,픵 푓, 휔+1 ( 푥푛 〉 ∗ 푚 °∈푤∧푤∈픞° ∈ 푤 ∧ 푤 ∈ 픵° 2 (( ~ 푉 ∈ 〉 ( (( ( and and ∗ ) realizes 푛 < 푘 : 〉 픞 픞 ℎ fromTheorem 4.10. . Of course,. Of for all . ( ℎ ) Note that for that Note ) 픞 ∈ ) 푛 and thus and 1 ) ∗ 0 푒 푉 ∈ ) 1 . Its property is realized by every application every by realized is property Its . 2 ) ° 푎푚 푎 ≃ 퐝푎푏푚푛 ) 푒 andas ∗ 1 1 ≡ 3 . 픠 = 픶 ⊩ 〈 such such that } ∗ 픡 = 픵 ⊩ ,픶 ℎ, 〉

휆푥 by , 푦∈푛+1 + 푛 ∈ ∀푦

) 푒 . 〉

∈ 푦 → 1 〈 픵 ∈ ( ,푘 푘, ) 푥 ] ] . ,

)

} iff . 푛

by 1). . 〉

Un “to be a successorof a be “to 푛 ∈ 푛 = 푚 . 1 + 휔 ∈ 푛 ( ( 푛 ∈ 푦 ∗ 픞 . ) ) . and

푛 = 푦 ∨ 푎푚 푏 ≃ 퐝푎푏푚푛 , ( 푚 ) ) . 푚 = ° 푚 WS WS . iff ” is actu- is ” 2019 푛 ≠ 푚 /20 ∎ .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. tion oftion bounded. For bounded. We need to find hs eiiin ed sm jsiiain Frt oe that note First justification: some needs definition This axiomatic constructive of properties Metamathematical Realizing the axiom of infinity is now easy:Truth in As for truth, for As Given Given BoundedSeparation Hence, gathering all the realizers in the proof of the lemma,

that that fromTheorem 4.10, On the other hand, other the On Let Strongcollection lection in 퐢 흓 ( 퐩 푉 ∈ 픞 ( ( 푘 푘 푉 ∈ 픞 ) 0 푉 1 ) ∗ 푚 ∗ 퐩 = , 퐂퐙퐅 and suppose, and ) °∈픞° ∈ 픠° ∗ 휙 ⊩ and and any formula 푓푔 픵 tothelatter formula yields a set 〈 ° 퐩 ∈ and ( 푓푔 푒 and hence and 픠 , ) Se 휆푛 . All ofthis shows that we take can ℎ 픠 , 퐢 with p 퐩 〉 흓 ( 퐝퐢 . 휙 푘푚 푘 ∈ ( ( 퐩 ) 픞 풓 1 Se ( 푥∈픞 푦 휙 ∃푦. 픞. ∈ ∀푥 ⊩ 푒 ) ( ↔ ∀푦 ∈ 푛 푛 ∈ ∀푦 ↔ 푠 = 푛 ↔ ∪ 푚 = 푛 ↔ 푘 휙 ∧ 픞 ∈ 픵 ⊩ 픵 = 픠 ⊩ ° 퐩 p is equivalent to equivalent is ) 푥∈픞.∃ .휙° 퐸. ∈ ∃푦 픞°. ∈ ∀푥 ( 1 휙 [ °∈ 픠° 퓈 1 − 푛 푔 ∪ 푚 ⊆ 푛 ( ( 픞 ) 푚 Se Se 휙 ) 휙 ⊩ ≡ ℎ ≡ 푒 ∗ , wedefine ) Se p p fr some for , we have that that have we , ) ( 퐩 = 휙 휙 퓈 푚 p WS WS ( ( ( ( ( 휆푘 휆푥 휙 푚 = 푦 ∨ 푚 ∈ 푦 픞 픞 ∈ 픵 ⊩ 푒 휙 ∧ 픞 ∈ 픵 ⊩ ℎ ( 픵 ) 1 − 푛 { ) ( ) 픵 )

{ [ 푚 = ° 픞 ∗ ) . 퐩 . 퐩 (퐩 퐩 . 푚 휆푛 means that that means ) } ( = ° }]

( 푒 ( 퓈 . . ByLemma 4.12 we can conclude that 퐩 3 ) ,푦 푥, { { ∧ ) ( ( ° 휙° 픞°: ∈ 푥 〈 〈 ( ( Se °∈픞 휙° ∧ 픞° ∈ 픵° 퐩 0⊩∀ 휔 ∈ ∀푛 ⊩ 푛0 푚 ,픠 푓, 퐥퐥 Se 푛 [ 퐥퐥 ( ∪ 푚 ⊇ 푛 푓푔 푥 ) ) ) p 푥 ,푦 푥, p . In particular, In . 퐸 )( ) 〉 푛∈휔 ∈ ∀푛 ⊩ 휙 ) 휙 ~ 푒 휙 ⊩ 푔 픠 , such that such 픞 ∈ (퐢 퐫퐥 ( 2 ( ( ) 픞 〉 픞 ] 62 픵 흓 푥 : set ) ) [ 푦∈퐸 푥∈픞.휙° 픞°. ∈ ∃푥 퐸. ∈ ∀푦 ∧ ) ) 〈 푒 ∗

) 〈( ( ∧ ∀푦 ∈ 푚 푚 ∈ ∀푦 ∧ by ,픠 푓, ( 퐂퐙퐅 and 1 휙 ∧ 픞 ∈ 픵 → ) ∈ 픵 → 퐩 푥 theories theories ~ 푘 ( ( { (퐢 푒 ( 푚 )

픠 ) 푚 ( 퐫퐥 and and 〉 } ) 0 픵° 흓 . istrivial and ) and hence by Completeness, by hence and 푠 . 픞 ∈

픠 , }] 푥 ( ( ] 휙 ⊩ 푔 ) Se 푚∈휔 푠 = 푛 휔. ∈ ∃푚 ∨ 0 = 푛 ) 퐩

〉 ( . To find find To . 1 + 푚 ⊩ Se 퐫푥 푛 ( ∗ 픞 ∈ p ℎ 퐫푥 ) p 휙 ⊩ 푔 ∧ 휙 tobe ) 푥∈픞.∃.휙° ∃푦. 픞°. ∈ ∀푥 2019 ( 휔 ∈ 휙 )) ( 푛 ∈ 푦 )( ∗ ( 픞 ( ( and and 픠 ) 픞 픵 퐫퐥 ( ) ∗ ) ) /20 퐫퐥 , Hence, .

i ide a e, s uniiain is quantification as set, a indeed is , 푥 .

푠 = 푥 ) ) 푒

) ( ( ) ∧ 푚 ∈ 푛. 푛. ∈ 푚 ∧ , note that note , , ) 푘 픠 .

) ( ( ) }

,푦 푥, 푚 1 ,

픠 = 픵 ⊩ ) . )

퐩푓 ( . ,푦 푥,

(

푘 ( Se ) ) 푚 ∈ 픵 ⊩ 푘 for some for . Applying strong col- strong Applying . 1 p ) 픞 ∈ 픵 ⊩ 휙 ) . (

픞 휙° ) 푉 ∈ Se ( 픠° ad ih with and p 픠 ) . As before, As . ∗ 휙 . By defini- By . . ( 픞 ) means 퐢 흓

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

Also, this this will dothe job. Indeed, we have We claim that if wedefine Again, invoking strong collection in and forand the second, where allwhere facts follow from definitions the of as for the firstline Finally, the axiom is realized iff but thisbut is clear from strong collection in Set and let and Let Subset collection By subset collection in ,픟∈푉 ∈ 픟 픞, 푥∈픞 푦 휙 ∃푦. 픞. ∈ ∀푥 ⊩ 푒 휓 ( ,푓 ,픲 푧 픲, 픠, 푓, 푒, ∗

[ 휙 ∀ 푉 ∈ 픲 푥∈픞.∃.휙° ∃푦. 픞°. ∈ ∀푥 beany formula. 〈 ,픵 푓, 〉 ∗ ) 픞 ∈ ,푓∈휔∧ 휔 ∈ 푓 푒, ∧ bethe formula ( ∗ 퐂퐙퐅 ,푦 푥, ∃ . 푦∈퐸 푥∈픞.휙° 픞°. ∈ ∃푥 퐸. ∈ ∀푦 〈 ) ,픡 푓, , there exist, there aset Co means that ( ,푦 푥, l = 퐵 〉 휙 휆푓 휆푓 ( 퐷. ∈ 푒푓 푥,푦 ) 퐩푓 . 퐩푓 . ∃퐶 → { ) 〈 ↓ ∧ ∃픡 ∃픡 ↓ ∧

∀ by 퐩 ( 퐂퐙퐅 ( ( 〈 푒푓 푒푓 ,픵 푓, 푒푓 푒푓

[ 픡 , 푥∈픞.∃ .휙° 퐶. ∈ ∃푦 픞°. ∈ ∀푥 휙 ⊩ ) ) 푥∈픞.∃ .휙° 퐸. ∈ ∃푦 픞°. ∈ ∀푥 , there is, there aset 〉 퐂퐙퐅 ( 〉 푥∈픞 푦∈ ∃푦 픞. ∈ ∀푥 ⊩ 푦∈ ∀푦 ⊩ 픞 ∈ 〈 ,푓∈휔∧ 휔 ∈ 푓 푒, : ( Co Co 퐩 ,푦 푥, 퐷 ( 푒푓 . ,픡 픵, 퐸 l l ∗ such that such 휙 ° 휙 ∗ ∃ . and and ) 픡 , ( ( ) ~ ad ∀ 퐷 ∈ and ∀푦 푥,푦 푥,푦 〈 )] 〉 ,픡 푓, Co ) ) 63 ∧ 푧 = ∧ 퐷 퐷 ∪ 퐸 = 퐷, = l 〉 휙 [ and completeness. and . 푒푓 ~ ∀ ( ( 푥,푦 퐷 〈 푒푓 ,픵 푓,

〈( such that such ↓ ∧ Co ) 푥∈픞 휙 픞. ∈ ∃푥 . ( 푒푓 휙 ⊩ ( ,푦 푥, 〉 l ,푦 푥, 푐 휙 〈( ) .∃ 퐷. ∈ , (

0 푐 푒푓 푥,푦 ( ) 픡 , 푥∈픞.휙° 픞°. ∈ ∃푥 . ) ,픡 픵, 푦∈퐶 푦∈픞.휙° 픞°. ∈ ∃푦 퐶. ∈ ∀푦 ∧

) ) 〉 휙 . 0 픟 ∈ ) 픡 , 〈 ) ,픡 푓, ( ( 〉 . ,푦 푥, ,푦 푥,

∗ 픟 ∈ ∧ 〉 ) ) 픞 ∈ ∗ ( , ,

} 푒푓 (

,푦 푥, ∗ . ) ( 1 푒푓 ) 휙 ⊩ ,

휙 ⊩ ( ,픡 픲 픡, 픠, ( ,푦 푥, ( ,픡 픵, ) ) ) ] ) )] , .

.

WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. is, e ed o hw ht indeed that show to need we First, 〈 axiomatic constructive of properties Metamathematical The last The implication follows because hand, other the On With With these ingredients, We can now define the witness: This meansThis Lemma4.12 another time, wecan conclude that We are now ready to find a realizer for the axiom. Let axiom. the for realizer a find to ready now are We

Note that we mayassume that 푣

푢 ∀픲 ⇔ ⇔ ⇔ ⇔ ⇒ ⇒ ⇒ 푤 ∪ [

∀픲 ∀푒 ∀픲 ∀푒 푥∈픞.∃ ° 휙° 픟°. ∈ ∃푦 픞°. ∈ ∀푥 푢 푐 ∀ ∀ 휙 픟. ∈ ∀ ∃푦 픞. ∈ ∀푥 ⊩ 푒 푤 , and and 푚 ∀ ∀ ∀ and and and and

푥∈픞 푦∈픟 휙 픟. ∈ ∃푦 픞. ∈ ∀푥 〈 〈 〈 〈 〈 〈 0 ,픠 푓, 픠 푓, 픠 푓, 푢 ,픠 푓, 픠 푓, 픠 푓, and [ ( 〉 ∀ 푒 푥∈픞.∀ .∃ 픷 ∈ ∃푦 픞. ∈ ∀푥 픞°. ∈ ∀푥 풲 ∈ 〉 〉 〉 〈 ) 〉 〉 〉 ,픠 푓, 픞 ∈ 픞 ∈ 픞 ∈ 푚 ≡ 픞 ∈ 픞 ∈ 픞 ∈ ∀ ∀ ∀ 1 〉 휆푓 . Continuing the chain of implications, 〈 〈 〈 ∗ ∗ ∗ ∗ ∗ ∗ 퐩 ,픡 푔, 퐩 ≡ 픞 ∈ 푧∈푤 ∈ ∃푧 . 휓 퐵. ∈ ∃푧 . ∃픡. . ∃픡. . ∃ . ∃ . 퐷 ∈ 푒푓 (퐩 퐩 . 푒푓 휆푔 푥∈픞 푦∈픟 휙 픟. ∈ ∃푦 픞. ∈ ∀푥 ⊩ 푒 〈 〈 ∗ 〉 퐩 퐩 픡 , 픡 , 푧∈퐵 휓 퐵. ∈ ∃푧 . 〈( ( 픷 ∈ 퐩 . ( 푒푓 푒푓 휆푒 ( 〉 〉 〈 ∀ 푒푓 ,푦 픲° 푦, 푥, ( ( 퐩 푤 ∈ 픷 ∈ 퐩0 . 푒푓 〈 픡 , 픡 , ,푦 픲 푦, 푥, ( 푢 ∗ 푢 푒푓 ,픠 푓, ) ( ∃픠 . 휓 . 〉 〉 0 퐥푔 ) ( 푢 ∗ ( 픡 , 픡 , 푢 ( 픷 ∈ 푤 ∈ ( ,푦 픲 푦, 푥, 〉 ,푓 ,픲 푧 픲, 픠, 푓, 푒, ∃ . ( ) 퐩푚 퐫 ( ∃ . 퐵. ⊆ 퐷 〉 ,푓 ,픲 푧 픲, 픠, 푓, 푒, 〉 ) 픞 ∈ 〈( ( ) (퐥 〈 푢 ∗ 픷 ∈ 푒푓 픟 ∈ 〈 ∃푣 → ( ,픠 푓, 푢 푑∈픢 ∈ ∃푑 → 푔 ( . ,픠 푓, 0 ,푓 ,픲 푧 픲, 픠, 푓, 푒, . ( ∗ ( ) ( 픲 ) ) ( WS WS 퐸 푧∈푤 ∈ ∃푧 . 푢 ∗ 〈( = 풲 퐥푔 1 푒 ∗ ) 〉

.휙° °. 〈( ∪ 퐶 = 퐸 〉 + 픡 , 푥∈픞 푦∈픷 ∈ ∃푦 픞. ∈ ∀푥 ⊩ ) ) ∧ = 픢 푒푓 ∧ 푉 ∈ 픢 픞 ∈ Using subset Using collection in )( 픲 푚 푒푓 픞 ∈ = 〉 ( ) ( 퐶 ∈ 퐫푔 ) 1 푒푓

푒푓 ( 픞 ∈ ) ) ∗ 0 ) { 〈 { ,푦 픲° 푦, 푥, and and ∗ 0 . 픡 , ( realizes ,퐸 퐸, 〈 〈 ( . ) ( ) 픡 , ∗ ) ,푦 픲 푦, 푥, 푤 ∪ 푣 ,픷 0, ) ( 픲 ( ) 푥∈픞 푦∈푑 휙 푑. ∈ ∃푦 픞. ∈ ∀푥 1 ∗ 〈( Frt oe ta clearly, that note, First . 1 〉 휓 . 푥∈픞.∃ 푣 ∈ ∃푦 픞°. ∈ ∀푥 〈( ) 〉 ∃푤 → ∧ 휙 ⊩ { 휙 ⊩ 푒푓 픟 ∈ ) 〉 + 푒푓 픟 ∈ ° 풲 ∈ 픷 픷°: 푧∈푤 ∈ ∀푧 ( 풲 ∈ 픷 : (( 푥∈픷 ∈ ∀푥 ⊩ 〉 ,푓 ,픲 푧 픲, 픠, 푓, 푒, ) ) ~

) ) ∗ 푉 ∈ 픢

푐 푔 0 ( ∗ ( 0 Implies . 푤 , 픡 , ∧ ,픡 픲 픡, 픠, n 푥∈픷 ∈ and ∀푥 ) 픲 ,픡 픲 픡, 픠, 픡 , 64 ∧ set 0 〉 ( ( 〉 〉 ( 푒푓 퐷 ∈ 푤 ∧ 퐶 ∈ 푣 : 푔 푢 픟 ∈ 푒푓 theories theories ∗ } ~ 픟 ∈ ∃ . ) , since

픲 } ) ) ) 1 푉 ∈ 픲

휙 . 픲 ) )

1 ∗ ) 〈 ) 1 푦∈픞 휙 픞. ∈ ∃푦 .

∗ ,픠 푓, 1 휙 ⊩ ∧ 푧∈푤 ∈ ∀푧 ∧ ( 휙 ⊩ ∧ 픲 휙 ⊩ ,푦 픲 푦, 푥, ( 푥∈픞.∃ ° 휙° 픟°. ∈ ∃푦 픞°. ∈ ∀푥 휙 . ( ( 〉 푒푓 ,푦 픲 푦, 푥, ∗ 푒푓 푉 × 휔 ∈ 픷 픲 ( 픞 ∈ and suppose and 2019 ( ( .∀ .∃ ° 휙° 픞°. ∈ ∃푦 픞. ∈ ∀푥 °. ,픡 픲 픡, 픠, ( ) ,푦 픲° 푦, 푥, ,픡 픲 픡, 픠, (so ) ,픡 픲 픡, 픠, 1 1 ) 퐂퐙퐅 ∗ 휙 ⊩

휓 . 휙 ⊩ /20 ) ( 픲 = 푧 ,푦 픲 푦, 푥, ) ∃ . 푦∈푑 푥∈픞 휙 픞. ∈ ∃푥 푑. ∈ ∀푦 ∧ } ) ( ) , there , there is aset )

) ,푓 ,픲 푧 픲, 픠, 푓, 푒, 푉 ∈ 풲

( ) ) 〈

( ,픡 픲 픡, 픠,

,픠 푓, ∗ 푦∈푣 ∈ ∀푦 ∧ ,픡 픲 픡, 픠, 〈 and and 퐩 ) 푒푓 〉

픞 ∈ ) 푥∈픞 푦∈픟 휙 픟. ∈ ∃푦 픞. ∈ ∀푥 ⊩ 푒 픡 , ) ∗ ) °∈퐸 ∈ 픷° ) b Lma .2 Invoking 4.12. Lemma by

〉

) ∗ (

퐵 ∈ 픲 ,푦 픲° 푦, 푥, .휓 푎. ∈ 푥∈픞.휙° 픞°. ∈ ∃푥 . ( for all for all 퐶 ) ) ,푦 픲° 푦, 푥, such that such ( ) ( ,푓 ,픲 푧 픲, 픠, 푓, 푒, ad hence, and ,푦 픲 푦, 푥, ) 〈 . ,픷 0,

( ) ,푦 픲° 푦, 푥, 〉 ) , 퐸 ∈

) ( )] ,푦 픲 푦, 푥, + .

) . 픷 )] 푢 .

= ) . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 휔 oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison Ⓜ sectionThis isdevoted to proving the numerical existence property for 4.6.1 r now are we structure, realizability our of soundness and completeness proved Having 4.6

Let Is valid in holds inholds Set induction certainly holds for conclude by set induction in all sets all 4.14 Lemma induction Set Proof that onlyif Let 훼 all all lemma, As As . If If . such thatsuch 푉 ∈ 픞 Theorem 4.15 푒푔 푒 ⊩∀푎 ⊩⊩ 푔

퐂퐙퐅 푥∈픞.∃ ° 휙° 픟°. ∈ ∃푦 픞°. ∈ ∀푥 푉 ∈ 픞 : If If : bea fixed point of

Proof metamathematicalof properties ≡ 푥 Disjunctionand Numerical existence property 푉 ∗ possesses some of the metamathematical properties discussedSection in 4.2. . 퐂퐙퐅 ( . This is true in particular for particular in true is This . 훼 휆푢 ∗ ∗ 퐂퐙퐅 , then , ∀픞 ⊨ 퐙 휙 ⊢ 퐂퐙퐅 : Let : 푢 . . But . But we can simply choose [( 푦∈푎 휙 푎. ∈ ∀푦 , hence wedo neednot tocare about truth. ( 휆푥 [( : Let 휙 〈 . 푥∈픞 휙 픞. ∈ ∀푥 ,픵 ℎ, be any If formula. be 푒푢 ( 푛 ) 〉 휙 ) ) 픞 ∈ 푔( ≃ 푔 ( ( . ( 휏 ,푦 픲° 푦, 푥, 푥 푦 ( ) ) ∗ 푧 be a formula with at most , then , ) ( ) 푥 휙 → 퐂퐙퐅 ≡ ) 휆푥 ) ) ∀푎 ⊩ 푒 휆푢 휙 → ∀푎 → ∃푑 ∈ 픢° 픢° ∈ ∃푑 → . ( 휙° that that 푉 ∈ 픵 푒푔 푎 푢 . [( ) , i.e.the formula ] ) ( ( ∀푦 ∈ 푎 휙° 푎 ∈ ∀푦 . The induction hypothesis is that is hypothesis induction The . , this , this completes proof.the 픞 휆푥 [( 훽 푎 휙° ∀푎. 푉 ) ∗ ] 푎̂ = 픞 for some for . ∀푦 ∈ 푎 휙 푎 ∈ ∀푦 ∗ , then by the completeness theorem, completeness the by then , 푧푢 푣 = 푑 ∀픞 ⊨ 휆푥 푔 ( ( ) 푥∈픞.∃ .휙° 푑. ∈ ∃푦 픞°. ∈ ∀푥 휆푥 . Weshow by induction that . ( which shows that shows which 푒푔 푎 픲 [( ( . ) , as 푒푔 푦 . 푥∈픞 휙 픞. ∈ ∀푥 푦∈픞 휙 픞. ∈ ∀푦 ⊩ ~ ( ) 푦 ) 훼 ∈ 훽 ) 65 ) 푣 휙° → 휙 ⊩ ) 픲 푥 휙 → 픢° ⊆ 퐶 ∈ ~ free. . Thus, by the induction hypothesis and the last the and hypothesis induction the by Thus, . ( ( 픞 ( 푎 ) ( 푥 . ) 푎

) ] ( ) ) If 푦 푎 휙° ∀푎. → ] )

휙 → . ( 퐂퐙퐅 ⊢ ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 ⊢ 퐂퐙퐅 푎 휙 ∀푎. → , ,푦 픲° 푦, 푥,

푥∈푎 휙° 푎. ∈ ∀푥 ( 픞 ) ] ) ( for all for ( 푎 푦∈푑 푥∈픞.휙° 픞°. ∈ ∃푥 푑. ∈ ∀푦 ∧ 푎 ) 퐂퐙퐅 푒푔 )

.

( 푥 휙 ⊩ : ) 푉 ∈ 픞 휙° → 푥∈픞.휙° 픞°. ∈ ∀푥 ( 푛 ( 픟 ) , then there is some ) ∗ for all for ( , then then , 푎 ) . We can therefore can We . ( 휙° 푥 푉 ∈ 픟 eady to pro to eady ( ) ,푦 픲° 푦, 푥, ( 휙° → 푥 WS WS ) 훽 forholds ∗ and ( 2019 ) 픞° )

) ∈ 훽 for ∈ 푛 /20 ve ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. As weAs know, Uniformity-rule is just aspecial case ofUnzerlegbarkeit. We therefore have: Hence, we immediately have: p disjunction the implies property existence numerical the 4.2.2, Section in noted have we As Let 푥 휓 ∀푥. 휔 .휙 휔. Ⓜ Lemma4. 4.6.2 Ⓜ axiomatic constructive of properties Metamathematical Set Ⓜ 4.6.3 Ⓜ Proof Proof tion of tion This means in turn that there is some is there that turn in means This By completeness the theorem, weobtain Proof 휔 = 휔 izing the infinity axiom we have shown that Theorem4.15 of Proof proves such that such then then then then with Theorem 4. Corollary4.19 Theorem 4.18 Corollary4.16 = 푠 푛∈휔 ∈ ∃푛 ⊩ 푡 ( : We may assume that assume may We : : Suppose : : 푥, (

푡 ⊩ ∀푎 휙 ∀푎 ⊩ 푡 ,푛 푥, weeven have 휔 { 휙° Unzerlegbarkeits-and Uniformity-rule Church’srule 휆푢 푒 , } ) ( If If If If 휔 ∈ ∀푥 ⊩ 푡 17 퐙 푥 푦∈휔 휓 휔. ∈ ∃푦 ∀푥. ⊩ 푡 ⊢ 퐂퐙퐅 픫 . ( . 푛 ( 푥 has to be one of the the of one be to has 휔 ∈ 푚 휔 ∈ 푚 〈 푡푢 ° : ) ,푚 푚, Let ) ) ) . 휙 ≡ 20 ( 휙 . 1 퐙 푥 푦∈휔 휓 휔. ∈ ∃푦 ∀푥. ⊢ 퐂퐙퐅 푎 . Then : If : Whenever ) 휙 ( 〉 : Whenever : If means that means 푛 then then , then ( ( 휔 ∈ 푛 ) 푥 퐙 푥∈휔 푦∈휔 휙 휔. ∈ ∃푦 휔. ∈ ∀푥 ⊢ 퐂퐙퐅 : Let : ti mas that means this , 퐙 휓 ∨ 휙 ⊢ 퐂퐙퐅 ) 푦∈휔 ∈ ∃푦 . ) 퐙 픞∈푉 ∈ ∀픞 ⊢ 퐂퐙퐅 . beany formula with all free variables shown. If

∗ 퐙 ⊢ 퐂퐙퐅 , then , then 푛∈휔 ∈ ∃푛 푡푚 퐙 푛∈휔 휙 휔. ∈ ∃푛 ⊢ 퐂퐙퐅 퐙 푥∈휔 ∈ ∀푥 ⊢ 퐂퐙퐅 푦∈휔 휃 휔. ∈ ∃푦 ⊩ 휙 . 퐙 푥 푦∈휔 휓 휔. ∈ ∃푦 ∀푥. ⊢ 퐂퐙퐅 퐙 푥 휓 ∀푥. ⊢ 퐂퐙퐅 휆푢 푡푚 휙 ⊩ 푡 with ( ,푦 푥, . 푛̅ ( , then 푦∈휔 ∈ ∃푦 ⊩ ( s and hence cannot depend on depend cannot hence and s 푡푢 ,푦 푥, ∗ ( WS WS )

( ,푦 푥, [ ) . 픞 ( 푛∈휔 ∈ ∃푛 ⊩ 푡 1 푡푚 ) ) 푉 ∈ 픶 for all for 푒 ≃ ( , i.e. , 퐙 휙 ⊢ 퐂퐙퐅 ) ( 푡 . By the soundness theorem, there is a closed applicationsoundnessclosed termtheorem, therea the By is . 푚 ) ) 퐙 휙° ⊢ 퐂퐙퐅 0 ( ( 0 .∃ 휔 ∈ ∃푦 . for some natural number 푦 , 푛 푥 휃 . 푛 = 푛 = ∗ ) ) 퐙 픞∈푉 ∈ ∀픞 ⊢ 퐂퐙퐅 푉 ( such that such ) . By the soundness theorem,soundness the By . ( ¬휓 ∨ ,푦 푥, . 푚 ∗ 푉 ∈ 픞 and and fr some for 휔 = 휔 ⊩ or 푦 , ~ 휓 . ) ( , then there is some natural number ,푦 푥, ) 휙 . ( ( ( 66 . 퐙 휓 ⊢ 퐂퐙퐅 set ∗ ( ,푛 픞, 푥 ( . This is true in particularintrue is This for . 푥̂ ( ) 푡푚 ) ) ,푦 푥, theories theories , ~ then , ° ) then 〈( ) ] ) and and hence, by Theorem 4. for all . 1 푡 ) 휔 ∈ 푛 ∗ ) . By the soundness theorem, there is some is there theorem, soundness the By . 휃 ⊩

0 [ . 픫 , 퐂퐙퐅 ⊢ ∀푥 휓 ∀푥 ⊢ 퐂퐙퐅 푦∈휔 휓 휔. ∈ ∃푦 ⊩ 푡 퐙 푦∈휔 푥 휓 ∀푥. 휔. ∈ ∃푦 ⊢ 퐂퐙퐅 〉 2019 ( 휔 ∈ and and 푚 푥 , i.e. 푛 , 픞 /20 . and and ) By 푒 ( .

퐂퐙퐅 ⊢ ∀푥 휙 ∀푥 ⊢ 퐂퐙퐅 푡 . We can conclude by completeness ) 푡 ⊩ ∀푎 휙 ∀푎 ⊩ 푡 Lemma 4.17, we conclude we 4.17, Lemma 1 ( ( 푥 푡 휙 ⊩ ) ( ) 푉 1 ,푦 픞, ∨ ∀푥 ¬휓 ∀푥 ∨ ∗ 휓 ⊩ ( 푛∈휔 휙 휔. ∈ ∃푛 ⊨ 푛 ) ] ) ( . As we know that know we As . ( B completeness, By . ( 푎 ( ,푦 푥, ,픫 픞, 푥̂ = 픞 ) 푥 10 , then ) ( . 푥 ) , ) . But by the defini- the by But . . 푉 ) 푒

. with , where , ∗ 푛∈휔 ∈ ∃푛 ⊨ ( 퐂퐙퐅 ⊢ ∀푎 휙 ∀푎 ⊢ 퐂퐙퐅 푛 ) . When real- When . 퐙 푥∈ ∀푥 ⊢ 퐂퐙퐅 푥 roperty. is a set.a is 휙 . 퐙 ⊢ 퐂퐙퐅 푉 퐶푍퐹 ( ( ∗ ∈ 푡 푎 푛 ∎ ∎ ∎ ⊨ ) ) 푡 . .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. which means number operator fromDefinition 4.4, 휔 4.7 Ⓜ 4.6.4 Theory Set Constructive of Analysis and Comparison Ⓜ Ⓜ soundnessof the theorem isbecomes clear, that this truth refers totruth inside Notice, that our realizability refers to truth (for example in the clause for implication) 4.7.1 structure r the adjusting slightly only while results further obtain to easy is it far, so work the With ¬휙 The soundnessThe theorem requires more work, as we need tocheck the additional setaxioms:

mean truthmean in we donot need to pay attention to the involved formulas to be bounded. ofboundedalmostproof theproof identicalseparation. to is Theonly This difference i Separationschema Proof that that Hence, Proof There is nothing to do to alter the completeness theorem:

휙 . Theorem 4.23 (Soundness) Theorem4.22 Theorem4.21 (

( 푛 퐙 ⊢ 퐈퐙퐅 : We togiveneed realizers axiomfor the schemas of (unbounded) separation and powerset. W my sue ht hr ae terms are there that assume may We : 푛

) Further results ) Te eod en (y lsia raoig, ht hr i sm term some is there that reasoning), classical (by means second The . . The first meansthatfirst for each The . Markov’s rule Markov’s Metamathematics of ( 푡 ( 푉 푝 ( ∗ ) 휃 ⊩ 푡 . 0 ) ( 퐈퐙퐅 0 푝 ) 0 = ) 1 : If : If , wecan easily transfer all proofs to this context. . ( 휙 ⊩ and and 푉 ⊢ 퐂퐙퐅 ∗

휃 ⊨ ( ( 푝 푡 ) ( , then ( 0 푝 푥∈휔 휙 휔. ∈ ∀푥 ) ) . 0 : For every theorem ) 1 퐈퐙퐅 퐙 휃 ⊢ 퐈퐙퐅 ( 휙 ⊩

( 휔 ∈ 푛 퐩 푛 퐙 푥∈휔 휙 휔. ∈ ∀푥 ⊢ 퐂퐙퐅 ( ( ) 푝 퐥푟 ° ) ¬휙 ∨ . 0 )( ) , either , . This shows that with 퐫푒퐥푟 ( 푡 푛 ~ and ) ) ) 푛∈휔 ∈ ∃푛 ⊩ 67 휃 ( ∧ 푡푛 of ( 푢 ~ ¬푛∈휔 휙 휔. ∈ ¬¬∃푛 ) ( with with 퐈퐙퐅 0 푥, 0 = { , there exists, there a closed application term 푒 휙 . } and and 푥∈휔 ∈ ∀푥 ⊩ 푡 ( 푥 ( 푛 ) ) ) . . (

푡푛 ≡ 푟 ( 푛 ) ) 1 ) 휇푝 휙 ⊩ , then 휙 . . ( ( 푛 푡 ( ( 푛 ) 푝 퐙 푛∈휔 휙 휔. ∈ ∃푛 ⊢ 퐂퐙퐅 ) ¬휙 ∨ ) , or , 푝 0 퐂퐙퐅 ) with 0 ( with ( 푡푛 . If wechange. If this to 푛 . In the formulation ) ) 0 and and 푛∈휔 ∈ ∃푛 ⊩ 푝 n thiscaseis that n 휇 0 ≠ being the least and and ¬푛∈ ¬¬∃푛 ⊩ 푢 ealizability WS WS ( 푛 ) ( . 2019 푡 푡푛 휙 . , such ) ( 1 푛 /20 ∎ ∎ ⊩ ) ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. tion 4.2). Wetion formulate our results in terms of the following theorem: to transform the proofs we conducted in Section 4.6 into proofs of the corresponding principle easy especially is it particular,In theories.the in proved be cannot properties, that some possesses may str realizability our that in leeway some gain we hand, other the On theorem. completeness the when realizing when Ⓜ Changing our realizability structure to not refer to truth inside 4.7.2 Ⓜ axiomatic constructive of properties Metamathematical 휔 Ⓜ Proof For example, let us show how to show the Uniformity-rule: Suppose UzP We realize the rule with a term a with rule the realize We zerlegbarkeit, Uniformity Church. and we can easilynow transform itinto a proof ofthe rule the same the term to realize For Powerset theorem and by and theorem Proof since in that for all for that 푛∈휔 푛⊩ ⊥ ¬푛 휔. ∈ ∀푛 We can now show that of °∈Pow ∈ 픠° 휓 . 푉 Theorem4.26 Theorem4.25 4.24 Theorem , 푉 ∈ 픞 ( ∗ : Let CP : Towards a contradiction, suppose that ,푛 픞, . If If .

Compatibility with Principles and and 퐈퐙퐅 ) ( Pow ⊆ 픠 ⊩ 푔 for all for all ∗ 픞 퐓 , we define define we , ) 푉 ∈ 픞 beeither , the ° and and hence MP , contradiction. : 퐵 → 퐴 ∗ 푉 푉 ∈ 픞 : : If 푉 : , 훼 Cons 퐈퐙퐅 푦∈휔 휓 휔. ∈ ∃푦 ⊩ 푒 푠 ∗ and and 휙 ⊨ 푉 퐂퐙퐅 ( has the disjunction and numerical existence property. It follows the rules of Un- of rules the follows It property. existence numerical and disjunction the has ∗ ∗ we now need not refer to truth inside truth to refer not need now we 픞 . Pow ( 퐈퐙퐅 휙 ⊨ ) 퐈퐙퐅 휆푔 , there is some is there , , then then , 퐵 → 퐴 푉 or 훼 ∗ 퐩푔퐢 . and and ( s are sets, so must be , then ) 픞 퐈퐙퐅 Cons ⇒ ) by . 풓 〈 퐂퐙퐅 ,픠 푔, . Given a proof ofthe property of the form

( realizes the axiom. ,푦 픞, 휆푥 Cons Pow 〉 WS WS are equiconsistent with the theories augmented by the principles 휏 . ( ) Pow ∈ 퐙 + 퐈퐙퐅 . We know that internalthat know We . mapping realizers of realizers mapping ( ( 픞 퐈퐙퐅 휔 ∈ 푡 ) ∗ If If ) = ( UP 픞 퐴 ⊢ 퐓 Cons ⇒ 퐙 휙 + 퐈퐙퐅 ) { with ∗ 〈 z + UzP + ad hence and 퐵. → 퐴 ⊢ 퐓 ,픠 푒, ~ Pow 〉 , then , then 픞 ⊆ 픠 ⊩ 푒 : 68 set 퐙 ⊩ ⊥ 푡 ⊢ 퐈퐙퐅 ( 퐈퐙퐅 is inconsistent. Then ( theories theories 픞 ~ ) CP ∗ 퐵 ⊢ 퐓 휙 + and and therefore

+ 퐩푔퐢 퐴 ) } MP and similarly and for 퐓 and to realizers of realizers to 휔 , 풓 2019 . But by definition of realizability, of definition by But . of the statement the of 퐂퐙퐅 is represented is by Pow ∈ 픠 ⊩ ) and similarly and for /20 Pow in in 푥 푦∈휔 휓 휔. ∈ ∃푦 ∀푥. ⊩ 푒

퐈퐙퐅 ( Pow 픞 ) , comes at the cost of giving up Pow = ° ( 퐙 ⊢ ⊥ 휙 + 퐈퐙퐅 픞 ( ) 픞 퐵 . Of course, course, Of . ) . The crucial point is that is point crucial The . is a well-defined element 퐂퐙퐅 °→퐵° → 퐴° ( 휔 퐂퐙퐅 픞 . and hence, hence, and ) 풫 ∪ . By the soundness ( . ,푦 푥, and simply use simply and ( °⊆픞° ⊆ 픠° 픞° ) . This means, ) Nt that Note . s (see Sec- 푛∈ ∃푛 ⊩ 푡 implies uctures 퐙 ⊢ 퐈퐙퐅 UZ ∎ ∎ ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. We have discussedWe have in that section4.2.6 following Corollary: 푥 휓 ∀푥. ⊩ 푒 above. inschema proof the follows the that see 4.18,we Theorem of proof tothe linesComparingthese Ⓜ Theory Set Constructive of Analysis and Comparison

휔 well-ordering of In light of this result, all weak counterexamples of sections 2.4.2 and 3.3 like the fou

some is there that means This , 픫 Corollary4.27 has to be one of the the of one be to has ( ,푛 푥, ) and therefore and 핆ℕ : 퐙 퐋퐄퐌 ⊬ 퐈퐙퐅 turn out turn tobeunprovable statements in 푛̅ s and hence both hence and s 휆푒 . 푉 ∈ 픶 ⊩ 푒 . ∗ ( such that such 푥 푦∈휔 휓 휔. ∈ ∃푦 ∀푥. 퐋퐄퐌 isincompatible with 픫 and and ~ 〈( 푒 69 ( ) 푒 ,푦 푥, 0 cannot depend on depend cannot 픫 , ~ 〉 ) ) 휔 ∈ → and and ( 퐈퐙퐅 푥 휓 ∀푥. and and ( CP 푒 ( ) ,푛 푥, 1 . Therefore, we immediately get the 퐂퐙퐅 휓 ⊩ ) 픞 ) . . . ( By ,픫 픞, Lemma 4.17, we conclude we 4.17, Lemma ) . But by the definition of definition the by But . ndation axiom or the WS WS dicated 2019 /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 5.1 Martin- on look closer a take will we 2.3, chapter in promised, As 5 Martin- like Expressions substitution. of rules following the down put We 5.1.2 pretation of w we chapter, this of section final the In application. some discuss briefly and rules all 5.1.1 phasizepresented that the theory presenteddefines asettheory on its own right. notation early his with stick us let now, For 5.2. section with starting we will so and work later typ exactly is “ Equality of types Transitivity Symmetry Reflexivity 퐵

( 푥

Martin- )

isa well-formed set under assumption the that Formulating Löf’sset theory Substitutionrules Rulesequality of 퐂퐙퐅 e theory, if we refer to sets as “types”. Martin“types”. as sets to refer we if theory, e Löf’s set theory set Löf’s into into

퐌퐋 퐌퐋 . It should be noted that all concepts are closely related to type theory. In fact, it 푏 퐴 ∈ 푎 퐴 ∈ 푐 = 푏 퐴 ∈ 푏 = 푎

∈퐴푏 퐴 ∈ 푎 퐵 퐴 ∈ 푎 퐵 = 퐴 퐴 ∈ 푎 퐵 퐴 ∈ 푎 푏 ( 푎 푏 퐵 ) 퐴 ∈ 푎 = 푎 퐴 ∈ 푎 = 푏 퐴 ∈ 푏 = 푎 퐴 ∈ 푐 = 푎 ( ( 푐 = 퐵 푎 퐵 ∈ 푎 퐴 ∈ 푎 푎

) ( ) 푎 퐵(푎) ∈ ( 퐶(푎) = ( ) 푎 푥 set ) ) 퐵 ∈ ( 푐 = (

푥 퐴) ∈ (푥 퐴) ∈ (푥 푥∈퐴) ∈ (푥 퐴) ∈ (푥 푥 푥 ( ) ~ ) 푥 ( ( 퐶(푥) = 퐵(푥) ∈ 푎 ) 푥 70 set ) )

퐵(푥) ∈ ~ 푥

isin the set

-Löf changed his notation accordingly in his inaccordingly notation his changed -Löf

∈퐴푏 퐴 ∈ 푐 = 푎 퐵 = 퐴 퐴 ∈ 푏 = 푎 ∈퐴퐵 퐴 ∈ 푐 = 푎 퐶 = 퐵 퐵 = 퐴 Löf’s set theory set Löf’s 퐴 푏 퐵 푥∈퐴) ∈ (푥 ”. ( ( 푎

푥 퐵 ∈ 푏 = 푎 ) 퐵 ) 퐴 = 퐵 퐵 = 퐴 퐶 = 퐴 퐴 = 퐴 set 퐴 set 퐴 푏 = ( 푎 ) should be read as follows: as read be should ( 퐵(푐) = 푐

) 퐵 ∈ 퐌퐋 ( 푥 퐴) ∈ (푥 퐴) ∈ (푥 푥 ( 푎 ( ) . We will describewill We . 푥

) 퐵(푥) ∈ ill give an inter- an give ill )

set

WS WS

to em- to 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. The right column explains equality between the constructed sets and their elements. In the case of case the In elements. their and sets constructed the between equality explains column right The aAs first example of rules allowing for a construction of sets, we discuss 5.1.3 Theory Set Constructive of Analysis and Comparison

introduction, two sets construct, given a set

as the setas the of all functions mapping The rules of rules The In theIn propositions- 퐴 how to give a function mapping The The symbol agree every on input. right explains when two such canonical elements are equal: This is the case if t of the the of translaterules nicely into logical rules about implication. suggestedAs in startingthe example, one role canonical elementcanonical family family Finally, the rules of describe elementshow of the type just defined behave. We will usually write usually will We to therules β 횷 횷 횷 횷 an an element -equality -elimination -introduction -formation

Π 퐵 훱 -type is to give an interpretation to universal quantification: One defines the defines One quantification: universal to interpretation an give to is -type ( -rules 푥 ) does not depend on Ap Π - 푏 reduction and γ -introduction declare how to give canonical elements of elements canonical give to how declare -introduction ( can be explained as follows: Given an element 푥 ) 휆푥 퐵(푥) ∈ as Π 푏 . -equality show how 퐴 - ( sets interpretation “ 푐 and for each ( 푥∈퐴 ∈ Π푥 ( 푥 푎 . The function ) ) . instead of instead ∈ 푐 Ap Ap 푏 퐴 ∈ 푎 - ( ) ( 휆푥 conversion of λ ( ,푎 푐, e 퐵 set 퐴 퐵 휆푥 푥 퐴 Π푥: 퐴 ∈ 푥 푥 ( 푏 . 푥 , wewill simply write 푏 . ( ) 퐴 ∈ 푥 Ap ) 푥∈퐴 ∈ Π푥 ( computes this element computes thisand returns

( 푥 and 푥 푏 ) 푥 ) ( in in into into 퐵 ( Ap ) ,푎 푐, 푥 푥∈퐴) ∈ (푥 ∈ 푎 , ( 푏 ↦ 푥 ) into into 푥 ( ( 퐴 ( ) ) ,푎 푐, 퐵(푥) ∈ 푥∈퐶 ∈ Π푥 ) Ap 푥∈퐴 ∈ Π푥 ) some set → 퐵 퐵 푏 = 퐵 ∈ ” plays same role the asits logical counterpart. Thus, the ( ( operates canonical on elements. Note they that correspond ) 퐵 푥 푥 ( ( , when we know that know we when , -calculus. 푥 푥∈퐴) ∈ (푥 ( ) 푥 ) 푥∈퐴) ∈ (푥 ( ( 푎 ( . This is done in giving a method computing for each set ) ~ 푥 푎 ) 푥 thus thus defined is denoted by ) ) ) ) ) 퐵(푥) ∈ 퐷 ) 퐵 : 71 퐵(푎) ∈ set ( ( 퐵 푥 푥 퐴 ∈ 푎 ~ ) ) (

are equal if

푥 ) 퐵 → 퐴 , the set

∈ 푐 instead of ∈ 푑 = 푐 ( 휆푥 ( 푥∈퐴 ∈ Π푥 퐶 = 퐴 푥∈퐴 ∈ Π푥 휆푥 ( 푐 푏 . 퐵 퐶 = 퐴 푥∈퐴 ∈ Π푥 is of a corresponding a of is . Ap ( Ap ( 푏 푥 ( 푥∈퐴 ∈ Π푥 푥 퐴 Π푥: ( ) and and for all ( ) Π 푎 ( 푏 ( ,푎 푐, ) 퐵 = ,푥 푐, 푥∈퐴 ∈ Π푥 ∈ 푐 -rules. They prescribe how to 퐵 ) ( ( 푥 . Thus, theThus,elimination . rules ) ( 푥 휆푥 he he two associated functions 휆푥 ) 퐵 ) 푥 ) ) ) ( ( 퐵 = ) . One can think of this set ()∈퐵(푥) ∈ 푑(푥) = ) 푏 . 푑 . ∈ 푐 = 푥∈퐴 ∈ Π푥 , we know it yields some 푥 퐵 푥∶퐴) ∶ (푥 ( ) Ap ( 푥 ( ( ) 푥 푥 푥 = ) 퐵 ) ) ) ( . . The column on the 퐴 ∈ 푥 , or in other words,other in or , ,푏 푑, ( ∈ ( 푥∈퐶 ∈ Π푥 set ) 푥∈퐴 ∈ Π푥 ( ( 퐵 푥 푥∈퐴) ∈ (푥 푥∈퐴 ∈ Π푥 ) (

) ∀푥 ∈ 퐴 퐵 퐴 ∈ ∀푥 , 퐵 ∈ 푥 퐷(푥) = 퐵 ) Π ( ) 푥 ( WS WS ) 퐴 ∈ 푏 = 푎 퐷 -set. If the If -set. 푎 퐵 ) ) ( ) ( 퐵 퐶 = 푥 푥 2019 ) ( ) (

푥

푥 ) ( ) ∈ 푥

푥 /20 to Π Π ) . - -

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. how its canonical elements are formed and elimination rules describe their behavior. Finally, equality Finally, behavior. their describe rules elimination and formed are elements canonical its how e rules Introduction sets. other from discussion under set the construct to how explaining rules, If we use use we If The way we introduced the the introduced we way The logic in aCurry-Howard-style fashion: ( Example Martin- directly to provability, we write provability,we to directly write We fication: n drvto te-ie ytx o id n lmn (pof o te set the of (=proof) element an find to syntax tree-like derivation a in be 퐴 퐶 → 퐴 ∀ ∀ ∀ . Now the -elimination -introduction -formation ( 푥∈퐴 ∈ Π푥 Löf’sset theory ) ) : Let us verify the logical axiom logical the verify us Let : diretlyapplying the → ) 퐵 -rules instead we get see how this proof of membership turns into a proof of in of proof a into turns membership of proof this how see get we instead -rules Π [ ( -rules take the following forms familiar frompredicate logic: → 퐴 ∈ 푥 [ 푥 → 퐴 ) . Let us translate the translate us Let . 퐴 prop 퐴 ( ( 푥∈퐴 ∈ ∀푥 퐶 → 퐵 푥푦푧 푥 휆푥휆푦휆푧.

( 푥 퐶 → 퐵 ( 푧 instead of instead 퐵 → 퐶 true 퐶 → 퐵 rp퐵 prop 퐴 ) ) ) ( true 퐵 푥∈퐴 ∈ ∀푥 퐶 → 퐵 ∈ → 퐴 Π ( ) ( Π 푥 ( ] -rules is paradigmatic: All rules in the following will contain formation contain will following the in rules All paradigmatic: is -rules 푦푧 푥 휆푦휆푧. 푧 퐵 푥∈퐴 ∈ ∀푥 퐵 1 ) -rules from above: ) ] 푥∈퐴) ∈ (푥 tu 퐴 ∈ 푎 true ( ( 퐴 true 퐴 1 ( ( 푎 푥 퐶 → 퐵 푦 ) ) ) ( 퐵 true true 푧 퐴 set 퐴 ( ( ) Π ( ) [ 푥 퐵 → 퐴 ) 휆푧 푧 iff there is some is there iff 퐴 ∈ 푧 퐵 -rules into the context of plain implication and universal quanti-universal andimplication plain of context the into -rules ) [ ) → 퐴 ∈ ) true 퐴 true 퐴 ( ( 푥 . . Capturing the constructive standpoint that truth correspondstruth constructivethatstandpoint the Capturing . 푥 → 푦 ( 푥 ) → 퐴 ( ( prop ( 퐴 → 퐶 퐶 true → 퐴 푧 푧 ] ( 푥∈퐴) ∈ (푥 ) ( 푧

2 ) ) ( 푥 ) →

( ) 퐵 → 퐴 퐶 true 퐶 (Π-elim) ] ( ( ) 푦 2 ( 퐶 → 퐵 ∈ prop 푦 ~ (→-elim) 퐶 → 퐵 ( ( ( 푧 퐶 → 퐴 ( 푧 72 ) 퐵 → 퐴 ) )

) )

퐶 → 퐴 ∈ ~ ) 퐶 ∈

→

)

→ → → (→-intr), → ) ) 퐴 ∈ 푐 ( true [ ) → -elimination -introduction -formation 퐴 → 퐵 true 퐵 → 퐴 퐶 → 퐴 ( → ( ( 퐵 → 퐴 [ ( ( [2 퐵 → 퐴 ∈ 푦 , i.e. there is some proof of the proposition the proof of some is there i.e. , 퐵 → 퐴 (→-intr), 퐶 → 퐴 (Π-intr), [2] [2] (Π-intr), ]

) ) true )

퐵 true 퐵 → [3 ) ) ] 3 ] 푦 →

(Π-intr),

( ( (→-intr), 퐶 → 퐴 ] 푧 ( 3 ) 퐶 → 퐴 퐵 ∈ (

[3] → 퐴 [ 퐴 → 퐵 true 퐴 true 퐴 퐵 true → 퐴

퐴 true 퐴 [1 )

퐴 prop 퐵 prop 퐵 prop 퐴 ] )

[ ) 퐴 ∈ 푧 (Π-intr), ) ( . One way to do this is this do to way One . 퐶 → 퐵 ] 2 퐴 → 퐵 prop 퐵 → 퐴 퐴 → 퐵 true 퐵 → 퐴

(→-elim) (→-elim) ]

[1] (퐴 true) (퐴 2 퐵 true 퐵 퐵 true

(Π-elim)

) (Π-elim) ) →

(

tuitionistic ( WS WS 퐵 → 퐴

2019 xplain )

/20 →

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Example The case where we start this procedure with a canonical element canonical a with procedure this start we where case The down down thein rule of jection as ( 퐵 If we have a set a have we If forrules with continue We 5.1.4 Theory Set Constructive of Analysis and Comparison The canonical elements of elements canonical The In the case of existential quantification, we can define can we quantification,existential of case the In ilarly, theilarly, right projection the assumptions of the rule. In this particular case, they should have included included have should they case, particular this In rule. the of assumptions the operates as follows: It computes It follows: as operates “

dexed by Another remark we need tomake isthat beginning rule with the of writenot them down explicitly in the following discussion. If If ity of the constructed sets and their canonical elements always follow the same We pattern. rules relate introduction eliminationand rules. Note that the rules of the right column explaini ,푏 푎, 횺 횺 횺 횺 × ( 퐵 -equality -elimination -introduction -formation 푥 ” plays the role of conjunction. Indeed, a proof of proof a Indeed, conjunction. of role the plays ” ( ) ) 푥 type

, where , where ) de nt eed on depend not does 훴 -rules : Let us set us Let : 푝 퐴 ( ∈퐴 ∈ 푥 ( . Wedenote this by 푐 ) 퐴 ∈ 푎 E ≡ 퐴 ) . For sake of the readability, we will not write down these obvious lines. and for each each for and ( and and ,푑 푐, Σ (,푦 푥 ≡ 푦) 푑(푥, -equality. ) 퐵 ∈ 푏 퐴 ∈ 푞 ( ( . By Σ 푥∈퐴 ∈ Σ푥 ,푏 푎, -sets that one can think of as generalized disjoint union of a family of sets. of family a of union disjointgeneralized as of think can one that -sets 푥 ( 푎 w write we ( and and ) ) Σ 푥∈퐴 ∈ Σ푥 푥 . Itthen substitutes the values into -equality it has the property of set of 퐵 ∈ 푏 = ∈ 푐 ) 퐵 (,푦 퐴 ≡ 푦) 퐶(푥, ( ( 푥 퐵 ∈ 푏 퐴 ∈ 푎 푥∈퐴 ∈ Σ푥 ) ) 퐵 퐴 are of the form the of are 퐵 × 퐴 ( ( some set some ∈ 푐 푥 푎 ) ) . isdefined. ) ( 퐵 for 푥∈퐴 ∈ Σ푥 in the rule of rule the in ( ~ E 푥 ( ) 73 ( Σ 퐴)퐵 ∈ (Σ푥 to find its associated canonical element of the form the of element canonical associated its find to 퐵 ,푏 푎, 퐴푏∈퐵(푎) ∈ 푏 퐴 ∈ 푎 e 퐵 set 퐴 ( ) 푎 )∈ 푏) (푎, 퐵 ∧ 퐴 ~ 푥 퐵 ( ) ) 푥∈퐴 ∈ ∃푥 ( 푑 , ( , we can form the disjoint union of the of union disjoint the form can we , ( 푥∈퐴 ∈ Σ푥 푥 E ,푏 푎, ) ) ( must be a pair a be must ( ,푑 푐, (,푏 퐶( ∈ 푏) 푑(푎, = 푎 I te rpstosa-es interpretation propositions-as-sets the In . ) Σ ( 푝 ) , where , ) 푥∈퐴 ∈ Σ푥 -elimination. We then obtain the left pro- leftthe obtainthen -elimination. We ) ( 퐵 ) ( 퐵 퐶(푐) ∈ ( ,푏 푎, 푥 ( ( 푥 ) ,푏 푎, 푑 푥 퐴) ∈ (푥 Π to be to ) ) toobtain set ) ( ( ) -introduction, we skipped some of (,푦 퐶 ∈ 푦) 푑(푥, 퐴 ∈ 푎 퐵 푥 ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 ) 퐸 = ) ( instead of a general a of instead set 푥 ( 푑 ( ) ,푦∈퐵 ∈ 푦 퐴, ∈ 푥 Σ∈퐴)퐵 ∈ (Σ ( ( ,푏 푎, ( and and

,푏 푎, ,푦 푥, (

,푏 푎, ) 푑 ) ) ) where ) ( 퐵 ∈ 푏 ( 퐶 ∈ 푑 , ,푏 푎, ,푦 푥, ( . Here, . 푥 ) ( ) ) ) 푑 = ,푦 푥, ) of ( (

푥 푎 퐴 ∈ 푎 ) ) ) 퐶 ( ) . The symbol The . ,푏 푎,

( ( ( ,푏 푎, do ,푏 푎, 퐴 type 퐴 푐 and and ) WS WS is written is therefore ng equal- ) 푎 = ) consists 퐵 ) 2019 . ( 퐵 ∈ 푏 . Sim- 푥 and ) in- /20 E .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. natively, we can rulesuse for before, we have that have we before, As an example, let us verify that this interpretation respects the logical rule logical the respects interpretation this that verify us let example, an As Martin- Given Given sets 5.1.5 퐵 witness a of + + + + ∃ ∃ ∃ ( -elimination -introduction -formation 푥 -equality -elimination -introduction -formation ) . If we. If again translate the rules, we have

Löf’sset theory + -Rules 퐴 and and 푎 and a proof a and 퐵 ( , wecan construct their disjoint union 푥∈퐴 ∈ ∃푥

푝 ( 퐵 퐴 ∈ 푎 푐

) ) 퐴 ∈ rp퐵 prop 퐴 퐵 ( 푏 푥∈퐴 ∈ ∃푥 ( of 푥 → whenever ( ) and and 푥∈퐴 ∈ ∃푥 true 퐶 true 퐶 true 퐵 ( 퐶 true 퐶 푑 퐵 + 퐴 ∈ 푐 푎 ) ) 퐵 ()∈퐶 ∈ 푑(푥) 퐴 ∈ 푎 ∧ . Indeed, if we have a have we if Indeed, . [ ( ( : 퐴 ∧ 퐵 true 퐵 ∧ 퐴 ) 푥 ,퐵 퐴, ∈ 푥 퐵 ) true ( ( 퐵 ∧ 퐴 ∈ 푐 푥 푎 ⬚ ) ) prop true 푥∈퐴) ∈ (푥 퐴 ∧ 퐵 → 퐴 true 퐴 → 퐵 ∧ 퐴 ( 푥 ( ) 푥 ~ ] prop )

1 퐴 true 퐴 true 74 . Thus, Thus, . D ( ( ,푑 푒 푑, 푎, 퐴 true, 퐵 true 퐵 true, 퐴

~ 퐴 set 퐵 set 퐵 set 퐴 )

( 푥∈퐴) ∈ (푥 D 푥 ∧ ∧ ∧ 퐴 true 퐴 퐵 + 퐴 ( ) 푗 푥∈퐴) ∈ (푥 푖 -elimination -introduction -formation 휆푐 ,푑 푒 푑, 푐, ( ( ) 퐶 ∈ 퐴 + 퐵 set 퐵 + 퐴 푎 푏 (→-intr), ()∈퐶 ∈ 푑(푎) = 푝 . ∈ 푐 ( ) ) 퐴 ∈ 푎 퐵 ∈ 푏 푖 퐵 + 퐴 ∈ 퐵 + 퐴 ∈ ( ( ( : (∧-elim) 푐 푥 푖 ) ( ) ( )

푥∈퐴 ∈ ∃푥 퐶(푐) ∈ 푥 [1 ) 퐴 → 퐵 ∧ 퐴 ∈ ) ) ]

)

( ) 푖 퐵 (

푎 ( ) 푥 퐴 ∧ 퐵 true 퐶 true 퐶 true 퐵 ∧ 퐴 ) ) 푒 퐴 → 퐵 ∧ 퐴 퐴 prop 퐵 prop 퐵 prop 퐴 , then , ( by 퐴 true 퐵 true 퐵 true 퐴 푒 푦 ( ) ( 푦 퐵 ∈ 푦 Π 퐶 ∈ ) ( -introduction. Alter- -introduction. 퐵 ∈ 푦 퐶 ∈ 퐴 ∧ 퐵 prop 퐵 ∧ 퐴 푝 퐴 ∧ 퐵 true 퐵 ∧ 퐴 ( ( 퐶 true 퐶 푗 푐 ) : By the example the By : ( ( ) ( 푦 푗 퐴 true, 퐵 true 퐵 true, 퐴 ) 퐴 ∈ ( ) 푦 )

) ) and and WS WS

2019 푞 ( 푐

)

/20 ) ∈

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. erator a type containing twoelements first casefirst it computes One might think that the disjoint uniondisjoint thethink that might One However, as Beeson remarks in [ in remarks Beeson as However, The The canonical elements of Theory Set Constructive of Analysis and Comparison out thatout without judgements of the form i.e. type, some to respect with elements two of equality the asserting judgments have we although that of use direct make not will We 5.1.6 In theIn context of logic, found itmore convenient toinclude these extra rules for the disjoint union oftwo sets.

sition”, setof proofs) asserting that Given Given a type

퐈 ∨ ∨ ∨ -formation -elimination -introduction -formation

D 퐼 decides whether the canonicalthe decides whetherassociated element to -rules 퐴 and and two elements I -rules one cannot even formulate arithmetic within 푑 + ( 퐴 ∈ 푏 = 푎 푎 takes therole of disjunction: If we write

) , in , in secondthe 퐵 + 퐴 0

I are the “labeled” elements of -rules, nevertheless, we will briefly discuss them in this section. Note section. this in them discuss briefly will we nevertheless, -rules, and and , we cannot formulate such judgement as propositions. Indeed, it turns 푎 4 and and ], one runs into difficulties in defining in such an such in defining in difficulties into runs one ], 푎 ()∈퐶 ∈ 푑(푥) 퐵 ∈ 푏 and and 1 and and 푏 퐵 + 퐴 푓 퐴 ∨ 퐵 true 퐶 true 퐶 true 퐶 true 퐶 true 퐵 ∨ 퐴 푒 of type ( 푏 ( 푥 are equal elements of 푏 ) ) = . may be eliminated in favor of favor ineliminated be may { ~ 퐵, 퐴, 퐴 e 퐴 ∈ 푏 퐴 ∈ 푎 set 퐴 75 D , I 퐴 prop 퐵 prop 퐵 prop 퐴 ( ( ,푑 푒 푑, 푏, ~ ,푎 푏 푎, 퐴, 푖푓 푖푓 푥∈퐴) ∈ (푥 푥=0 = 푥 푥=1 = 푥 I 퐴 ∨ 퐵 prop 퐵 ∨ 퐴 ) 퐴 ∨ 퐵 true 퐵 ∨ 퐴 퐴 ∨ 퐵 true 퐵 ∨ 퐴 ( ) ,푎 푏 푎, 퐴, ()∈퐶 ∈ 푒(푏) =

( 퐵 true 퐴 퐶 true 퐶 (퐴 true) (퐵 true) (퐵 true) (퐴 is the set (or in this case, rather the “propo- 퐵 + 퐴 ∈ 푐 푖 퐴 true ( , . and and 푥

) 퐵 ∨ 퐴 퐴 ) ) set : 퐵

퐌퐋 denoted by for ( is of the formof is 푗 . ( 푏 퐵 + 퐴 )

) ( 푒 푥∈ℕ ∈ Σ푥 (

푦 ) we have rules: the ( 퐵 ∈ 푦 퐶 ∈ 푖 ( 푎 2 ) ( ) 푖 and and 푗 ( 푓 )

푓 ( 푎 ( 푦 , so Martin-Löf so , ) 푥 ) or ) ) , where where , 푗

( 푗 푏 WS WS ( ) 푏 . The op- ) . In the In . 2019 ℕ 2 /20 is

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. each each The The type The The 5.1.7 If indeed, Martin- In In the elemination rule, We set It then returns the corresponding element As an example, let us verify example,let us an As toequality.up

contains an element, thenelement, an contains Given Given ⊥ ℕ ℕ ℕ ℕ 퐈 퐈 퐈 -equality -elimination -introduction -elimination 풌 풌 풌 풌 -equality -elimination -introduction -formation ℕ 푘

푘 , wepostulate the existence of a 퐴 ∈ 푥 Löf’sset theory -rules are the only rules so far guaranteeing the existence of sets without any assumptions. any without sets of existence the guaranteeing far so rules only the are -rules ℕ ⊥ ≡ ℕ 푘 ℕ 퐴 ∈ 푏 = 푎 -rules -rules 0 corresponds to the empty set and takes the role of , weknow that 0 and observe and rule the , then

푅

푘 I 퐴 ∈ 푏 = 푎

( op I ∈ r ,푎 푏 푎, 퐴, ( 푥∈퐴 ∈ ∀푥 erates as follows: It executes ( ,푥 푥 푥, 퐴, ) contains a canonical proof . Finally, the equality rule says that saysequalityrule the Finally, . ) I ( ) 푘 ,푥 푥 푥, 퐴, and thus and -element set: 푐 ℕ ∈ 푐 푚 ) . true ~ R 푘 푐 휆푥 푘 0 saying that all elements of elements all that saying 76 ( 퐶 ∈ ∈ r . 푚 R 푘 ~ 푐 , for for ( 푐 ( ,푐 푐, 0 I ∈ r = 푐 0 0 ( 푐 , … , 푥∈퐴 ∈ ∀푥 ) ,…,푘−1 − 푘 , … 0, = 푚 ,…,푘−1 − 푘 , … 0, = 푚 퐶 ∈ I ∈ 푐 I ∈ 푐 I ∈ r 푐 0 ⋯ 푐 ⋯ ∈퐴 ∈ 푏 = 푎 퐴 ∈ 푏 = 푎 to find out to which element 푐 , … , 푚 true ⊥ ℕ 퐶 true 퐶 ( r 푘−1 ⊥ of this fact. On the other hand, if 푘 0 ( ( ( ℕ ∈ ,푎 푏 푎, 퐴, in in the proposition-as-sets interpretation. ,푎 푏 푎, 퐴, ,푎 푏 푎, 퐴, set ) 푘−1 ) 푘−1 ( ) ⋯ 푐 ⋯ I ,푎 푏 푎, 퐴, ( 푘 푐 = ,푥 푥 푥, 퐴,

) 퐶 ∈ ) 퐶(푐) ∈ ) ) 푚 푘−1

) ( r 퐶(푚 ∈

1 − 푘 ) is the only element of onlyelementthe is . . . 퐶 ∈ ( ) 퐴 1 − 푘 ) are equal to themselves.to equal are

)

푚 it converges. WS WS I I ( ( 2019 ,푎 푏 푎, 퐴, ,푎 푏 푎, 퐴, For /20 ) )

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. would now like to form to like now would The rational numbers, as well as basic operations on them are obtained from obtained are them on operations basic as well as numbers, rational The The The 5.1.8 Theory Set Constructive of Analysis and Comparison We give two examples two give We 5.1.9 The

¬¬퐴 of of rule usual the is This R ehr t yields it wether two rewrite rules. The successor rule tells us that there is no proof of the statement th statement the of proof no is there that us tells rule successor The rules. rewrite two successor ℕ ℕ ℕ ℕ ( ,푑 푒 푑, 푐, 푠 -equality -elimination -introduction -formation ( ℕ R 푥 isproved as usually:

) in the above rules gives us the possibility of recursive definitions: If If definitions: recursive of possibility the us gives rules above the in -rules postulate -rules the existence of the setof natural numbers: . ℕ 퐌퐋 ) will be computed recursively by recursively computed be will -rules in action 0 or not. If If not. or – Real numbers andtheaxiom of choice to ex falso quodlibet falso ex ℝ see how to work in in work to how see as the set of all Cauchy-sequences all of set the as 푐 cnegs to converges [ 퐴 true 퐴 → 퐴 . As usual, we can define negation as negation define can we usual, As . ( ()푒 퐶(0) ∈ 푑 ℕ ∈ 푎 ()푒 퐶(0) ∈ 푐 ℕ ∈ 푐 →⊥ 퐴 ] ( 푒 1 ( ( R →⊥ 퐴 푚̅, 푅 ( true ⊥ () ,푒 푑, 푠(푎), 퐌퐋 0 ()푒 퐶(0) ∈ 푑 ) ~ , ⊥true →⊥ ( . The first is a construction of the set of real numbers. real of set the of construction a is first The . R ) 푚̅, 푒 푑, 77 [ ( ⊥true →⊥ 퐴 →⊥ ,푑 푒 푑, 푐, R ~ ) ( ) R ,푑 푒 푑, 0, I true ) 푒 = ( ) ( ) ,푠 ℕ, (→-intr), [2] (→-intr), ,푑 푒 푑, 푐, . In this sense, the rules for rules the sense, this In . returns 푠 ( ( ℕ ∈ 푛 ,푅푎 ,푒) 푑, 푅(푎, 푎, (→ ℕ ∈ 푥 ℕ ∈ 0 ( ℕ set 푥∈ℕ 퐶 ∈ 푦 ℕ, ∈ (푥 ) 푛 ] ℚ → ℕ 푥 2 - ) 퐶(0) ∈ 푑 = intr),

) ) ( (→-elim) ℕ ∈ 0 , ,푦 푥, 퐶(푐) ∈

)

[1] 푥∈ℕ 퐶 ∈ 푦 ℕ, ∈ (푥 푥∈ℕ 퐶 ∈ 푦 ℕ, ∈ (푥 퐶 ∈ 푑 )

∈ ⊥ . To accomplish this, we will use use will we this, accomplish To .

( (

퐶 ∈

,푦 푥, ,푦 푥,

) ( ( ) 퐶 ∈ ) 0 푠 ( 퐶 ∈ 푠 ( 퐶 ∈ ) 푥 ( Otherwise, . ( 푥 ) 퐴≡퐴→⊥ 퐴 ≡ ¬퐴 푠 ) ( ) ℕ ∈ 푐 ( ( ) 푠 ( ℕ 푠 ( 푎

푠 ( 푠 ( in the usual way. We way. usual the in ) 푥 ( 푥 ( ) 푥 ) 푥 ) ) ) ) i executes it ) ) )

at ℕ -equality define -equality 0 . The rule rule The . is a successora is 푠 = 푐 WS WS ( 푐 푚̅ 2019 t see to ) and → 퐴 /20 Σ -

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. terminology. From now on, we will refer to sets as sets to refer will we on, now From terminology. crease the expressivity of our system. As announced at the beginning of the chapter, slightly chan Martin- 5 of interpretation an give to order In 5.2 instead ofinstead “ in functions all from Cauchy-sequences the separate to rules form the set nepeain ti means this interpretation, to a set of pairs of set a to { where Note that cific cific constructive. Interestingly, in roles and strengths differentin versions of set theory, sometimes extending theory the to be highly non- As another example let us consider the axiom of choice. It turns out that this axiom takes very divergent un ot o e eial: Suppose, derivable: be to out turns Here is how aderivation tree would look like: 퐵 we can define the function Martin-Löf himself changed his terminology in this way. way. this in his terminology changed himself Martin-Löf :퐵 퐴: ∈ 푥 ( 푥 푐∈ [푐 )

ℕ ∈ 푛 ) 퐶 Additional rules ( Löf’sset theory ( 푥∈퐴 ∈ ∀푥 ,푦 푥, ( 푥 Cauchy ∈ 푝 such that such ) 퐴 set 퐴 ) } 휆푥 { 푐 for with this set. our In case wecan set :퐵 퐴: ∈ 푥 ( 퐀퐂 ) 푝(푐 . Cauchy 푎 ∃ 퐵 ∈ (∃푦 ) ( ” 퐴 ∈ 푎 ∃ 퐵 ∈ (∃푦 ∈ ,푝 푎, ≡ 푝(푐 5 (

and “ and 푥 ( ( ) 푥∈퐴 ∈ ∀푥 휆푥 푎 ( 푚∈ℕ ∈ ∀푚 ∈ ) (

( ( 푎 ) ( 푥 푥 푥 Uig rjcin functions, projection Using . , where where , ) 푝(푐 . 푦 ) ) 퐵 ∈ ) ) ) is a modulus of convergence, i.e. a method computing to each } 퐴 ∶ 푎 ( )퐶 , i.e. the set of all elements of ≡ 푥∈퐴 ∈ Π푥 ( ( 푦 ( 푓 ) 푥 ( ,푦 푥, ) ( ( wehave 퐴 ∈ 푎 푒∈ℚ ∈ ∀푒 tobe ) )퐶 ” instead of “ 푎 푦∈퐵 ∈ ∃푦 ),

퐌퐋 ) 푎 ) 휆푥 ( ) ] is an element of element an is ,푦 푎, 퐵 1 , itsstrongest formulation 푞(푐 . ( and and 휆푎 푥 휆푐 ) ) ) ( (Σ-elim) ( ( . 푝 . 푥 퐂퐙퐅 ( (Π-intr) ≡ ℝ → 0 > 푒 | 푥 ) 푥 휆푥 [ ∈ 푐 ( ) ) 퐴 ∈ 푎 푝 ( )∈(푓∈ (∃푓 ∈ )) 푐 퐶 푛 푝(푐 . 푎 into into (

(

( ) 푎 , 퐴 ∈ 푎 (

,푦 푥, 퐵 ∈ 푥∈ℕ→ℚ → ℕ ∈ Σ푥 [ 2 푥∈퐴 ∈ ∀푥 푥 − ) ] ] ) (

2 . This shows that 푥 ( ) 퐌퐋

( ) 푛∈ℕ ∈ ∃푛 (Π-elim) (Π-elim) ( 푎 ), → ” and say that” and “ ~ 푛 + 푚 ) 퐴 휆푥 we need to discuss some further rules which will in- will which rules further some discuss to need we . Thus, Thus, . ( ( ) 78 and and 푥∈퐴 ∈ Π푥 푓∈ ∃푓 ( 푞(푐 . types 푦∈퐵 ∈ ∃푦 푝

퐴 ~ ( )( 푐∈ [푐 ) ) 푐 | satisfying some condition ( Cauchy 푝 ( 푚∈ℕ ∈ ∀푚 푒 < 푥 ( 푎 푎

) ) 푥∈퐴 ∈ Π푥 – is a proof of proof a is ) ( 퐵 )∈ )) (

) 푥∈퐴 ∈ ∀푥 ( ,푝 푎, and we will correspondingly write “ write correspondingly will we and ( . 휆푥 푥 푥 퐵 ∈ ) ) ℚ → ℕ 푞(푐 . ) ) 퐀퐂 푎 휆푐 ( 퐶 )( ( 푎 푐 ) 푥 ( 푥∈퐴 ∈ ∀푥 ) 푞(푐

( ( 푎 ) ( . ) | is is typeof 퐵 ∈ 푎 ,푦 푥, 푥 ∃ 퐵 ∈ (∃푦 ( , )

( ) 푥 휆푎 ( and and ( ( 푥 ) 푛 . Generally speaking, if we want to want we if speaking, Generally . ∃ 퐵 ∈ (∃푦 ∈ 푥∈퐴 ∈ Σ푥 푎 ∈ ) ) ) ) 푝 . ) ) By . ) 푎 푝(푐 (푎. 퐶 ∈ ) 푥 − ( (,푓 퐶(푥, 퐵 ( 푥∈퐴 ∈ ∀푥 ( 푞 푐 ( 푥∈퐴 ∈ ∀푥 (

푎 ( ( 푦 퐴 ( 푐 푎 ) ) ) 푛 + 푚 ” instead of “ Π ( 퐵 . In the propositions-as-sets the In . )퐶 ) ( 푎 ) ( -elimination, 푥 ( , ) 푦 ( 푎 ) ) 휆푎 ) ,푦 푥, ) ) ) 퐶 ) (,푝(푐 퐶(푥, )퐶 푎 푝 (푎, 퐶 ∈ (Π-intr) ( ad e a identify can we and ) 푞 . 퐵 ( ,푓 푥, ) | ( 푎 , this will correspond ] ,푦 푎, ( 푒 < ) 1 푐 )) ( ( , 0 > 푒 )

푥 푎 [ ) 1 ( (Σ-elim) ) ) ] ) 푥

푎 ) ) . ( )

, [ ∈ ) isan element 푐

) 퐴 ∈ 푎 푐 ( a some spe- ( 푎 (Π-intr) (Σ-intr)

푎 WS WS ) 퐀퐂 ) ) ) ] ∈ 4 퐴 퐴 . Thus, . 2019 , (Π-elim) (Π-elim)

ge our

[ ( 4 ty 푦∈ ∃푦 ]

pe /20 ”

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. for “well operation symbols to terms. Clearly, this has recursive character and would yield an empty resultempty an yieldwould characterrecursiveand has thisClearly,terms. symbols to operation form the term algebra given by 푠 In general, let general, In qa t te rt of arity the to equal given constantsgiven w Suppose, algebra: from example following the discuss us let rules, the stating explicitly Before case will help us in understanding. i nt ae n fntos f arity of functions any have not did however, the recursive character ofrecursivecharacter the however, As noted, the rules discussed so far are enough to develop a constructive set theory on its own 5.2.1 Theory Set Constructive of Analysis and Comparison stants. In our example, we need to fix values fix to need we example, our In stants. recursivelyfunctiondo we a defineHow Writing these ideas asrules, we have (we write { tem. In the following we will thus discuss the missing ingredients. We start with

ductive step, i.e. we need rules of the form of of appropriate functions prevent confusion, as in order to interpret the language of languagetheinterpret to inorderas confusion, prevent containing these sets and the using 1 ( 퓦 퓦 푠 + 퐴 퐴 ( 2 , -introduction -formation . In our example,. In some terms are represented in the following way: ” 푎 (we could thus add another line to the table we discussed in chapter

) + 푏 + } 풲 . We can then define the term algebra term the define then can We . - order”). Although we will need only a special instance of these rules, understandi -rules -rules ) and so and on. 퐴 푎 be a set of operations and for each operation each for and operations of set a be and and ∗ 푓 푏 I or example our In . and and , the unary operationunarythesymbol , 푔 . . ,푏 ,+ 푠, 푏, 푎, 풔 ( term 풔 풂 ( ) 풂 풃 ∈ 풂 퐂퐙퐅 풃 +

-signaccordingly. )

0 : It consist of terms, where “term” is defined as result of applying Tak t te, xmls f em are terms of examples them, to Thanks . , as there is no way to conduct induction on sets within the sys-the within sets oninduction conductto way no is there as ,

퐹 = 퐴 퐹 on thealgebra term on ( 푠 ( 퐹 { 풯 푡 ,푏 ,+ 푠, 푏, 푎, mt ucin퐵푎 풯 → empty function 퐵(푎) empty function 퐵 ( sup ~ as all functions from functions all as ) :퐵 푏: 퐴 ∶ 푎 푎 ) ) 79 푓 = and 1 ( ,푏 푎, + corresponds to ye퐵 type 퐴 푠 ~ sup 푠 ↦ ( and the binary operationbinarythe andsymbol } ,퐹 푡, ) 퐹 and and to label the tolabel function 1 ( ( ( ( 푠 풲푥 푏 ,푏 푎, ( 푎 퐂퐙퐅 푎 ↦ 푡 ) ) ) . Next, we define how define we Next, . 2 , 퐵 ) 퐴 : ) and and

( +

∶ we will introduce a special type speciala introducewill we ∗ ( 푎 ) 풯 ( ( 푎 퐵 푏 ↦ ) of 푏 ? First, we declare its value on all con-allonvalue itsFirst, wedeclare ? 풲푥 ) ( ) 퐵 = 퐹 푥 → 퐴 풯 → ) (

퐴 : , let , type 푡 퐵 ( ( 푥∶퐴) ∶ (푥 1 ( 풲푥 ( 푏 푥 ) 푡 +

∗

퐵 ) ) ) 퐵 type 2.3 ( ∅ = 퐴 : 풯 → 푥 2 ( ) ∗ ) . This will be convenient to ) ) :퐵 푏: 푔 = 퐵 be a set with cardinality with set a be ,

, where , 풲 퐵 ( 푥 ( ( ( -rules (here, ) 푠 푎 푡

) 푎 1 퐹 ) 푡 , = , behaves in the in- the in behaves 풯 → 2 푏 { ∗ ∗ 퐹 , , 1 + is an operation an is 푠 푠 ( ) ng ng the general ( } . We can thencan We . 푡 푎 and and 1 ) ) WS WS 퐹 , , . It misses, 풲 푠 ( ( 푉 퐵 푡 stands 푎 2019 2 in ( ) ) e are e if w if + ) 푏 + for ) 퐌퐋 /20 = e ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Martin-Löf Intuitively, 5.2.2 in a paradox like Russel’s. like paradox a in all typesall constructible from To 1 ∶ 푐 2푥 the the inducive step. If we define operator our (in case Note technically that speaking, we could include base the case (the case for operations with arity 푼 푼 푼 퓦 퓦 , 푠 ofmultiplication by fix ideas, let us think of -elimination -introduction -formation ( as the successor function and and function successor the as -equality -elimination 풲푥

푈 퐴 : -rules ’sset theory ) 퐵

푈 (

푥 can can be thought of as the “type of all types”. )

퐹 퐵 ∶ 푏 ( 2 ⋅ ) Thus, Thus, we would define

푇 = What we can do however, is to define to is however, do can we What T 풯 ( ∶ 푐 ℕ 푎 ( as coding terms in the language of arithmetic. Say, we interpret u 푎 ) 푑 푏), (푎, sup and the and ) ( ( 푑 ,푑 ⋅, → 풲푥 to be the function computing all inductive steps, and ) ( + ) we) can write 풲푥 퐴 : as addition function and we want to denote by denote to want we and function addition as ) ℕ 퐴 : 퐵 퐹 푘 ( 퐹 ) ( ) 푥 without using 퐵 푏 + 푎 ( ) 푑 = 푠 ( 퐹 퐹 ( 푥 ( ( 푡 ) 푎 푏 ) ( ) ) ) ) ,푏, 푎, ~ 푠 = 푎, = 퐹 = 푠 + 푡 = 푥∶퐴 퐵 ∶ 푦 퐴, ∶ (푥 80 휆푣 ( (

푏 푥∶퐴 퐵 ∶ 푦 퐴, ∶ (푥 푡 ~ T . 퐵 푈 ∶ 퐵 퐴 푈 ∶ 퐴 1 ) , ) 푈 ∶ 퐵 푈 ∶ 퐴

( ( 풲 푏 However, such a definition would clearly result 퐹 + 푏 ) ( ( ( ℕ -rules. , 푥 퐴 Π푥: 푥 퐴 Σ푥: T 푣

푘 ( ( ) 푈 ∶ 퐵 + 퐴 푡 ( ,푑 푐, 푑 , ∶ 푈 푈 ∶ 2 푥 푈 type 푈 퐴 type 퐴 푈 ∶ ℕ 푈 ∶ 퐴 ) ) ) ) . )

( ) ) 퐵 퐵 → for all for all 푑 푥 푈 (u 푎 푏)) (푎, 퐶(sup ∶ 퐶(푐) ∶ ( ( ) ( 푥 as the type of type the as 푥 ( ,푦 푧 푦, 푥,

→ 풲푥 )

푑

) 푥∶퐴) ∶ (푥 퐴) ∶ (푥 ( 푈 ∶ 푈 ∶ ( ( ( ,푦 푧 푦, 푥, 푥 푥 푘 풲푥 퐴 : ) )

) 푈 ∶ 푈 ∶ ) 퐶 ∶ 퐴 : 퐵

) (

( ) 퐶 ∶ 푥 sup 퐵 ) ∶ 푧 , ( ( 푥 sup ( small types small ) ,푦 푥, ∶ 푧 , ( 퐹 푣 퐵 Π푣: ( T ,푦 푥, )) the function the for the recursion ( 푣 퐵 Π푣: )) ( 푥

) 푎 WS WS – ) ( as basically 퐶 푥 ) ( 2019 ) 푦 0 0 퐶 ) into) ( , 푣 ( 푏 ↦ 푥 푦 ) /20 as ) ( ) 푣

) ) )

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ( 푑 5.3.1 Finally, Finally, we are ready to define the type of 5.3 Theory Set Constructive of Analysis and Comparison to get used to the notions: It is clear that definitionthis needs some justification. Let us, however, start with some simple examples cursion a by define we types,appropriate into theory set of language the in propositionsinterpret To 5.3.2 on recursion use often will we following the In ing ing

휆푥 푥 푉 Π푥: 푽 푽 푽 푽 ( ,푓, 퐴, 푦 . -equality -elimination -introduction -formation 푉

(

-rules (we write 푥 Interpreting ) ) 휆푣 퐶 The universe The The interpretation The in the in second case). Now define we { ( T . 푓(푥) 푥 ) ‖ ‖ ( Spoe w hv defined have we Suppose, : ( ( ̃ 푓 풙∈휶 ∈ ∀풙 풙∈휶 ∈ ∃풙 ( | ‖ ‖ ‖ ‖ 퐴 ∶ 푥 푣 ‖ ‖ ‖ ∀ ∃ 흍 → 흓 =휷 = 휶 ) 흍 ∨ 흓 흍 ∧ 흓 휷 ∈ 휶 풙흓 풙흓 푑 , ‖ ⊥ ) } ( ( ‖ ) ) ) 풙 풙 = 흓

흓 , where , where { ‖ ) ) ‖ ‖ 푉 ‖ ‖ 푓(푥) ( ( 퐂퐙퐅 ‖ ‖

휆푥

풙

풙

) ) 푓 . ‖ ‖

|

( 퐴 ∶ 푥

in 푥 푑 ) is such that ( and and 퐌퐋 푥 훼̅Π푥: } 푑 푉 → 퐴 ∶ 푓 푈 ∶ 퐴 instead of

T { ̅ ) ̅ 푓(푥) ( ̅ ( { ̅ 푦 훽 Σ푦: 푓(푥) ̅ ̅ ̅ 퐹 푑 푉 ∶ 푐 ̅ sets | ̅ ( 푑 퐴 ∶ 푥 ̅ 훽 ̅ | sup ̅ ( )‖ ̅ 퐴 ∶ 푥 ) ,푓, 퐴, ̅ , ̅ fr all for 훼̃ ≡ 푉 ̅ ̅ } ~ ( ( ,푏 푎, 푥 퐴 = } 휆푣 81 ) 푑 , ( 훽 = 푥 푈 풲푥: ) Σ:훽 (Σ푥: T . Π:훼̅)(Π푥: 푉 Σ:훼̅)(Σ푥: ) 퐴∶푈 ,푧∶ 푧 푉, → 퐴 ∶ 푓 푈, ∶ (퐴 ): ): ~ (with (with 푓 = 훽 Π:푉) (Π훼: Σ:푉) (Σ훼: 푑 = i te olwn wy o eie function a define to way following the in ( ̃ 푉 → 퐴 ∶ 푓 푈 ∶ 퐴 푓 ( ‖ ‖ ‖ 푦 ( 휙 휙 휙 푣 ) ( ̅ { ) ) ‖ ‖ ‖ ‖ 푓(푥) ,푓, 퐴, ) ‖ T 푥 ‖ ( ‖ ( 푑 , ℕ 훽 = 훼 → ( + 푥 × × . From the 푑 휙 ,푓 푧 푓, 퐴, 휙 ‖ 퐵∶푈 ,푧∶ 푧 푉, → 퐴 ∶ ℎ 푈, ∶ (퐵 ‖ ,푑 푐, 0 ) ( ) ( ( 휙 푉 type 푉 휙 ‖ (

‖ for ,푦 푧 푦, 푥, ‖ ) 훼̃ 훼̃ | 휆푣 푦 훽 Π푦: 휓 휓 ( 퐴 ∶ 푥 휓 ( ( 퐶 ∶ ) ( 훼 훼 ̃ ‖ ‖ 푥 ‖ 푥 T . ( 퐶(푐) ∶ ) ) )

)

퐴 ∶ 푥 푥 ) ‖ ‖ ( )‖ 퐶 ∶ )‖ ( ( ) ) {

̅ } ) ,ℎ 푧 ℎ, 퐵, 푓 ‖ 푓 푥 = (

푉 ∶ (

( ( 푥 훼̅Σ푥: 푣 풲 푥 { W te set then We . 푓(푥) ) ) ( 푑 , -rules we can infer the follow- | in the first and and first the in 푣 퐴 Π푣: 퐴 ∶ 푥 ) ) )

퐶 ∶ ) ‖ | 퐴 ∶ 푥 훼̃ 퐶( ∶ ( } ( ) { ) 푥 퐶 ℎ(푥) . ( ) ( { } 푣 퐵 Π푣: 푓 푓(푥) 훽 = ) ( 푣 | 퐵 ∶ 푥 ̃ 퐹 ) ( ) | ) 푦 ( 퐴 ∶ 푥 ) 퐶 { )

푓 ‖ ( } ( ℎ

) 푥 ( 푑 } WS WS ) 푣 ) n easy re- easy n ( | ) 퐴 ∶ 푥 ,푦 푧 푦, 푥, ) ) 2019

} ) ) ∶ 퐹 /20 = =

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Indeed, thus beidentified. Indeed, with It isdefined using recursion with ℕ Martin- rules andrules finally axiomsall of as well as axioms logical all for expressions validating find must we theorem, this prove To Ⓜ Ⓜ We then set Let us now justify the definition of Definition5.2 and thisand type stands for As As a slightly more interesting example, let

Examples By observations these and aneasy induction onthe structure of formulas we obtain: canonical empty function and if we define we if and function empty canonical resent the empty set and we will prove this in section 5.3.4. If 5.3.4. section in this prove will we and set empty the resent ‖ 2 m emp = emp , where Theorem5.3 Lemma5.1 ( 휆푥 Löf’sset theory 푑 ( . ( ‖ 퐹 : Let : ,ℎ 푧 ℎ, 퐵, 훽 =훽 = 훼 ̃ ( ( ( 0 ̅ { ‖ 0 푓(푥) ‖ 푒 , =훽 = 훼 : For each restricted formula ) . Note these that lines of proof do not work with some other type : sentenceA ℕ ∶ 푒 푝 : All theorems of emp = 푑 ) ) ‖ , ) ( | 푈 → 푉 ∶ 퐴 ∶ 푥 ,ℎ 푧 ℎ, 퐵, 퐹 ≡ 휆푦 0 ‖

푉 → 퐹 ≡ ( . and and ( } 훼 ) ) ( ‖ ≡ ≡ 0 and and we have with )( ̅ 훽 = 훼 be the canonical empty function. Intuitively, function. empty canonical the be ( (,푓, 푑(퐴, ≡ = 훼 푒 , ( 휙 휆훽 훽 훽 푎 )( 푥 훼̅Π푥: ̃ 휆훽 푝 ( ) ( ( ) 푥 . 훽 1 훼 ≡ 퐂퐙퐅 [ )∶ )) . 1 ) ) ‖ ( 1 [ 푥 , … , 푒 , 푥 퐴 Π푥: ( . 훼 , … , ( emp = 퐂퐙퐅 푝 ) 푥 퐵 Π푥: 푥 훼̅Π푥: 푈 → 푉 ≡ 퐶 ( , which we will do thein following. asabove, 휆푣 ‖ ( 푦 훽 Σ푦: 푥 ℕ Π푥: 훽 = 훼 푛 are valid. 퐹 . ) 푛 ) ( ) ) ) is ( . Intuitively, 푦 훽 Σ푦: ( ( ̅ )‖ ∶ 푓 푦 훽 Σ푦: 푦 훽 Σ푦: 1 valid ‖ ( )( ‖ 훼̃ 푉 ∶ 훼 푣 . This type clearlyy aims to mimic the axiom of extensionality. 휙 휙 ( 푒 푦 ℕ Σ푦: ) ̅ ()≡T ≡ 퐹(푥) and ) , 푥 ( 푝 ) 푦 ̅ 퐹 ̅ ) ) ‖ 훼 ) iff there is an expression similarly, then for then similarly, ()(훽 푧(푥) 퐹 ( 휙 1 be given by 훽 = ~ 푓 ( 훼 , … , ‖ 2 훼̃ ( has small has type. ) 푥 ̃ 82 ( ‖ ( ) 푥 훼 푦 m emp = emp ) ̃ ) ( 푛 ) ~ and and ) ( (훽 ,푑 푥, ‖ ) 푦 (훽 ̃ ‖ ) × (

̃ × ) 푦 ( ) ( ( 훽 푦 ) : : 훼 푦 훽 Π푦: × ) ̅=ℕ 훼̅ = ) represent the sets 1 ( × ) 푦 훽 Π푦: ,…,훼 , … 푉, ∶ ( ‖ 푦 훽 Π푦: 푒 ̅ ) ( × 푝 ( 푦 훽 Π푦: 1 푒 푥 푥 훼̅Σ푥: , where 푝 ̅ ) ℕ ∶ ( ( 푦 ℕ Π푦: ≡ 푥 퐵 Σ푥: ̅ ) 푛 0 ( ̅ ( ) ) 푥 퐴 Σ푥: 푎 m = emp 푉 ∶ 휆푥 ( → ‖ 푥 훼̅Σ푥: ( 훼̃ 훼 ) 2 푒 . ( 훼̃ ) ()(훽 푧(푥) )( ( 1 푦∶ℕ ∶ Σ푦 . 푥

( 훼 , … , 푝 ) 푥 푥 ℕ Σ푥: 0 ) 퐹 , ) ) { 휆푦 퐹 훽 = { ( ∅ 푒(푥) 푓 퐴 emp = ( } 푒 . ̃ 훼̃ ( 푛 instead theone of ̃ and and 0 1 ( 푥 ) ( ( ) 푝 ) 푦 푦 ) such that such 푦 푥 ‖ ‖ | ) ) ) ℕ ∶ 푥 ) ) 푒 m emp = emp we have that that have we ) ‖ (훽 ) ( ] and and { . 푥 (훽 . ̃

,∅ ∅,

) ( ̃ 푦 0 ( 푒 = ) } 푦 } 푉 ∶ 훽 should rep- should ) ) and and should ] ) WS WS ( . inference

푦 ) by ‖ ‖ 2019 is the is ,

훽 ℕ 푒 /20 푝 ̅ 0 = . ∶

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. rtto. eadn lgcl xos w tu cnet usle wt te axioms the with ourselves content thus we axioms, logical Regarding pretation. propositions the in as same the is connectives logical the of interpretation our that Note 5.3.3 Theory Set Constructive of Analysis and Comparison Ⓜ To give a validating expression hensible: will be rather technical. We will skip the technical details to make the argumentation a Finally, we need to verify the axioms of identity. Naturally, this is done via recursion and f

‖( esting esting forms (in (EI), take (EI) and (UG)rule. Ponens Modus the establishedhave already inference,we of rules for As set

construction should be carried by out a kind of double recursion we used to define member left the symmetricallydefinedto is “…” where As a final example, us give a validating expression for expressionvalidating a give us example, final a As 훽 훼̃ = (푥) o tastvt, assume, transitivity, For 푝 to make direct use of recursion: Let recursion: of use direct make to Then 푒푙 above, a validating expression formula the of structure the on induction easy an by can, expressionswe above the by that Note ( 2 → 휙 푞 ( Lemma5.4 푟 ( ,푥 푐, 0 푐

( )( 훼 푒푙 ( ) Validating 푦 ) 휒 → 휓 1 훾̅ ∶ 푦 = ) ̃ ( = ) ( 푐 푦 ) and 휆푥 ) 훽 ∶ . Thus, 푤 (,푟 ((푥, . ) : For any formula 0 ) ̅ and ≡ → and 푝푟 푎푐 푏 휆푎휆푐. 휆푐 ( ( 퐇퐏퐋 0 ,푦 푐, ( 푐 ( (푒푙 . 푎 휓 → 휙 0 푝푟 푝푟 훼̃ does depend not on (훼̃ ) ( 푏 ( 푒 2 1 ( 푥

,푐 푎, 휙 푝 ≡ 2 ( ( ( ∶ 푎 푐 ( ) ,푐 푎, 푐 ( 푐 푐 0 of 푥 ) ,푒푙 푐, ) ) 0 ( ) ) 푥 푟 (푥, , ) ) ,훽 훼, ( ‖ 푝 ≡ 훽 , (훼̃ ∶ → ) → 푦 = 푥 ∶ ∶ 푞 휙 ̃ ‖ ∶ 1 ( ( ( ‖ 푟 ‖ ( 훽 = 훼 푐 푦 ( ) 푥 0 휙 ( 휙 → ∀푥 휓 ∀푥 → 휙 휒 → 휙 ‖ 푉 ∶ 푐 , 푐 )( ) 푞 ( ) ≡ 휓 there is avalidation expression ) 0 ) 훼 훽 , ( 푦 ) ( ( 푞 ( ) ̃ 푡 , 휆푐 ) 훼̃ 푐 푝푟 ( ∶ 푐 ( ) ∶ ) ( ( 0 푦 푐 ) ( ( . ‖ 푥 Te for Then . 푒푙 ) ) ) ( ‖ 훼 훽 (훼, 휙 )‖ ) ) ∶ ) ,푦 푐, 휆푦 훼 = 훼 ‖ 1 ( ) (푥) ( 훾 ∈ 훽 ∧ 훽 ∈ 훼 푥 )) wehave dealt with before. ( 푐 푐 . ) 푥 ) ‖ 푐 ) ( . ) ) ) 0 )‖ 휙 ↔ ): ̃ 푥 훽 훼̃(푥) = 푒푙 : : ‖ ∶ ∶

(훼̃ ‖ ‖

1 we may assume I w g o t set to on go we If . 훽 ‖ ( ~ ( ̃ ( ()=훾 = 훽(푥) 푐 UG ( 푦 훼̃ = (푦) 푒푙 푦 ) 83 ) 푐 ∶ ∶ 푐 ) ( ) 푒푙 , ) ,푦 푐, ̃

. ~ ( ‖ 2 푦 ‖ 푤 ( 훽 = 훼 ) ) , and set and , ( ) ,푒푙 푐, of the pair. Note that technically speaking,thistechnicallythat Notepair. the of 휆푎 0 푥 훽 → ( 훽 , 푦 ) ∶ ̃ 푏 . ‖ ) ( 1 ̃ ‖ ‖ 푦 . W ilteeoedfn Wewill therefore define ‖ 푦 훼̃ = (푦) ( ( 훾 ∈ 훼 → 훾 ∈ 훽 ∧ 훽 ∈ 훼 ) Atgte, e il eie with define will we Altogether, . 푐 푎 ) and and ) ( ) 푟 푏 ) ∶ 푎 ( 0 푒푙 ∶ ( (푝푟 ) 푝푟 ( 푒 푎 훼̃ 1 휙 ‖ ) ‖ ( 훽 ∶ 푦 푐 푝 ≡ (푐) 푥휙푥 휓 → 휙(푥) ∃푥 ( of ,푦 푐, 휙 ( ∶ ∶ 푥 푥 1 ( ) ‖ ( ) 푐 ) ∀푥∀푦 ∀푥∀푦 푒푙 푐 ) ‖ 휓 ) ) ∶ ) ‖ ̅ Lt s abbreviate us Let . 2 ‖ ), … , 푝푟 , , 푐 )≡푞 ≡ 푥) (푐, ) ‖ ( 푒푙 ̃ 푥 훼̃훼̃(푥) = 푝 2 [ ( 푐 훼̅ ∶ 푥 = (푐) ( → 푦 = 푥

푐 ,푒푙 푐, ) ) ‖ and

1 ( ‖ ‖ ( ( EI 훽 = 훼 . Here, we do needdo we Here, . 푐 푞 ( ) ) 푥 ( ‖ bit more compre- 푝푟 )

( 푞 ) 휙 → 휓 ∧ 휙 )). 휙 ‖ ( 1 -as-types inter- -as-types with . Thus, we will 푐 ( 푐 푞 ≡ (푐) ‖ or this reason 푥 )

. ) ) (푥) WS WS 휙 ↔ 푒푙 푝푟 ) ( 2019 ( ,푦 푐, ( then , 휙 ( ‖ 푝 ,푦 푐, inter- 푦 푡 find ( and 0 ) 푐 ) )] /20

) ) as ≡ ) . ∶ .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Set Empty set eiiin Lma 5.5 Lemma & Definition Before starting to validate axioms, the we a need simple observation: 5.3.4 Martin- Ⓜ Now if Now if we have impossible, since there is no no is there since impossible, On the other hand, let hand, other the On 푝 such such a way to satisfy this axiom: Let The The validation of extensionality will of course exploit the fact that the definition of Extensionality 훼 훼̃ ing thating Proof

훼 validates Given setsGiven Pair istion symmetric. Now if Now if Infinity ℎ { type Let Union ̃ 푓(푥) ( ( ( ∗ ∗ 푏 훼 ( 푥 ( Lemma 5.6 Lemma 푥 푥 ) 푚 ≡ 푒푚푝 ) ) • • 훼 ( ) ) (

Let : ‖ 훼̅ ∶ 푥 =

| b ast n define and set a be 푦 푗 ℕ ∶ 푥 ∶ ∶ ∶ ∶ ∃푧 ( ) Löf’sset theory Validatingtheaxioms of Or 훼 Either 1 ∶ 푐 ‖ ‖ ‖ 푆 ) ( ‖ . Using a . Using validating expression for transitivity of equality, we can validate 훼̃ 훼̃ ( ‖ ) 푧 ∈ 훽 ∧ 푧 ∈ 훼 (where 훼 푝 훼 ( ( ‖ ¬ { 2 훼 훼 = 푥 푥 푓 ( 푆 ∈ 훽 ) } with be a set and define define and set a be and and ( 푐 ) ) isthe successor of . Then, Then, . ( 푥푥∈푒푚푝 ∈ 푥 ∃푥 ) 푝 푥 : For each set each For : 훼 ∈ 훼 ∈ ( . Note that 푗 = ) 푐 | ∶ 푐 ℕ ∶ 푥 ) ( 훽 푙 ‖ ‖ 푞 훼 ( is defined on 푖 = , define , define and thus and

1 ( ) ‖ ( 푏 ) 훾 ‖

,푥∶훼̅ ∶ 푥 푉, ∶ 훼 훾 ∈ 훽 ( ) , then 0 ∗ ̅+ℕ 훼̅ + ∶ 푥 ) ( } ∶ ‖ ) 0 , where , 푉 ∶ 훿 ‖ . ) ) ̅+ℕ 훼̅ + ∶ 훼̃ = 훿 ‖ I w set we If : ∶ ∶ ≡ 훾 ‖ 푆 푓 . , then 훼 ( ( ‖ 1 , 훼 (푐 푝 푐 we can find an expression an find can we ℕ ∶ 푥 훾 ∈ 훼 . Then ∶ 푎 ) ) { ( ) 푔(푧) 푓 1 푅 = 푠 = 훼 퐵 + 퐴 푥

, i.e. , ( ‖ is the canonical empty function empty canonical the is . ) 훼̃ 푝 훿 ∈ 훽 0 푆 ‖ ‖ ( ( that could be used to validate to used be could that ( 2 ( Te atr yields latter The . 훼 퐂퐙퐅 | 푞 and and 푐 훼 푥 훼 ( 퐴 ∶ 푧 and and ( ) 훼̅ ∶ 푥 as ,훼 훽 훼, 푐, ) ) ) 훼 푐 ) in in fact stands for = ) ) ≡ ∗ ‖

푙(푖 ∶ and and 훾 ( ≡ ( } . But then But . { 훼 ,푦 푥, where , 훽 ∗ ‖ ℎ ( ) ( 휆푥 ∗ 훼 = 훽 be sets and 푎 ( , i.e. 1 ( 훼 ) 푥 ) ) . ∶ 푏 푎 = ) )( and and ) ( ∶ ∶ ) ,푟 푥, ~ 푦 ℕ ∶ 푓 ‖ ‖ ‖ ∶ ) 훾 ∈ 훽 84

훼 ∈ 훿 | ≡ 퐴 0 ‖ and ̅+ℕ 훼̅ + ∈ 푦 ( (훼̃ 푞 훼̃ ,푞 푦, ( ~ 2 ( ( 푐 푥 푝 ( 푥 푉 → ) ‖ ∶ 푐 ‖ 푙(푗 푥 훼̅Σ푥: ) ( ( ) ∀푥 . Therefore, . 푆 ∶ ∶ 푐 . Then . 푞 ) 훽 ∈ ( ) ( ( with with ‖ 훼 ) ‖ ( 푏 with with 푏 훼̃ = 훽 푆 ∈ 푥 ) ∀푧 ∀푧 ) ) ∶ ∶ ) 1 ‖ and validate the formulathevalidate and 푏 = ) ) ̅ 훼̃ } ‖ ̅ ( Σ:훽 (Σ푦: ≡ , where where , ̅ ( ( 푧 훼̃ ∈ 훽 ̅ 푥 푝 훽 ∈ 푧 ↔ 훼 ∈ 푧 푟 ) ̅ 푓 ( 0 ̅ ( ) ( ℕ 푥 훼̃ ∶ 푦 = ( 훼 ) 푎 and a bfr, i.e. before, as ‖ 0 )‖ 0 ) ) (훾, 푒푚푝 ∈ 훽 ) ( 푉 → 훿 ∶ 푧 = ↔ which shows which 훼 = 푥 ̅ ) ) 푔 ℎ ‖ ( ( ̅ ‖ ̃ ( ̅ 훾 훼 = 푥 ∨ 훼 ∈ 푥 훼̃ ( . If If . and and ̅ ( and and ( 훼 ∗ ̅ 푥 ( ,푦 푥, ̅ ( 푥 ) ̅ ) ̅ with 0 ( 푉 ∶ 훼 ( ) with ) 푖 푥 ) ) ( 훾 , 훽 = ( 푓 ) ‖ ) 푥 ,훼 푥, ‖ ( . Then for ∗ ) 훼̃= 1 . This shows that that shows This . ̃ ( , then then , ) 푟 푞 ( ) 1 0 ‖ 푞 푦 ( ∗ 훼̃= ( ̃ ) ( 훼̃ = 훽 훽 = 훼 훽 = ( ( 푎 푙(푝 ) ) 훼 ( 푥 푞 ‖ ) ) 푥 ) ) ( ) . The other direc- other The . is of the desired the of is ) ( ∶ ) ( 푆 푏 ) ∶ . Let ∶ 푐 ( 푥 푐 ‖ ̃ . ( ) 푦 ∶ ) ) ‖ 훼 ( 훿 = 훽 ) ‖ ) ,푞 ), for 훼̅ ∶ 푥 푥 ( 훼 = 훼 ‖ ‖ is tailored in ) (by 푧 ) 훼̃ 푒푚푝 ∈ 훽 푠 = 훾 ) ( ( WS WS ( 푦 bethe set 푐 ∶ 푥 ̃ 훼̅ ∶ 푥 ) we have ) ( Σ ) . ( ) ‖ 푧 ‖ -rules). 훼 2019 훼 ∈ 훿 ) ∶ ̃ then , ‖ ) ( say- 휆푥 ‖ and and 푧 ‖ ∈ 훽 ‖ ) /20 is . 푥 . =

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

ecnnw set now can We Combining these cases,two we find the validating expression The otherThe direction is similar: Let Altogether, 푆 have have shown how to validate equality equality to validate On the otherthelet hand,On on Let Bounded separation

type by Lemma 5.1 and Let Strongcollection validating pression gether with On the other hand, for hand, other the On 휆푥 ‖ ( 훾 ∈ 훽 Δ 푝 . ℕ ( • • • • 훼 훼 푛 (

: be a set and set a be be a set, ( 푐 ) 휆푥 ( ) ‖ If If If validate 푠(Δ If 푥 . The last equality shows that, for any we have we . ) 푞 푖 ∶ 푐 푖 ∶ 푐 ℕ ∶ 0 = 푛 ( 푠 = 푛 ) ( ,푞 푥, ( 훾 ∈ 훽 . Now for all 푐 푚 휆푐 Ⓜ ) ( ( ) ∶ 푏 푏 ( (푙(푝 D . 휙 푆(Δ = ) ( 푐 ) ) Lemma ‖ Δ 푚 ( with with with with . )( 휙 ,푦 푥, ( ≡ 휔 ) 푛 푝 푥 ( , then , 휙 w cn use can we , 훽 ) ( ) 푠 푐 ) ) a bounded formula. We set We formula. bounded a ) ∅ = ( ( a formula and , ) ‖ ∶ 푏 ∶ 푏 ( { 푐 Δ 휆푦 푚 5.4 tovalidate ℕ ∶ 푛

Δ = will yield a validation for validation a yield will ) ( 푉 → 퐴 ∶ 푔 ∶ 푐 Δ (푙 , 훼̅ ∶ 푥 ( . 푛 ) . ‖ ‖ 푛 ( ) ( ( ) 훼 ∈ 훽 훼 = 훽 ,푣 푥, 푛 from the lemma to validate ) ,푞 푦, ) ( | ‖ ) . We can easilycan validateWe . 푝 ℕ ∶ 푛 휔 ∈ 푠 휙 ∧ 훼 ∈ 훽 ( 푒푚푝 = ( ) ( ( 푐 훽 = Δ 푐 ‖ ) 훾̅ ∶ ‖ )( ( . ) , then , then 푚 푛 } 푞 , is defined by ̅ ), we), have ∶ 푐 푦 , where , , where where , ) te fact the , ) ) ( and wethevalidatingcan use and expression theaxiomfor of Empty set to ) 푐 ∶ 푐 푆 = ‖ ) ∶ ∶ ) ) 휙 ∧ 훼 ∈ 훽 훼 = 훽 ∨ 훼 ∈ 훽 푝 ( 푞 , ) ‖ 푆 = ‖훽 ∶ 푏 훽 ( ‖ 푏 ( ) ‖[ ( Δ ) ‖ ( 푥∈훼 ∈ ∀푥 Δ 푞 we have have we ( ( 푐 훼̅ ∶ 푥 = is defined by recursion on recursion by defined is 푥∈훼 ∈ ∀푥 푛 ( ) 푞 푐 ) Δ ) ℕ ∶ 푛 ) ( validates the implication from left toright. ) ( 푐 푔 . We can thus combine this with validating expression for ( 푛 ∶ ̃ ~ )( 훽 ( ( ) ‖ ) 훼 ≡ 훾 ( ) with with 푥 ∃푦 휙 ∃푦 훼̃ = 훽 ‖ Δ(푠 = 85 ,푣 푥, )( , . ) 휙 ) ( ‖ ‖ 휔 푦∈훽 ∈ ∃푦 푝 1 ∶ ( Δ . ( ~ { 훼̃ ( ) ) ∗ 푚∈휔 ∈ ∃푚 ( ‖ 푔(푢) ( ‖ 푝 ( ) 푞 ( 푛 ,푦 푥, ( ( 푠 . (훼̃휙 ( 푥 ( 훼̃= 푚 ) 푥 ( 푐 푏 ) 푛 ) ) ∨ 0 = )‖ ) ) ‖ ) ) ) ) | 푆(Δ = ) 퐴 ∈ 푢 ‖ ∶ 휙 , where , ) ( ( and all in all, we would have expressions have would we all, in all and 훼̅ ∶ 푥 = 푥 . Define ‖ 푥 ) ∶ ( ) 휆푐 훼̃ = 훽 푛=푠 = (푛 ) ,푦 푥, for ‖ 훽 , ( (푙 D . 푆 푚∈휔 ∈ ∃푚 ̃ } ( ( , where , ) ( 푥 Δ 훼̅ ∶ 푥 푚 ] with ( ) ∶ 푣 ( ( ∧ 푥 ) ) 푛 ( 푚 ≡ 훽 )‖ ) ‖ 푝 [ ) ) ( ad vldtn epeso for expression validating a and ) ( . This shows. This that and and ‖ ℕ and also ) 푥∈훽 ∈ ∀푥 푐 ) . 휔 ∈ 휙 푞 Δ ) as { ≡ 퐴 ) ( 푏 ( ( (푝 , 푝 훼̃ ( 푛 푥 ∶ 푣 ‖ Δ ( ( ) 푐 . But in the last lemma we 푥 ) ( ( ‖ ) )( | 푠 = 0 ) 푙 훼̅ ∶ 푥 ) 푥∈훼 휙 훼. ∈ ∃푥 ‖ )‖ ( ) 푦∈훼 ∈ ∃푦 푞 휙 푐 ∶ 푒푚푝 = ( ) . We can use this to- this use can We . ( 푏 ‖ ( ) 푚 훼̃ } 푞 , ) 훼̃ = 훽 , where we set ) ( 훽=푆 = ‖훽 ∶ ‖ ( 푥 푙 using recursion using ) ) ( 휙 and )‖ ( 푐 푥 ( ) ( . Then for ) 푥 ) ,푦 푥, ‖ 푙 , ) WS WS ̃ is a small a is ‖ Δ ( ( . The ex- The . 푐 ) ( 훼 ]‖ 푠 ) ) 2019 ) ( ( . 푛 . 푥

) ) ≡ 푏 ) ‖ /20 ∶ 푐 . ∎ =

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Martin- sets Given Subset collection

훾 Naturally, this is validated recursion: using Let us abbreviate induction Set Suppose, we have defined and forand

we have that This showsThis that ‖ ∗ ∀푥 ∈ 푎 ∃푦 ∈ 푏 휙 푏 ∈ ∃푦 푎 ∈ ∀푥 ( 휆푥 푝 . Löf’sset theory ( 퐵 ∶ 푏 휆푥 ( 푐 ( 푥 푞 (푥, . 푥 훼 ) , and and ) ) validates ( 휆푏 푐 ( ( ,푦 휂 푦, 푥, 훽 . 푥 휆훼 define , )

) ℎ . , ) ) ( 휆푦 ‖ 훼 훾 ∈ 훿 W define We . )( ℎ 푦 푞 (푦, . such that such 푏 ≡ 훾 ≡ 퐵 ) does the job. . If . If ( { 푐 ( 푔 휆푥 훼̅ ∶ 푥 ( 훼∶푉 ∶ Π훼 ( 푦 푏 푧 ℎ . ) ( ) 푏 ) 훼 | ℎ ( ( )∶ )) 휆.푝 (휆푥. 훾 ≡ 훿 ( ̅→훽 훼̅ → ∶ 푧 ) ( 훼̃ 훼 훿 = 훼 ( ( ) ) 휆푥 ) 푥 ( ∶ ∶ ‖ ̃ ) ( = ), then), 푥∈훼 푦∈훿 휙 훿. ∈ ∃푦 훼. ∈ ∀푥 ) ℎ . 푥∶훼̃ ∶ Π푥 ‖ ( 휙 휆푏 푏 ( ( ) 훼̃ ~ ̅ } 푏 . 훼̃ ( ∶ , where where , ( 푥 86 ( ( ( ) 푥 푞 ) 푥∶훼̅ ∶ Π푥 훼 푐 ‖ ) ( ) ( ) 휙 )‖ 푐 ( ~ 푥 ( 푏 ( ( ) 푥 휆푥 훼̃ ) ) → ∶ ) ) ≡ ) ( ) ) ℎ . 푥 ≡ 푔 ‖ ‖ ∶ ) 휙 휙 ( ‖ )‖ ‖ { 훼̃ ( ( 휙 ( 휙 ≡ 퐵 훽 ,푦 푢 푦, 푥, 훼 ̃ 훼̃ ( 휆푧 ( → ( 푥 ) (푝 훼 ( 훼̃ ‖ ) . 푥 ) ‖ { ) ( , ) ( ‖

‖ 훽 ( 푥 휙 ̃ )‖ 푐 . ∀푎 푏 )

) (푧 ( ( ) 훿 , 푦∈훿 푥∈훼 휙 훼. ∈ ∃푥 훿. ∈ ∀푦 ∧ 푥 , 훼

→ 퐵 ∶ ) ( ̃ ( ) ) 푥 ( ∀푦 ∈ 푎 휙 푎 ∈ ∀푦 ) ‖ 푥 ) ) ). | ) 훼̅ ∶ 푥 | 휂 ,

훼̅ ∶ 푥 )‖ ‖ . This shows. This } 휙 } Let . ( ( Te, y em 5.5 Lemma by Then, . 푦 훼 ) ) ‖ 휙 → . Since for 휂 be a set and and set a be ( ( ,푦 푢 푦, 푥, 푎 ) )‖ WS WS , i.e. ) ‖ 푉 ∶ 훼 .

2019 , /20 ∶ 푐 ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. In In chapter 2.2 thus obtain sequences of rational numbers. Similarly, the closed unit interval In order to obtain the Uniform Continuity Theorem (every function (every Theorem Continuity Uniform the obtain to order In 6.1 ological model that we will in theuse independence and consistency6.4-6.6. proofs in Brouwer’ in role from independent is central mathematics, a plays that Theorem, Bar Decidable the that section, this in show will We 6 Theory Set Constructive of Analysis and Comparison 6 Here, nodes labelled bylabelled nodes ean topological space. We will motivate the usage of topological semantics by generalizing the classical Bool- toolOur topological main will be semantics, where logical each statement is interpreted asopen set a in

ous), Brouwer showed in a proof of rather metamathematical fashion the Decidable Bar Theorem Bar Decidable the fashion metamathematical rather of proof a in showed Brouwer ous), from which obtainedhe as acorollary the Fan Theorem form Continuity Theorem: and thorough analysis and discussion of discussion and analysis thorough and criat Proof: Fan-Theorem6.1 theorem and the Uniform Continuity Theorem: ers in accepting in ers finity, we will remain agnostic about this proof and follow Heyting, Kleene, Trolestra, Dummet and oth 퐴 퐴

Brouwer himself admitted this in footnote 7 in [ in 7 footnote this in admitted himself Brouwer

( 퐵 ∉ 푢 ∨ 퐵 ∈ 푢 -valued semantics to Heyting-valued semantics in section 6.2. In section 6.3 we will describe the top

Topological Semantics and Independence of Bar induction Barinduction of andIndependence Semantics Topological

Bar induction in Brouwer’s mathematics 푄 bar ( 푢 ( ) ,퐴 퐵, be the formula bethe 푄 ( (〈 BI we defined, in Brouwer’s spirit, the real numbers ( ) 퐅

〉) D means ) BI ) ) : . Although constructively unproblematic, not asit the requires acceptance of actual in- , which is exactly the consequence of [ BI 퐷 2 푖 itself as constructively sensibleconstructivelyas itself 푘 퐷 , implies 푖+1 2 푘 휔 ⊆ 퐴 ] for

푧∈휔 훼⊇푢 푥≤푧 훼 푧. ≤ ∃푥 푢. ⊇ ∀훼 휔. ∈ ∃푧

푘 < 푖 휔 퐅 . and

[ [ fan a(,푇) bar(퐵, ∀푢 ∈ 퐵 푄 퐵 ∈ ∀푢 bar 푢∈휔 ∈ ∀푢 and 퐈퐙퐅 ( ( 푇 ,휔 퐵, 퐵 ) . We will start with a brief discussion of the Bar Theorem in 6.1 in Theorem Bar the of discussion brief a with start will We . i a eial br in bar decidable a is ∧ 0 > 푘 <휔 휔 ] ( BI ) ( 푢 푧∈휔∀ 푥≤푧 훼 푧. ≤ ∃푥 푇 ∈ ∀훼 휔 ∈ ∃푧 → ∀푘 ∈ 휔 푄 휔 ∈ ∀푘 ∧ ) 퐷 11 . With these definitions we can now prove Brouwer’sprovedefinitionsthese now can Uni- With we . ∧ and and ] ] ~ 87 퐅 | 6 , see [ see , and adopting it as an axiom. For the original prooforiginal the For axiom.an as it adopting and 푥 ( ~ ∗ 푢 퐵 ∈ 퐹 퐅 . 〈 : . Itsatisfies the twoconditions of( 푘 70 〉) ] or [ or ]

푄 → 퐵 ℝ [ as members of the spread i.e. , 0,1 50 ( 푢 ]. From ]. ) ] ) can be given as binary spread with | 푓: ] 푥 훼∈퐴∃ .훼 휔. ∈ ∃푛 퐴 ∈ ∀훼 푄 → 퐵. ∈ [ 0,1 (〈

] BI ℝ →

〉) 퐷 , we can derive the Fan the derive can we , ,

is uniformly continu- uniformly is | 푛 퐵 ∈ 푆 BI of Cauchy- WS WS and 퐷 ) and) we 2019 푢∈ ∀푢 BI /20 ∎ D s - - .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. → mantics. Recall, that the Boolean algebra Heyting-valued semantics 6.2 induction Bar of Independence and Semantics Topological 7 validityof this of Because Definition 6.3 Definition a speaking, Bluntly logic. intuitionistic Heyting-algebra is aBoolean algebra, for but without semantics similar a give to want we if Heyting-algebras variables into into variables most natural semantics for intuitionistic logic and can be seen as generalization of Boolean- of generalization as seen be can and logic intuitionistic for semantics natural most we conclude that there is some is there concludethat we 푓 퐵 푓 Theorem6.2 value of its expansion as element of the spreadthe ofexpansion aselement its of value with respect to respect with Sketch of proof which showsthat seen by seen the following truth table: The The rest of the chapter is devoted to showing that in the context i.e. both Fan theorem and Uniformity Theorem can beproved without interpretation. For example, the law of excluded middle excluded of law the example, For interpretation. 0 → 푎 Actually, it remains to explain why why to explain it remains Actually, 푘 푘 0,1 and and ( ( 훽 훽 . As it is known, formula is classically derivable iff its truth value is value truth its iff derivable classically is formula known, is it As .

| ) 푛 , if only, if . A . A Heyting-algabra is a Heyting-valued semantics ) ¬ . Translating this into the language of analysis, this means that for all . A Boolean-valued interpretation ( : Given 훼

퐙 퐂 퐅 + 퐖퐂퐍 + 퐈퐙퐅 { : A A : 푏 | 0,1 푛 , , 훽 = ≔ 푏 → 푎 Heyting-algebra } 푓 . The The . isuniformly continuous. 푓 | 푛 , we define the functions . Effectively, we can write truth value truth 퐋퐄퐌 ⋁ (in the literature also referred to as “Heyting { :푥∧푎≤푏 ≤ 푎 ∧ 푥 푥: ): Every function):

Boolean algebra in every Boolean algebra, we will have to pass from Boolean algebras toalgebras fromBooleanpass to have algebra,will Boolean everywe in 푚 |

푦 − 푥 ( such that for all for that such | 휙 푦 − 푥 ,01∧∨ ≤ 0,1,∧,∨, 퐻, of a formula in the propositional language is then computed inside computed then is language propositional the in formula a of WS WS ( 푢 ) | 2019 | ofpropositional logic is a mapping from the set ofpropositional ∀푣 ≡ 2 < 2 < } 퐵 always exists. We define the define We exists. always 0,1 /20 푚+1 푚+1 풂 ퟏ ퟎ iff is defined on the set 7 (

푓 → 푣 = 푢 ) 푓: means that we may assume that that assume may we that means is a bounded lattice such that the that such lattice bounded a is 푆 푓 푎=1 = ¬푎 ∨ 푎 ~ → ¬풂 푘 . By . [ 0 1 푓 : 0,1 88 | ,훽 훼, 푘 퐋퐄퐌

[ 푓

0,1 ( ( 훼 ] 퐖퐂퐍 푥 ~ ¬풂 ∨ 풂 ) ℝ → and some and ] ) required tohold. 푓 = 푘 ℕ → 푓 − ퟏ ퟏ ( 푢

퐋퐄퐌 , for each for , iscontinuous. for all for all 푘 ) ( ( as 푦 푓 =

훼 ) | , holds true in classical logic, as is easily is as logic, classical in true holds , | 푛 푓 푘 퐈퐙퐅 2 < ) 푘 ( 푚 < 푛 . By ( { 푎 푣 훼 0,1 . ) , the assumption of 푘+1 ) ) 훼 , 푓 = 퐅 }

, there is someis there , with the usual operations of , applied to theformula BI , - , we have we ,

algebra semantics”, pseudo-complement ( 퐷 1 훼 . under every Boolean-valued every under ) 푘 푘 푥 , i.e. , there is some | 푚 푦 = 훼 pseudo-complement ofpseudo-complement 푓 푘 | 푛 assigns to assigns | 푛 푚 훽 = such that such . . BI 퐷 | 푛 of is too strong, [ 푓 → 푚 28 va 푎 such such that as 푓 ]) are the 푘 lued se- lued 푓 its ( 푘 훼 푎≔ ¬푎 ( | 훼 푛 ∧ 푘 ) ) , -th ∨ ∎ = = 푎 ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Theorem6.5 ofresult Given this completeness result (actually soundness suffices) we arereadyanother to give independence Proof be derived in propositional intuitionistic logic. ⟦ Ⓜ Theory Set Constructive of Analysis and Comparison how to verify the axiom the verify to how proof of Sketch valued interpretation. with

e u so te E-ue If DET-rule: the show us Let 푥 valued interpretation. Indeed, for ≤ 푎 variables of A ⟦ cial rolecial of the two-valued algebra vanishes): With this definition we now have a similar completeness proof as in the classical case (however the spe- We saythat a formula 휓 휙 and and we aredone. Heyting-valued interpretation Heyting-valued ⟧ ⟧ Theorem 6.4 퐻 퐻 ( : Consider the Heyting-algebra the Consider :

푝 푎 → 푏 of 1 = ( 퐻 all all formulas of ) means that means 퐋퐄퐌 = ) . But this is clearly the case, as ℒ 2 1 : The of law excluded middle ( for apropositional variable into into a Heyting-algebra : The completeness proof is very similar to the classical case. For soundness, let us show us let soundness, For case. classical the to similar very is proof completeness The : : : A formula is derivable in intuitionistic propositional logic iff it is valid in every Heyting- ⟦ 휙 휙 ℒ ⟧ is valid isvalid under aHeyting-valued interpretation iff into into → 푝 퐻 ⟦ ≤ ¬푝 ∨ 푝 ( ⟦ 퐻 푝 → 푞 휓 of a propositional language propositional a of : ⟧ ⟦ 퐻 휙 ⟧ ,푏∈퐻 ∈ 푏 푎, , and , and we immediately conclude 퐻 ⟧ 퐻 ) 푝 = , i.e. , 퐻 퐻 1 = . As usual, this interpretation extends naturally to a mapping ⟦ with the underlying set underlying the with ⟦ ⟦ 휓 → 휙 ( 휓 ∧ 휙 휓 ∨ 휙 퐻 ⟦ 푝 and , b ⟦ ) 푎 → 푏 → 푝 , ¬휙 퐋퐄퐌 ¬푝 ∨ ⟦ ⟦ ⟦ y the definition of “ ⊥ 푝 ⊤ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟦ ( 퐻 퐻 퐻 퐻 퐻 퐻 퐻 휓 → 휙 ~ 푝 → 푞 ) does hold ) not allin Heyting-algebras.Hence, cannot it is the largest element ( 퐻 푝 = 0 = 1 = = = = = 89 ) ⟦ ⟦ ⟦ ⟦ = 휙 휙 휙 휙 (

퐻 ~ ) ⟧ 2 1 ⟧ ⟧ ⟧ ⟧ ⟧ ) 퐻 퐻 퐻 퐻 퐻 퐻

¬ ∨ 1 = ∧ ∨ → 0 → 푝 = ⟦ ⟦ 2 1 ⟦ 휓 휓 = ℒ , then then , 휓

( ⟧ ⟧ = 퐻 ⟧ 퐻 퐻 ) 퐻 → is a mapping a is 2 1

→ {

”, 0, = 0 ∨ 1 = 푞 ( we have that 2 1 ⟦ 푞 휓 1 , ( 퐻 푥 ⟧ } ) 퐻 with . 2 1 ad h sa odrn. Then ordering. usual the and 푝 → 1 = 1. ≠ ⟦ 휙 ( 퐻 ⟧ 푏 ≤ 푎 ∧ 푥 But .

퐻 푝 ↦ 푝 ) ) 1 = 1 = → 푎 ⟦ . ( for every Heyting- every for 휓 → 휙 퐻 ( ) . Our a is such an 푐 → 푏 of propositional of ⟧ 퐻 ) WS WS 1 = = ⟦ 2019 휙 iff ⟧ ↦ 휙 퐻 ∧ 1 /20 → ∎ ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Con as usually: usually: as Proof 푅 values We define an define We 퐵 Definition 6.7 the symbols the to addition In models. classical consider and further bit a semantics classical of mimicking the take We 6.2.1 induction Bar of Independence and Semantics Topological ⋁ ∧ for all for all index sets eiiin 6.8 Definition tions from a given domain given a from tions realtions Lemma6.6 Before wepass onto first-order logic, we prove somebut easy, useful facts about Heyting-algebras: f osat symbols, constant of By soundnessof Heyting-valued semantics, 0,1 ( 퐻 -distributive law ) 5. 3. 4. 5. 1. 2. 3. 4. 1. 2. . If these these large infima and suprema always exist, we will have topostulate this for Heyting-algebras: 풟 : :

푡 푛 ⋁ ∧ 푎 Using 2 thein third step, Since [( If Obvious from the definition. [ If If 1 = 푎 → 푎 → 푎 Heyting-semantics forlogicfirst-order 1 → 푎 푑∈풟 푡 , … , 퐻 → 푏 → 푎 Rel 푏 ≤ 푎 푏 ≤ 푎 푏 ≤ 푎 ( : anyIn Heyting-algebra ⊥ ( 푐 → 푏 휙 and functions and ( 푐 ∧ 푏 ∧ and and ( if the arity if of 푛 푐 ∧ 푏 푎 → 푐 ( , 퐻 : A Heyting-algebra is called are ) , then , then , then Gvn frtodr language first-order a Given : 푑 ∨ - ∧ ) , term . 퐼 ) ) for existential and existential for → ( ⊤ )] , : ∧ 푏 ≤ = 푐 → 푎 퐻 퐻 ∈ 푝 ) are and and 푐 ∧ 푏 ≤ 푎 ∧ -terms, then so is 푐 → 푏 ≥ 푐 → 푎 1 = 푏 → 푎 푏 → 푐 ≤ 푎 → 푐 푏 ≤ 푎 ≤ 푐 ∧ ( by the following rules: All variables of variables All rules: following the by 푏 → 푎 푅 ↦ 푅 퐻 )] ( ¬ and and 푐 → 푏 -formulas and so is so and -formulas = 푎 ∧ , one interprets functions, relations and quantifier symbols as functions func-functionsas symbols quantifierand relationsfunctions, interprets one , 푅 ) Fun 풟 ( ∧ is 퐻 { onto itself, functions from tuples of elements of elements of tuples from functions itself, onto . ) ( ) 푞 hence , o rlto symbols, relation of , we define a define we , 푐 → 푎 푛 , we also have → 푎 = 푎 → 푎 = 1 푖 [( 푐 ≤ 푐 ∧ 푏 = 퐼 ∈ 푖 : and and 푏 → 푎 . . WS WS 퐻 ) ⋀

퐹 and for and any } 퐹 푑∈풟 → 푎 2019 ) ( 퐻 ⊆ (퐻) 퐻 푞 ⋁ ∧ 푝 ∧ ) 풟 : (푡 ( 휙 andhence, ( /20 푐 → 푎 complete . 푐 ∧ 푏 1 ( 푖∈퐼 퐋퐄퐌 푛 푑 푡 , … , Heyting-valued interpretation Heyting-valued 푅 푏 → 푐 ≤ 푎 → 푐

) 풟 → ( for universal quantifiers. While in the Boolean-algebra the in While quantifiers. universal for = ℒ 퐻 ~ ( ) )] 푖 ) cannot bederivable. 푏 ∧ 푎 푛 ( ≤ ⋁ = 90 푡 ,푏 퐻 ∈ 푐 푏, 푎, ∧ ) , if , if aritythe of if it is a complete lattice and it satisfies the following 1 ( , if ( ( 푡 , … , Var, 푖∈퐼 푏 → 푎 퐹 ↦ 퐹 푎 ∧ 푎 ~ ) 푐 → 푎 ≤ 푐 → 푏 Fun ∈ 퐹 = ( 푞 ∧ 푝 Con 푛 ( ) ) ) 푎 → 푎 . . , whenever , ( = ∧ 퐻 the following the rules hold: e,Fun Rel, , ) 푖 Var ( [( o fnto smos uh that such symbols function of ) 푐 → 푎 and and the arity of , 푏 → 푎

) are ∧ 퐹 is ( . ) 푏 → 푎 ) Te te drcin olw from follows direction other The . ) 퐻 푛 푎 ∧ o variables of -terms and so are all all are so and -terms 푡 .

1 of 푡 , … , ] ) ∧ ∧ 1 = ℒ [( to be a mapping a be to 푛 푐 → 푎 are 퐹 풟 . ( 퐻 푏 → 푎 to 퐻 ) - Var formulas -terms, 푎 ∧ { 0,1 ) constants , ] 푏 → 푎 = } 푐 ∧ 푏 = and the truth the and Rel ∈ 푅 푎 are defined ( 퐻 푎 푎 ↦ 푎 ) . ( . for 퐻 ) Con 풟 ∈ and ∈ 푎 ( 퐻 ∎ ∎ ) , ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. guaranteed. and and hence We verify the axiom 풟 oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison cific domain cific 8 Ⓜ Proof valued interpretation. Note that that Note of arity the write Heyting-interpretations of propositional logic to first-order logic first-order to logic propositional of Heyting-interpretations 푐 Var order logiccan be carried over tothe intuitionistic setting, see [ therefore come as no surprise, that the proof of completeness of this semantics known from We say that a formula a that say We Whether or not the expressions for universal and existential quantification are well-defined, are quantification existential and universal for expressions the not or Whether 1 a domain. 푐 , … , Theorem 6.9 Theorem . : Let us again exemplar 휙 푛 ( 푥 풟 ∈ ) 퐻 if the if variable ⟦ 푅 풟 ∀푥 휙 ∀푥 . Note that the Tarski-semantics is just a special case of this definition, where frua my r a nt e ebr of members be not may or may -formulas Th is . Although it is certainly true if true certainly is it Although . e verification of the axiomcan be now done onein line: : A formula is derivable in intuitionistic firsr-order logic iff it is valid in every Heyting- every in valid is it iff logic firsr-order intuitionistic in derivable is formula A : 푛 ( . If If . 푥 ) 휙 → 휙 ∀푥 휙 ∀푥 ⟦ 푅 and and ( 휙 ⟦ ⟦ 푎 ( ∀푥 휙 ∀푥 ∃푥 휙 ∃푥 푐 푥 ( ( ⟦ 1 푥 ⟦ ⟦ 푥 ) 휓 → 휙 푎 … , occurs freely in 휓 휓 ∧ 휙 휓 ∨ 휙 ⟧ 1 ) il 푥 … , ⟦ are ⟦ 1 = 휙 → y verify some first-order axioms: Let

¬휙 ( ( ∀푥 휙 ∀푥 ⟦ ⟦ 푥 푥 푛 ⊥ ⊤ ) ) ) by Lemma 6.6. 푛 ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ 퐻 ( ) 퐻 퐻 퐻 퐻 퐻 퐻 퐻 퐻 퐻 푐 -formulas, then so are so then -formulas, ( is ) 푥 푅 = 0, = = 1, = ⋁ = = ⋀ = = = , where ) ⟧ valid 푑∈풟 ⟦ ⟦ 푑∈풟 ⟦ ⟦ 휙 휙 휙 휙 ⋀ = (

퐻 ⟧ ⟧ ⟧ ⟧ ⟦ ⟦ ) 푑∈풟 풟 퐻 퐻 퐻 퐻 under a Heyting-interpretation iff Heyting-interpretation a under 휙 휙 (푎 휙 is a set, our set, isa ∧ ∨ 0, → → ( ( 푐 and and 푑 푑 ⟦ 1 ( is free for ⟦ ⟦ 퐻 휙 ⟦ ) ) 휓 휓 ) 휓 ⟧ ⟧ ~ ( 푎 , … ,

⟧ ⟧ 퐻 퐻 푑 ⟧ 퐻 퐻 휙 . 퐻 ) 91 ,

, ,

⟧ ( ,

푑 ≤ 푛 ( ~ ) 퐻 for the result of substituting 풟 ⟦ ) 푥 ) , , ) 휙 will be a proper class. class. proper willa be in in 퐻 휓 ∧ 휙 ( 푐 W cn o etn te eiiin . of 6.3 Definition the extend now can We . for ( 휙 퐻 ) and and suppose the existence of 푎 )⟧ , 69 1 휓 ∨ 휓 푎 … , ]: = 8 : For simplicity, we write we simplicity, For : 퐻 ⟦ be a complete Heyting-algebra and 휙 푛 , ( ¬휓, ∀푥 휙 ∀푥 ¬휓, a ∪ Var ∈ 푐 ) ⟧ ,

⟦ Con and and 휙 ( 푐

1 풟 ∈ 푑 푐 … , ∃푥 휙 ∃푥 depends on the spe- the on depends 푛 퐵 = 퐻 ) . As usual, we usual, As . for classical first- ⟧ 퐻 ⟦ 푥 ∀푥 휙 ∀푥 WS WS 1 = 푥 ( 퐻 0,1 in in ) for 2019 . It will fr all for ( 휙 푥 . ) ∈ 푥 ⟧ /20 is

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. In In the sections 6.4-6.6 we will prove that 퐈퐙퐅 But weBut know class asclass well, contradiction. Then Then Hence, Proof In In section 6.3.2, we will show that Heyting algebras give rise to Heyting-valued models of 0 1 we can define astrictly increasing class function can definecan a set function. Then there is a least Ⓜ induction Bar of Independence and Semantics Topological Ⓜ Proof Suppose, holds: UG-rule the that show We Heyting-algebra 퐻 domain properis a (hierarchical) class: We will need the next lemma, to show that the interpretations of quantifiers exist, even if the ⟦ ∀푥 휓 ∀푥 and and for all theorems of ad domain and Theorem 6.11: Lemma 6.10 . These results. These rely the on following theorem: : If : If : Let Cons ( ⟦ 푥 퐅 퐺 ) 퐓 ⟧ ⟧ is a bijection between the proper class isinconsistent, then, 퐹 and thus, and = ( be an increasing function, the proof for decreasing functions is similar. If 퐙퐅퐂 ⟦ 퐖퐂퐍 퐅 ⊢ 퐙퐅퐂 ) : Let: 풟 퐻 Cons ⇒ 퐻 e a interpret may we , Let (as witness (as of a formula 퐺 ⟧ , where where , 퐅 ⊢ 퐙퐅퐂 퐻 is a complete Heyting-algebra and ⊢ 퐙퐅퐂 ( 1 = 퐻 ⟦ ⟦ 퐈퐙퐅 Cons 1 + 훼 휙 → ∀푥 휓 ∀푥 → 휙 ⊥ 퐓 be acomplete be Heyting-algebra and 퐺 퐺 ⟧ be any first-order theory over the language . Hence, Theorem. Hence, 6.11 will yield . In sections 6.4-6.6 we will investigate a special Heyting-algebra, where 퐻 ( ( ( 퐓 0 휆 ( 0 = ) ) ) 퐙퐅퐂 ⟦ ) 푐 . 휏 is free for free is 퐺 ⋁ = 0 = 퐹 = ⟧ 퐻 퐻 핆ℕ ∈ 훼 훼<휆 ) and hence and ( ⊢ ⊥ 퐓 퐻 1 = ( 푥 Cons ⇒ 훾 , )

) ⟧ , , where 훾 퐻 WS WS ( 1 = 훼 for each the and by and hypothesis, 푐 such that such ) as we please and hence and please we as 푥 2019 , , for limit ordinals 휆. ( . in in BI 퐈퐙퐅 퐅 0 ⊢ 퐙퐅퐂 퐷 ∃! 푥 휃 푥 ∃! /20 퐅 ⊢ 퐙퐅퐂 휙 is not needed to prove the uniform continuity theorem from ¬BI + and occurs free in neither in free occurs and ⟦

휓 → 휙 퐹 ~ 핆ℕ ( :핆 퐻 → 핆ℕ 퐺: 푥 isconstant above ⟦ 퐷 92 퐻 ) is ⊥ and and ) such that such ) Uniform continuity theorem + latwt 훼 and 퐹 > least with훾 ⟧ 1 = ( 퐻 ~ 푐 ) 1 = ⟧ 퐻 Im 1 = , showing :핆 퐻 → 핆ℕ 퐹: 퐻 ( byrecursion: 퐺 .

(i.e. ) 퐻 ⊆ ⟦

휙 ⟦ ⟧ 휙 . This shows. This that ℒ 훼 ¬Cons ≤ ⟧ of set theory. Suppose that in , i.e. an increasing an or decreasing (class) ≤ ⋀ 휙 푑∈풟 ⟦ orem 휏 휓 nor 퐹 ( ( ( ( 퐙퐅퐂 훾 ⟦ 푐 훽 ) 휓 ) ) 휓 ⟧ 퐹 > ( 퐹 = ) hls n n complete any in holds ) 푑 . In particular, for given for particular, In . of . ) ⟧ ) ( 퐓. . W conclude We . (

훼 훼

) 퐹 퐻 )

is never constant, for all must bea proper 퐈퐙퐅 underlying , i.e. 훼 ≥ 훽 ⟦ ⟦ BI ⟦ 휏 ℒ 휙 . ⟧ 퐷 퐻 we ⟧ ⟧ ∎ ∎ ∎ = = ≤

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Example check check that for Definition 6. Definition definitions: The most important examples of a Heyting-algebra will be topological spaces. Let us recall the f 6.2.2 Theory Set Constructive of Analysis and Comparison The elementsThe of Proof: Set Proof It iseasyto check that this forms acomplete lattice. Moreover, the Definition 6.14 Definition sp such thatsuch

sion andsion the operations 6.13 Lemma

Proposition 6.15 ℬ

′ ectively. ℬ ⊆ 3) 2) 1) 1) 2) : Note that an arbitrary intersection of open sets it is not guaranteed to be open. Hence, we can

with For For For For 풯 ∈ 푋 풯, ∈ ∅ Topologies examples as Heyting-algebras of Equivalently: For all ⋃ For For : For any set any For : 푋 = ℬ = 풯 풯 풯 ,푈∈ℬ ∈ 푈 푂, 푓푖푛 : Every topology defines a complete Heyting-algebra, if we interpret the order as set-inclu- as order the interpret we if Heyting-algebra, complete a defines topology Every : = 푈 ′ ′ 12 풯 ⊆ { 풯 ⊆ 풪 ⋃ 풯 ⊆ . : Let : : Let : 풯 : Let ℬ iff iff iff iff ⋃ : are called ′ ⋃ with with there is . ℬ | , ℬ 푋 풯 ′ 풯 ⋀ 풫 ⊆ ℬ

orequivalently, if for all ′ be a set. A A set. a be 푋 be a topological space on space topological a be ′ int = 풪 ℬ ⊆ , 풯 ∈ 푂 풯 { 푓푖푛 1 ,푋 ∅, ′ 푂 ∨ ℬ . } 푂 ( finite: . 1 ′ 푋 open open ∈ 푥 푈 ∩ 푂 ∈ 푥 푂 ∈ 푥 ∩ 푂 ∈ 푥 } 푂 , ℬ ⊆ 2 ) and and ( . Then . Then ⋂ 2 푂 = 푈∈풰 ⋃ ℬ ∈ 풪 sets with with topology 푂 and and ⋂ ) 풫 1 1 . ( 풯 Pseudo-complements are given like this: and 푈∈풰 ( ⋃ 푂 ∪ 푂 → 푈 ∩ 푂 푓푖푛 . 푋 ℬ ′ 0 = 푈 ∩ 푂 푈 ∈ 푥 for some ) isthe basis of atopology iff are topologies on topologies are 풯 ∈ 2 푈 2 푂 ∈ 푥 and and or

= ) . .

0 topological space topological ⋃ 풯 ∈ 푂 for some 푂 1 ⋃ ~ 1 푂 ∩ { 푋 푈 :푂∩푂 ∩ 푂 풯: ∈ 푂 ℬ 푂 ∧ 93 . Then Then . 0 ′ and 2 . 풰 ∈ there is 2 ~ 푂 = 푈 푂 ∈ 푥

0 ℬ 1 풰 ∈ is called a called is 푋 푂 ∩ on on ℬ ∈ 푈 , called the called , there is 1

2 푂 ⊆ . 푋 with with ⋁ ∧ or simply or 2 } .

ℬ ∈ 푈 -distributive-law asholds, basis 푂 ⊆ 푈 ∈ 푥 trivial for with topology and 풯 푂 ⊆ 푈 ∈ 푥 if for all all for if 1 푂 ∩ discrete 2 is a set a is . 풯 ∈ 푈 . topology re- topology WS WS 풫 ⊆ 풯 ollowing there is there 2019 easily ( /20 푋 ∎ ∎ )

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. The following example on the space the on example following The oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological Example 6. role for us: ℕ lying tolying right the of Example 6. archetypeThe a of topological space is Euclideanthe topology of the real line: According to Lemma6. to According We can also think of think also can We ordering.Let check us that this defines a topology eie mpig n h ohr ieto: ie a Heyting-algebra a Given direction: other the in mapping a define ∗ is best thought of as a tree. The The tree. a as ofthought best is 2) 1) 1) 2)

( ⋃ ⋃ Let Let ,푏 푎, ⊇ ℬ 푈 푢,휆 푈 ∈ 푤 17 16 ) ∩ 푈 : Wedefine for : L ℕ = ⋃ 푢,휆 ( {( ,푑 푐, et 푢,휆 ,푛+1 + 푛 푛, ∗ = = ℬ is clear. ) 푈 ∩ { 휆 = 푢 푈 : } 푢휆 { 푣,휇 { ∪ 13 ( ( ( ( ∅, ( )| as nodes with length at least at length with nodes as ,푏 푎, ,푏 푎, ,푑 푎, ,푏 푐, ,푑 푐, , w.lo.g. { , we can define for eachdefinecantopologyfor we , ℤ ∈ 푛 ℕ ∈ 푣 ) ) ) ) ) ℕ ∈ 푢 f푎≤푐<푏≤푑, ≤ 푏 < 푐 ≤ if 푎 , | f푎≤푐<푑≤푏, ≤ 푑 < 푐 ≤ if 푎 , f푐≤푎<푏≤푑. ≤ 푏 < 푎 ≤ if 푐 , f푐≤푎<푑≤푏, ≤ 푑 < 푎 ≤ if 푐 , ,푏∈ℝ ∈ 푏 푎, } if ∗ ℝ = | 푢 ⊇ 푣 푑≤푎o r푑≤푐 ≤ 푎 or 푑 ≤ or 푏 푎 ≤ 푑 푣 ⊇ 푢 and 푣 푢 ⊇ 푣 ∗ 푈 WS WS and and . 푢,휆 ℕ } then . Let 2019 can be pictured as set of all nodes of sequences extendingsequences of nodes all of set picturedas be can ∗ of finite sequences of natural numbers will play an important an play will numbers natural of sequences finite of ℕ ∈ 휆 /20 max{푖 = 푖 ℬ 푈 ∈ 푤 ( is the basis isthe of atopology on

ℕ 푖 푣,휆 with ~ ) 94 푤,훾 풯 휆 > 푣,휆 on 휆 ~ 푈 ⊆ | 푖 , dom ( dom푢

푖 휇,휆 ℕ 푣,휆 푢,휆 ∗ 휆 } : ) (

. Then for ( for theleast 푖 푢 푢 푈 ∩ = 푢 푢 = 푈 ,풯 푋, )

and coming aftercoming and 푢,휆 푣,휇 ) (an (an infinite through path

a Heyting-algebra a .

max = 훾 퐻 푣,휆 , and and , ℝ o( 휆) ∩ dom(푣 ∉ … … since { ,휇 휆,

퐻 ∈ ℎ

휆 } in the lexicographicalthe in w ae we have 퐻 w define we , ( 푋,풯 } ) 푢 . . We can alsocan We .

), define), 푢 푂 and ℎ = Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. contained in contained ea closed are referred to as referredto are closed 푀 Let topological space topological A A set on topology can be found introductory in books on the topic, like [ In this section we will recall some basic definitions and facts about topological spaces. All of it a 6.2.3 Theory Set Constructive of Analysis and Comparison

set can be written asarbitrary of union finite intersection ofmembers of { A functionA is called all all e aiy e ta a function a that see easily We Definition 6.19 Indeed, Indeed, one must pass fromgeneral topologies toa certain subcategory of Heyting-spaces, see [ Also, it is clear that both constructions are inverses of one another. To show that both categor Theorem6. tinuous at { a finite subcover Proposition 6.20 restricted to basic or subbasic open sets. We have following the simple fact: Proof :푤≤ℎ ≤ 푤 퐻: ∈ 푤 :푓 푋: ∈ 푥 , i.e. ch ch open cover of 푈 ∈ 푦 ( ,풯 푋, : Let :

푈 cl is called neighborhood of Morefacts about topology ⋃ ( . With the definition of the pointwiseimage the ofdefinition the With . ( ) 푀

푥 be a topological space. We call complements of open of complements call We space. topological a be 푥 풪 ℎ∈퐻 ) ) iff for each neighborhood be an open cover of cover open an be 18 푈 ∈ = 푀 푂 : Heyting-algebras and topological spaces are in 1-1 correspondence. } . Wemay invoke Proposition 6.15 toshow that the ⋂ ℎ , i.e. , : } Let : Let { 푂 ⊇ { 풯 ∈ :퐴coe 푀 ⊇ 퐴 ∧ closed 퐴 퐴: 푓 ( ,풯 푋, 푀 −1 int continuous ( 푋 1 has has a finite subcover, i.e. ,풯 푋, :푋→푌 → 푋 푓: [ for all all for 퐻 = 푂 ( ) 푀 is called compact iff compact called is ] 푋 clopen. 풪 ∈ 푂 : ) ) and and and and = 푓 [ 풯 ∈ 푂 ⋃ 푋 be a continuous function. If if itis continuous at For any set anyFor { ] 푂 ′ ( :푂⊆푀 ⊆ 푂 풯: ∈ 푂 푓 } 푌 ,풯 푌, 푓 ⊆ ℎ of 1 푥 . Then Then . i cniuu if riae o oe st ae pn sets: open are sets open of preimages iff continuous is 푌 iff iff there is an open set 푂 ∩ . Also, it is clear that our attention in both characterizations may be may characterizations both in attention our that clear is it Also, . 푌 푋. [ } ) 푂 푂∈풪 (note that arbitrary intersections of closed sets are always closed). be topological spaces and ⋃ Then ℎ of 2 { ′ 푂 = 푓 푓 푓 −1 푋 ⊆ 푀 −1 ( 푥 풪 ℎ [ } 푋 [ ) 푂 1 ′ and its and ( 푂 there is some neighborhood must be must openan cover of ∧ℎ is compact. A set A compact. is ∧ 풯 ⊆ 풪 ] ~ ] 풪 ∈ 푂 : ] 2 , we defineits we , . Wethus have: 95 = 푥 for each 푂∈풪 ⋃ 푓 ~ closure [ } 퐴 ⋃ ′ is an open cover of cover open an is ] 푓 푀 ⊇ 풪 푂 = [ 푋 푓 with with iscompact, then so is { −1

푓 cl 푋 ∈ 푥 ( [ ( 푂 푥 푀 interior ) 푋 ∈ 푥 sets ) ] 푈 ⊆ 푂 ∈ 푥 풯 ⊆ 풮 푂 퐴 ∈ 푥 : ] → ) 47 to be the least closed set containing set closed least the be to . ℎ ⊆ form abasis of atopology

( ] or [ closed ∃풪 푂∈풪 . A function ⋃

is called is int } 풮 ′ ′ we can write this as this write can we 푓 35 . .풪 풪. ⊆ 푂 ( ( . Sets that are both open and open both are that Sets . 푀 푋 푈 . A set . ].

푋 ) of ) , since . By compactness, there is there compactness, By . to be the largest open setopen largestthe be to ′ 푥 fnt ∧ finite subbasis such such that :푋→푌 → 푋 푓: 푓 푀 [ is called 푋 ] . 46 ⋃ iff every open every iff ]. ]. is called 풪 푓 ies are dual, ′ ( WS WS compact 푀 ⊇ 푦 푓 푓 ) nd nd more 풯 [ −1 푈 퐻 푂 ∈ 2019 on ] [ ) 푂 . The 푂 ⊆ con- ] for /20 iff 퐻 ≔ ∎ . :

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. 6.3 induction Bar of Independence and Semantics Topological Using the ideas from section 6.2, we will define Heyting-valued models for the set th set the formodels Heyting-valueddefine will we 6.2, sectionfrom ideas the Using 푉 Allowing the functions to take values ain complete Heyting-algebra,define we the universe open cover of cover open set in set in Lemma 6.22 Lemma Proof Proposition 6.21 functions. Sticking to our to Sticking functions. We will define an underlying domain lowing idea: Note that for any set rather than rather comp a inside values take to them generalizing(and functions such all of class the the concepts and proofs are slight adaptations of the treatmentof adaptations of the proofsslight areand concepts the { (classically speaking). We define for all sets 푉 We may thus as well then Definition6.23 Proof that isthat left toshow is continuity at there issome 퐴 ∖ 푋 . In interpreting our only relational symbolsrelationalonlyinterpreting our In . ( 2 )

퐵 → : Let : For each each For : 푓 푋 Heyting-valued interpretation of } iscontinuous iff it is continuous on isafinite cover of and define and 0,1 퐴 . Actually, this definition recursion isby on ordinals be a closed subset of the compact space 퐵 : Let : 0,1 푚 푋 푂 ∈ 푥 : We say that a sequence ) is the fact, that the domain of domain the that fact, the is ) such that such for all . By compactness, there is a finite subcover finite a is therecompactness, By . : closedA subset ofa compact topological space is compact itself. ,푌 푋, 푉 푉 푉 0 훼 (2) ( ( , continuity of continuity , 2 2 identify be topological spaces and spaces topological be ) ) = ∅, = = 퐴 훼∈핆ℕ { 2 ⋃ 픞 . -valued case, we therefore define the class the define therefore we case, -valued

| safnto ran ∧ a function 픞 is

퐴 푉 with its characteristic function. What hinders us to set our domain to be 훼 푚 > 푛 (2) 퐴 휕푂 . WS WS

, all information about this set is carried by its characteristic function 푓 at . 푉 , 2019 ( 푥 푥 퐻 ( 푓 푛 follows from the continuity of continuity the from follows 푥 ) ퟙ ( which will be a reflection of the hierarchical system of all sets 푛 휕푂 퐴 푈 ∈ 푥 /20 ) 푥 ( ) 푛∈휔 푥 : =

= ∈ ) . ퟙ and and ~ ( = { 퐴 cl 픞 푔 푔 푋 ⊆ ℎ does itself not consist of such Heyting-valued valued Heyting-valued such of consist not itself does ) ( { ( and 96 ( 0, 0, 1, 1, 푂 푥 푥 푋 ⊆ 퐈퐙퐅 ) ) = )

. Let converges to , , , , ~ { 푂 ∖ if if , and constructingand , 0,1 ℎ if if 푥∉퐴. ∉ 푥 푥∈퐴, ∈ 푥

continuous functions continuous 푥∉푂, ∉ 푥 푥∈푂, ∈ 푥 . } 풪 휉<훼 dom 훼: < ∃휉 ∧ be an open cover of 풪 ′ (possibly containing (possibly

훼

: 푋 ∈ 푥 퐙퐅 and Boolean-valued models in [ inBoolean-valuedmodels and iff for each neighborhood 푉 ( 푔 푉 ( 0,1 픞 , for , ( ) 퐻 ) to be the class of functions of class the be to ) 푉 ⊆ , we are guided by the fol- the byguided are we , ∖ 푋 ∈ 푥 푌 → 푋 퐴 휉 ( 2 , then ) } lete Heyting-algebra Heyting-algebra lete 퐴 ∖ 푋 ,

. Let . eory cl ∪ 풪 ( ). But then But ). 푂 푂 ) 퐈퐙퐅 , from , { be an open an be 퐴 ∖ 푋 푉 . Much ofMuch . ( 퐻 )

푈 } by ℎ is an of . All . 풪

5 ′ ∎ ∎ 푥 ]. ∖ ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. logical logical truth oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

elements ofthe given domain of the setof consideration only, i.e. Also, it is reasonable to wish, in the of boundedcase quantification, tobe toable restrict our attention to How do we interpret the interpret we do How

these twothese observations, we should set for equality set-membership: and We will later see that this is compatible with the interpretation of unbounded quantification. Co List of List clausesHeyting-valued interpretation of language of settheory. For sake of the completeness, a we give listfull clauses of all ofthe Heyting-valued interpretation of the 푦∈픳 푦 = 픲 픳. ∈ ∃푦 ↔ 픳 ∈ 픲 푉 푉 훼 (퐻) ( ⟦ ⟦ 퐻 ⟦ 픳 ∈ 픲 픳 = 픲 ) ∀푥 휙 ∀푥 = ⟦ = ⟦ ⟦ 휓 → 휙 ⟦ ⟦ ∈ 휓 ∧ 휙 휓 ∨ 휙 훼∈핆ℕ 픳 = 픲 픳 ∈ 픲 { ⋃ - and - ⟦ ⟧ 푓 ⟧ ¬휙 ( ⟦ ⟦ | ⋁ = 푥 ⋀ = safnto ran ∧ a function is 푓 ⊥ ⊤ ) ⟦ ⟦ 푉 ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ 픵∈dom ⟦ ⟦ 픵∈dom 푥∈픳 휙(푥) 픳. ∈ ∀푥 푥∈픲 휙(푥) 픲. ∈ ∃푥 훼 픳 = 픲 픳 ∈ 픲 (퐻) = 픳 ⋁ = 0 = ⋀ = 1 = = ⋀ = = = = weshould have -relation? To incorporate the axiom of extensionality as well as the as well as extensionality of axiom the incorporate To -relation? 픵∈dom 픡∈푉 ⟦ 픵∈dom ⟦ ⟦ ⟦ .

휙 휙 휙 휙

( ( 픳 ⟧ 픲 ⟧ ⟧ ⟧ ⟧ ⟧ ) ( ) [ 퐻 = [ = ∧ ∨ → 0 → 픳 ) 픲 ( ( ⟦ ( ( ⟦ ⟦ 픳 픲 ⟦ 휙 ⟦ 픵 ⟦ ) 픵 ) 휓 휓 푦∈픳픲=푦 = 픳 픲 ∈ ∃푦 ) 푥∈픲푥∈픳∧∀ 픲 ∈ 푦 픳 ∈ ∀푦 ∧ 픳 ∈ 픲 푥 ∈ ∀푥 ( 휓 ) (

픲 ⟧ ⟧ ⟧ ⟧ ∧ 픡 ( → ⟧

( 픵 ) ⋀ = =

픵 ⟦ ) ⟧ ) 픵 = 픲 ~

⟦ ∧ 픵∈dom 픵∈dom(픲) 픳 ∈ 픵 → ( ⟦ 97 푓 ⋃ 퐈퐙퐅 픵 = 픲 ⟦ ) 픳 ∈ 픵 ⟧] 휉<훼 dom 훼: < ∃휉 ∧ 퐻 ⊆ ~ ⟧] (

. 픳

) ⟧ ⋀ ∧ [ [ ⟧ . 픲

픲

⟧ 픵∈dom ( ( ) 픵 푥 ⋀ ∧ ) ) → 픵∈dom ∧ ( ⟦ 픳 ⟦ ) 휙 휙 [ 픳 ( ( ( ( 픵 픵 픳 픵 ) ) ) ) ( ⟧] ⟧] 픳 ⟧ → . , (

, ( 픵

푓 ) ⟦ 픲 ∈ 픵 ) → 푉 ⊆ ⟦ 픲 ∈ 픵 ⟧] 휉 ( 퐻 ,

) } ⟧ , )

WS WS mbining 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Now suppose, 1. holds for all Proof 3 Holds 3 Holds by symmetry of definition.the Proof similarly for infima. Ⓜ interpretat our about results technical) rather (but useful some gather will we section this In 6.3.1 induction Bar of Independence and Semantics Topological Ⓜ eventually be constant. The case for infima is similar. proper The classes. following results show that this definition is reasonable: suprem taking are we since justified, are quantification unbounded for expressions i example, For section. last the of definitions) resulting the (and cussion All of the results are slight adaptations of [ of adaptationsslight are results the of All

Lemma6. 6.24 Lemma 8. 7. 6. 5. 4. 1. 2. 3. : 1and 2are shown by simultaneous induction: Suppose 2. Holds for : The function The :

⟦ ⟦ ⟦ ⟦ ⟦ ⟦ usefulSome facts about ⟦ ⟦ 푦 휙 ∧ 픵 = 푦 ∃푦. 픳 = 픲 픳 = 픲 픳 ∈ 픵 픶 = 픵 픟 = 픟 픟 ∈ 픞 픟 = 픞 25 ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ⟧ ∧ 픟 ≥ ∧ ∧ ∧ = 1 = : For Fr ah formula each For : ⟦ ⟦ ⟦ ⟦ 픴 = 픳 ⟦ 픴 ∈ 픶 휙 픴 = 픳 ( 픞 = 픟

⟦ ⟦ 픞 ( 푥∈픲 휙 픲. ∈ ∀푥 푥∈픲 휙 픲. ∈ ∃푥 :핆 퐻 → 핆ℕ 퐹: ,픟 ,픶 ,픳 푉 ∈ 픴 픳, 픲, 픶, 픵, 픟, 픞, ) 픲 ( for ) 푦 ⟧ ⟦ ⟧ ⟦ ) ⟧ ⟧ ⟧ 픟 ∈ 픞 ⟧ ∃푥 휙 ∃푥 ≤

≤ ≤ ≤ dom ∈ 픞 = ⟦ ⟦ ⟦ 휙 ⟦ ⟦ 픴 ∈ 픵 픴 ∈ 픵 픴 = 픲 ( ( ( ⟧ 휙 ( 푥 푥 푥 푉 ∈ 픟 픳 ⋁ = ( defined by defined ) ) ) ) 픵 ⟧ ⟧ ⟧ ⟧ ) 푦∈dom

⟧ ⟦ ( ⟧ ⟧ ⋁ = ⋀ = ⋁ =

픟 픟 = 픟

⟧

훽 WS WS 휙

) 푉 with with 픵∈dom 픡∈푉 픵∈dom

( ( 푥 ( 퐻 ( 퐻 2019 ( 픟 ) ) 퐻 ⟧ ) )

, there is a least a is there , ) ( and formulas and ( ( ⟦ 픟 ⋀ = 픟 훼 < 훽 픲 픲 휙 ( ) ) 퐹 /20 5 ( ( 픞∈dom 푦 ( ]. Some of them will be the justification of the informal dis- informalthe of justificationthe be will them of Some ]. 픲 픲 ( 픡 ) 훼 ( (

) ∧ 픵 픵 ) ⟧ ) ) and and let ~

⟦ = ∧ ( → 푦 = 픞 픟 98 ) ⋁ ⟦ ⟦ 휙 픡∈푉 휙 ( ( ~ 픞 ( 픵 ⟧ ) 훼 픵 ( 휙 ) dom ∈ 픞 ) 퐻 ) ⟧ → , wehave ) ⟧ 픟 ≥ ) 훼 ⟦ )

휙 ⟦ sc that such

픟 ∈ 픞 ( ( 픡 픞 ) ) ⟧ ( ∧ i icesn. y em 61, t must it 6.10, Lemma By increasing. is ⟧ 픟 ) ⟦ 1. = . 픞 = 픞 ⋁

픡∈푉 ⟧ t is a-priori not clear that the that clear not a-priori is t 픟 = ( 푉 ∈ 픟 퐻 ) ⟦ ( 휙 픞 ( 훼 ) ( 픡 . 퐻

) ) ⟧ , then, = a and infima over infima and a ⋁ 픡∈푉 훼 ( 퐻 ) ⟦ ion of ion 휙 ( 픡 ) ⟧ and 퐈퐙퐅 ∎ .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

showninduction:arenumbers 4-6 by The all Assume for that 푉 And And 5, Hence, and similarly,and And And finally 6, And And altogether, Then, Then, we can infer 4, 훽 ( 퐻 ) ⟦ , 픳 ∈ 픵 ⟦ ⟦ 픶 = 픵 ⟧ 픳 = 픲 ⟦ ∧ 픳 = 픲 ⟦ 픴 = 픳 ⟧ ⟧ ∧ 픲 ∧ ⟧ ⟦ ∧ ⋁ ≤ ⋁ ≤ = ⋁ ≤ 픴 ∈ 픶 ⟧ ( ⟦ 픵 픶∈dom 픶∈dom ⟦ 픶∈dom = 픴 = 픳 ) 픴 ∈ 픵 ∧ ⋁ = ⟦ ⟧ 픴 = 픳 ⟦ 픷∈dom 픴 = 픳 = ( ( ( 픳 픳 픳 ⟧ ⟧ ) ) ) ⟦ . ( (

⋀ ≤ ( 픶 = 픵 ⟦ ( ⟧ 픴 ∈ 픶 픳 ⋀ ( ( 픵∈dom 픴 ( ⟧ ⋁ ∧ 픞∈dom 픶 ) ≤ ( 픶∈dom ) ( ( ( ⟧ ⟦ 퐼퐻 퐼퐻 퐼퐻 → 픶 = 픵 ⟦ ⋁ ∧ ⟦ ⟧ ⟦ ( 픳 = 픲 픳 = 픲 픳 = 픲 픲 1 2 3 ∧ ( 픷∈dom ⟦ ) 픳 ) ) ) ( 픴 ∈ 픶 ) (

⟦

픲 ( 픳 픶 = 픵 ⟧ ) 픳 ( ( ⟦ ⟦ ⟦ ( 픵 ⟧ ∧ 픳 ⟧ 픶 = 픵 ⟧ 픶 = 픵 픲 ∈ 픵 픞 ) ( ( ∧ 픴 ∧ ⟦ ) ∧ 픶 → ⟧ 픷 = 픶 ) ⟦ → ) ) ⟧ ( ⟦ ⟦ 픲 ∈ 픵 ) ⟦ 픴 = 픳 픴 = 픳 픳 ∧ ∧ ⟦ ⟧ ~ ⟧ ⟧ 픶 = 픷 픴 ∈ 픵 ⟦ ≤ ⟦ ∧ 픴 ∈ 픞 ∧ ∧ 픶 = 픵 퐼퐻 ( ⟧ 99 ⟦ 픶 ⟦ ⟦ ⟧ 픴 ∧ 1 픳 = 픲 픴 ∈ 픶 픷 = 픶 ) ⟧ ⟧ ⟧ ∧ 픶∈dom ~ ⟧ ∧ 픴 ∧ 픲 ≤ 픴 ≤ ) ⟧ ⟦ ⟧ ( ⟦ ⋁ ) 픴 = 픳 ⋀ ∧ 픷 ) 픶 = 픵 픳 ∧ ) ) ⟧ ⟧ ⋁ = ⟧ 픶∈dom ( ( ) ( ≤ 픵 픷 ≤ ( ≤ 픷 ≤ 픳 ) ) 픶∈dom ) ) ) ( ⟦ ⟦ 퐼퐻 → ⟧ ⟧ ⟦ ( → ⟦ 픷 = 픵 ,픳 푉 ∈ 픴 픳, 픲, 픳 ∈ 픵 ) 픶 픴 ∈ 픵 픴 ∈ 픶 2 ≤ ( ) ⟦ 픷 ⟦ ) 픷∈dom 픴 ∈ 픵 퐼퐻 ∧ ( ( 픳 ∈ 픷 픳 픴 3 ⟧ ⟧ ) ⋁ ⟦ ( , . ( ⟧ 픶 = 픵

⟦

⟦ ⟧ 픷 , 픳 ∈ 픵

픴 = 픳 ) ( ⟧ ∧ ⟧ 픴

→ 훼 ) ⟦ ( ( 퐻 픶 = 픵 ⟧ ⟦ ⟦ ⟧ ) ) 픷 = 픵 픳 ∈ 픷 and all and ⟧ ∧ 픳 ∧ ⟦ ⟧ 픴 = 픳 ) ⟧ ⟧ ( ) ⋁ = 픶 픴 ∧ ) = 훼 < 훽 픶∈dom ∧ ⟧ ⟦ ( ⟦ 픷 픴 = 픲 ≤ ) 픶 = 픵 ) 퐼퐻 ( = and for and 픳 3 ) ( ⟦ ⟦ ⟧ ⟧ ⟦ 픴 ∈ 픵 픴 ∈ 픵 . ) 픴 ∈ 픵

WS WS 2019 ,픶 ∈ 픷 픶, 픵, ⟧ ⟧ ⟧

. )

/20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. with the unbounded case: interpretationboundedthe definitionquantificationof ourof that shows lemma next The ⋃ Proof Proof Ⓜ induction Bar of Independence and Semantics Topological Ⓜ 7 is shown by induction on is shownthis: like where thewhere last step is by Lemma6.25. For 2., we apply this lemma again together withLemma 6.6: Finally, for 8, we 7:use and and hence, on the other hand, clearly ⟦ ⟦ { 픳 = 픲 푥 휙 ∧ 픲 ∈ 푥 ∃푥. 훼 Lemma6. Lemma6. 2. 1. 푥 : For 1., we compute : For each For : dom ∈ 푥 :

⟧ ⟦ ⟦ 푥∈픲 휙(푥) 픲. ∈ ∀푥 푥∈픲 휙(푥) 픲. ∈ ∃푥 ∧ ⟦ 휙 27 26 ( 픲 ( ( dom ∈ 푥 ) 푓 : : Let 푥 = = ⋁ = 픲 ⋁ = 픲 ⋁ ( ⋁ = ) 휓 → ) } ⟧ ⟦ ⟦ 픵∈dom 픡∈푉 픵∈dom . Hence, 픳 = 픲 픳 = 픲 ⋁ = ⟦ :푉 푓: ⟧ ⟧ 푦 휙 ∧ 픵 = 푦 ∃푦. ( (퐻) 픡∈푉 픲 = = ) ( ( ( ( ( ⟧ 픲 픲 ⟧ ⟧ ⟦ 푓 ⟦ 퐻 (퐻) ) ) 푥 휙 ∧ 픲 ∈ 푥 ∃푥. 푥 휙 → 픲 ∈ 푥 ∀푥. ∧ 픲 ⋁ ( ) ∧ ∧ ) ⟦ there is some (least) some is there dom ⟦ 휙 픵∈dom 휙 퐻 → ( ⟦ 픡∈푉 ⟦ ⟦ 휙 ∧ 픲 ∈ 픡 픵 휙 휓 휙 ( . The base cases are 4.- ) 픵 ⟦ ( ( ( (퐻) ) ∧ 픳 = 픲 픳 픲 픲 ( ⟧ bea (set) function. Then ) 푓 ) ) ( ⟦ ⟧ 픲 = ( ⟧ 푦 휙 ∧ 픵 = 푦 ∃푦. ) 휓 → ) 푦 ( ≤ WS WS 푉 ⊆ 픵 ) ⟦ ⟧ ( ) ⟧ 픵 = 픵 퐼퐻 ( 픵 ∧ ∧ ( 픡 ⋁ = ( ) 훾 2019 푥 ( 픲 ( ) ⟦ ⟦ ∧ ⟦ 퐻 푥 ⟧ ) ) 픡∈푉 휓 휙 픵 = 픡 ) ⟧ ) ⟧ ⟧ ⟦ ⋁ = . ( ( ⟧

픵 = 픡 ∧ ∧ 픳 /20 픲

( 퐻 픡∈푉 ) ) ⟦ ⟦ ⟧ ) 휙 ⟧ ⟦

휙 휓 → , (

휙 ∧ 픵 = 픡 ~ (퐻) 푦 ∧ ( ( ⟧ 훼 픳 픵 ) ∧ ) ⟦ ) ) 푥 ( ⟧ 100 ( 휙 ⟧ ⟧ ⟦ such that such 6., all other cases except for “ 픲 픲 ⋁ = isbounded bythe above supremum. 픲 ∈ 픡 ( ∧ ) ⟦ 픡 ⟧ 픵∈dom 휙 ⟦ ) ~ 픳 = 픲 ⟧ ≤ ( ) ( 픡 ⟧ 픡 푉 ∈ 푓 ) ⟦ 픲 ⋁ = ) ∧ ⟧ 휙 ⟧ ( ) 픲 ⟦ ( ⟧ 픵∈dom ) ⋁ ≤ 픳 휙 푉 ∈ 푥 픲 ⋁ ( ⋁ = ≤ ) ( ( ( 픡∈푉 ⟧ 퐼퐻 퐻 픡 픵 픡∈푉 ) → ) ) . ⟧ ( 훼 ( ⟦ ∧ 픲 퐻 ) ( (퐻) 푥 ⟦ 픳 = 픲 ) 퐻 ) ⟦ 휓 ⟦ ) 휙 휙 . By replacement, we can form can we replacement, By . ( ( 픵 ( 픳 ( 픵∈dom ) 픵 ) 픵 ⟧ ) ) ⋁ ( ∧ ⟧ ⟧ ⟧ . ∧

= = ( 픡∈푉 ( ⟦ 픲 ⟦ ⟦ 휙 ) 푥∈픲 휙(푥) 픲. ∈ ∃푥 휙 (퐻) ( → 픲 ( ( 픵 픵 ⟦ ) ” are easy. Implication ) ) 휙(픡) ∧ 픵 = 픡 ⟧ ⟧ ∧ → ,

⟦ 픵 = 픡 ⟦ 휓 ( 픲 ⟧ is compatibleis ) ⟧ ,

⟧ ∧ ) ⟧ ∧ ⟦ ) 휙 ⟦ ( 휙 픡 ( ) 픲 ⟧ = 훾 ) ) ⟧ ∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. and and hence, 푉 And And hence, Let Pair Extensionality 6.25. Itremains6.25. to show validity ofset axioms, which we will do thein following. been has equality of axioms of Validity 6.2. in rules and axioms first-order of lidity oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison Ⓜ 6.3.2 Proof

On theOn other hand, for any

⟦ ( 픲 ∈ 픶 퐻 Theorem6.28 ) 푉 ∈ 픲 . ⟦

: We have given examples of validity of propositional axioms as well as examples of proofs o proofs of examples as well as axioms propositional of validity of examples given have We :

∀푥 ∈ 픲 휙(푥) 픲 ∈ ∀푥 ⟧ Soundnesstheorem for ⋀ ∧ 훼 ⟦ ( 푥∈픲 휙(푥) ∈ ∀푥 퐻 픵∈dom ) 푉 ∈ 픳 , ⟦ 푧 픳 ∈ 푧 ↔ 픲 ∈ 푧 ∀푧. ( = ⋁ ≤ ⋁ = 픲 ) : Every theorem ⟧ ( 훽 ⟦ 픷∈dom 픷∈dom ( 픲 퐻 휙 ⋀ = = ( ) ( 픵 and let and 픵∈dom 픶 ⟧ ) ⟦ ) 푥 휙 → 픲 ∈ 푥 ∀푥. ( ( ⋀ = → = ⟧ 픲 픲 . ) )

( 픵∈dom ( ⟦ ⟦ 픲 ( 휙 푥 휙 → 픲 ∈ 푥 ∀푥. ⋀ ( = 픲 푉 ∈ 픶 픲 ( ) ( 픷 ( ( { 픵 ) 픷 픲 ,픳 픲, ⟧ ) ) ( ∧ ( 픷∈dom ⟧ 픲 = ∧ 픵 ) ) ⟦ ( ) ( } 퐻 휏 픷 = 픶 ⟦ 픲 ⋁ ( = 픲 ( ⟦ 퐈퐙퐅 → ⟦ 퐻 of 픷 = 픶 ) ( ( , ∀푧 푧∈픲 푧∈픳 픲 ∈ 푧 픳. ∈ ∀푧 ∧ 픳 ∈ 푧 픲. ∈ ∀푧 ) 픵 ( ( be defined on defined be ⟦ 픲 ) 푥 픷∈dom 퐈퐙퐅 ) 휙

( ( → ⟧ ) 픳 ∈ 푧 ↔ 픲 ∈ 푧 픲 ( ⟧ ⟧ ∧ 픵 ( . ( ⟦ ⋀ ( ∧ are valid, i.e.

) 픷 푥 ( 휙 ⟧ ) ( 픲 ) ) 픲 ( ⟧ → ( ) 픵∈dom 픵 ⋀ ≥ . 픷

) ) ⟦ ⟧ ( ~ 픵∈dom → 픳 ∈ 픷 ) 픷 ) ≤ 101 ( ∧ ⟦ 픲 휙 ) ⟦ ⟦ { ( ⟧ ) 픲 ∈ 픶 ( ( ,픳 픲, 픷 = 픶 픲 ) 픲 ⟧ 픷 ) ( ) ~ ⋀ ∧ ( → 픵 ⟦ ⟧ ⟦ } ) 휏 ) 픷∈dom as 픲 ∈ 픵 ) ⟧ ⟦ ⟧ → ⟧ 픳 = 픲 ⋀ ( ∧ ) ⋁ ≤ 1. = → 1 ↦ 픲 ⟦ 휙 픷∈dom ⟧ ⟦ ( 픳 ) 휙 (

) → ⟧ ⟧ 픵 ( 픵∈dom ( ) 픳 픶 1. = ⟧ and and ⟦ ( ) ( ) 휙 픷 픲 ⟧ ) ) ) ( ( ⋀ ≤ ) → 픵

( ⟦ 픲 ) 1 ↦ 픳 휙 ) ⟧ 픶∈푉 ( ⟦ ) ( 픲 픲 ∈ 픷 픷 ⋀ ≥ ( ) ( 픵 퐻 ⟧ ) ) 픵∈푉 . By Lemma 6.26, Lemma By . ∧ ( → ⟧ ⟦ ⟦ ) 픲 ∈ 픶 (퐻) 픷 = 픶 = ) ⟦ 휙 ( ⟦ ( 픲 ∈ 픵 픵 ⟧ ⟦ ⟧ ) 픳 = 픲 → ⟧ ) shown in Lemma in shown ) ⋁ ≤ ) ⟦ ⟧ 휙 픷∈dom → ⟧ ( , 픶

⟦ ) 휙 WS WS ⟧ { ) ( ( 픲 ,픳 픲, 픵 ) ) ⟦ 2019 ⟧ 휙 } ) ( ( f va- f 퐻 픶 ) /20 ) ∎ ∎ ⟧ ∈

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ⋁ element of element and and hence For a set Empty set We have that for This showsThis Let Union Hence 푤 oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological We have ⟦ ) 0 = ∅ 푤∈프 푤 ∈ 픵 프. ∈ ∃푤 ⟧ . This is. This a well-defined element of 푉 ∈ 프 ⟦ 프 , we have ∈ 픴 푥 ( 훼 ( 픴 ⟦ , define by recursion 퐻 ⟦ ∈ 픴 푉 ) ) ∃푤 ∈ 프 프 ∈ ∃푤 { W give We . ( 픴 ∧ ,픳 픲, 퐻 ) { . We show that show We . ⟦ ⟧ } 프 ⋁ ( ⋁ = ⋁ = ,픳 픲, ( 푥∈ ∀푥 dom ∈ 픵 ( 픵 프 ⋁ = 퐻 ) 픴∈dom 픶∈ ) } ( 프 ≤ 픴∈dom ↔ ( 푤 ∈ 픵 ⋃ 퐻 { Un ) dom

⟧ ( ( Un ( ⟦ 픴 ( ( 프 ⋁ = 픲 = 픴 ) ( 프 ( 픴 ) 프 ) 픴 ( ∈ 픵 → ) 픵∈dom 프 ) ) ) ∧ ∃ .푥∈푤 ∈ 푥 퐴. ∈ ∃푤 :픴∈dom , 픶∈dom ) 픴 ⟦ as ( ⋀ = 픴 ∈ 픵 푥̂ = 픴 ( ⟧ ∅ ̂ 픵 ( 픵∈dom ) ∨ is a suitable witness for the empty set axiom: Since axiom: set empty the for witness suitable a is ) Un dom { 픲,픳 ( ∧ ( ⟦ ≤ 픴 WS WS { 프 픳 = 픴 〈 ⟧ ( } ) ⟦ )} ( 푦̂, 1 프 ⟦ 픴 ∈ 픵 퐻 ⋁ ≤ 푉 ( ( ( 픴 ∈ 픵 ) 2019 ) Un Un ( Un

( ) ⟧ 픴 퐻 픴∈dom { 〉 ,픳 픲, ( 푥 ∈ 푦 : ) 1 = ) ⟧ ( ( 프

프 ⟧ 프 ⟧ ) by /20 픴 ∧ ) ⟧ ⟧ ) ⋀ = 프 ⋁ = ) )( } ( and and hence . ) ( Lemma 6.26. Also, we have 1 = ⟦

퐻 ( 픵 푤∈프 푤 ∈ 픵 프. ∈ ∃푤 픵∈dom 픴∈dom ( 프 = ~ ) ) } 픵 )

. ( ∧ ) ( It is shown by induction on the rank of . ⋃ 102 픵 프 ∧ ) ⟦ { ( 픵 = 픶 ∧ dom ⟦ ( 픴 ( Un 픵 = 픶 프 ⟦ ) ~ ) 픵 = 픴 ( ∧ 프 ( ⟧ ) ( ⟦ 픴 ) ) ⟧ 픴 픴 ∈ 픵 ( ) ) = Un ) dom ∈ 픴 : ⟧ ⟧ ⋁ ( ⋁ ≤ 픴 ⋁ ( ∧ ⟦ → ( = ∈ 픵 프 픴∈dom ⟧ ⟦ ( )( ) 픶∈dom ∧ 1 푤∈프 푤 ∈ 픵 프. ∈ ∃푤 Un = 픵 ) ⟦ ( ⟦ → ( 푤∈프 푤 ∈ 픵 프. ∈ ∃푤 ( 프 픲 = 픴 프 ( 프 픴 ) ) ⟦ ) ) ⟧ 푤∈프 푤 ∈ 픵 프. ∈ ∃푤 } ,

픶∈dom

and and ( 픵 ⟧ ) ) ∧ ∨ ⟧ Un ( ) ⟦ 픴 ( 픵 = 픶 ∧ 1 ) 1. = ( Un 프 ⟧ )( ⟦ = (

픳 = 픴 프 ⟧ ⟧ 픵 Un ) ) ) dom )( = 푥 픵 ( that this is an ) 프 ⟦ ⟧ ( ∧ ∃푤 ∈ 프 프 ∈ ∃푤 )( ) ∅ ̂ , ⟦

) 픵 픵 = 픶 ) ∅ = .

and ⟧ ( ) ∈ 픵 Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. We can now verify the infinity axiom: Ⓜ Theory Set Constructive of Analysis and Comparison Ⓜ

Now for Proof

We show that Infinity For any Proof On theOn other hand, ⟦ Lemma 6. Lemma 6. 픰 ∈ 픵 : Itis clear that : Indeed, ⟦ ∀푦 ¬ ∀푦 픲 휔 ∈ 푛 ⟧ 0 = 푛 ⋁ = ( ∅ ∈ 푦 픶∈dom 30 29 , 휔̂ , we already, we know that ⟦ : For all : Define for 푛∈휔 푚∈휔 푠 = 푛 휔. ∈ ∃푚 ∨ ∅ = 푛 휔. ∈ ∀푛 does the job: ̂ )⟧ ( 픰 픲 푉 ⟦ ) ⟦ ¬0 ⋀ = ⟦ ¬ ⋀ = ( 1 + 푛 ( ∃푚 ∈ 휔 휔 ∈ ∃푚 푥∈휔̂. ∈ 푠 ∀푥 ̂ 픰 퐻 푦∈푉 픲 푦∈푉 ) 휔 ∈ 푛 ( ⊨ 픶 ) ( ( ⟦ 퐻 퐻 ( 푉 ∈ 픲 픰 = ∧ 푠 ) 푛∈휔̂. ∈ 푠 ∀푛 ) ( ⟦ , 푛̂ ( 픶 = 픵 ⟦ 푛 ̂=푠 푛̂ = ( ⟦ ̂ ) 1 + 푛 푥 ⟧ ∅ ∈ 푦 ̂ 1. = 휔̂∈ ( ) 퐻 1 = 휔̂∈ ) ⟦ ⟧ , ̂ 푥 픰 ∈ 푥 ∀푥. ( ⟧ )

픰 푚 , sothe result follows transitivity. by ( ⟧ 푠 = 픲 ⋁ = = 푛 ⟧ ⟦ ⋀ = ) ∪ 픲 = 0 ) ̂ ⋀ = )⟧ 푚∈휔 ( ( 휔̂∈ 픰 푦∈푉 ∅ = 푛̂ 픲 푛∈휔 ⋁ = ( ) 픲 ⟧ ⟦ 픲 ( ) 푚∈휔 ⟧ 퐻 푠 { ) ( 1 = ( ↔ 〈 ∧ ( ) 휔̂ 1 = ∧ 푚 ( ,1 픲, ~ 푛̂ ¬ ( ( ) ⟦ ( ) ⟦ 푛∈휔 푚∈휔.푛=푠 휔̂. = ∈ 푛 ∃푚 ∨ ∅ = 푛 휔. ∈ ∀푛 푛̂ . ⟦ 픲 = 픵 픲 = 푥 ∨ 픲 ∈ 푥 103 ⟧ ̂ =푠 푛̂ = 푚̂= . For 〉 ∅ ∈ 푦 ) } ⋀ = . Then . Then → 푛∈휔 ⟧ ~ ̂ ⟧ ⟦ ( 1 + 푚 = 푛 ) 푠 ≥ ⟧ 푚̂ ( ⋁ ∨ ( 0 → 푛̂ ⟦ ) ⟦ ⟦ 픶∈dom ) ∅ = 푛 ⟧ 푠 픰 ( 휔̂∈ 픲 ≥ 푛̂ ) 푠 = ) ) ⟦ ¬⋁∅ ⋁ (¬ ⋀ = ⟧ 1 + 푚 1 + 푛 = ⟧ ( ⟧ ̂ 픲 1. = ) 푦∈푉 , ( ) ∨ ( 픲 ⋀ = ̂ 픲 ) ⟦ ( ( ⟧ 푚∈휔 푠 = 푛 휔. ∈ ∃푚

푛∈휔 퐻 픶 ) 1 = 푠 = ) ⟧ ∧ ⟦ 1. = 푠 , or in , or in moredetail, ( ⟦ ( 픵∈dom 푚̂ 픶 = 픵 푛̂ ) )

⟧ 휔̂∈ ( 1. = 푚 ⟧ ( ∅ ̂ ) ) ) ⟧ ) = ( ̂

. 푚

( ⟦ 픵 ) 픲 = 픵 ) ⟧ ) ∧ .

⟦ 픵 = 푦 ⟧ ∨ WS WS ⟦ 픲 ∈ 픵 ⟧ ) 2019 ⟧ /20 .

∎ ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological For Separation and therefore,and Proof Claim 2 which shows the claim. Claim Claim 1 Proof hence On theOn other hand, define for Let Powerset dom ⟦ 푉 ∈ 픞 푥∈Pow ∈ ∀푥 ( : : For any 푉 ∈ 픞 ⟦ Pow Sep ∈ 픶 ⟦ 픵 ⟦ : : ′ 픞 ⊆ 픵 ⟦ ⟦ ( 픵 ⊆ ( 퐻 픵 → 픞 ⊆ 픵 픵 = 픵 → 픞 ⊆ 픵 ( ⟦ 픞 퐻 ) 픵 ∈ 픶 ∧ 픞 ∈ 픶 ) and a and formula ) ) ⟧ ⟧ . ByLemma 6.26, 휙 ( 푉 ∈ 픶 ad define and = ⟦ 픞 ( ∧ 1 = 픵 ⊆ 픵 ∩ 픞 푥

) ⟦ ) ⟦ 픞 ⊆ 푥 . 픵 ∈ 픶 ( ∀푦 ∀푦 픞 ′ ) ( = 퐻 ⟧ Pow ∈ ⟦ ( ) ′ 픵 ∈ 푦 픞 ⊆ 픵 (Sep ⋁ = ≤ , we have ⟦ ⟧ ⟧ ′ ∃푥 ∈ 픞 픞 ∈ ∃푥 ⟧ ⟧ ′ ⋁ = 픶∈dom(Sep ⟦ ⋀ = ⟧ = ⋁ ≤ ⋁ = 1 = 픵 1 = 픷∈dom ′ ( ′ 픵∈dom ⟧ 휙 ⟦ 픷∈dom 픷∈dom 픞 픵 ⊆ 픵 ∈ 푦 → 픵 ∈ 픶 = 1 ∧ ( ) . Pow 푥 ⟧ 푉 ∈ 픵 . Putting these things together, ( ) ⟧ 휙 Pow 1 = ( , ( 픵 ∧ ( ( ( Pow ′ set ⟧ ( 픞 픵 휙 ) 푥 ′ 픞 ) ( ⟦ ⟦ ( ⋁ ∧ ) ( . ( ) 푥 WS WS 픵 ) ( ( 픵 ⊆ 픵 픵 퐻 ) ⟦ ) ′ 픞 ⟦ ( 푥 = 픶 ∧ Sep ′ 픷∈dom ⟧ ( 픷 = 픶 ) ( 픞 픞 픷 = 픶 ) by , 픵 ⊆ 픷 ) ) 1 = ) ) ) 픵 2019 푉 ∈ ( ′ 휙 ∧ Pow ′ ⟧ ⟧ = ( ( . Furthermore, ⟧ 푥 ⟦ 픞 ⟧ dom ∧ ≤ ( /20 ) 픷 = 픶 ) ∧ 퐻 { ) ( ( 픵 ∧ 휙 〈 ⟦ ( ⟧ ) 픞 픞 ⟦ 픶, ⟦ ( . 픞 ⊆ 픵 픞

) ( 픵 = 픵 푥 픷 = 픶 = )( ′ 픷 ( ~ ) ⟦ = ( ) ( Pow ⟧ 픵 ∈ 픶 픷 픵 ⟦ 픞 ) ∧ ) ) 104 푥( ∃푥 { )( ) ⟧ ⋁ = 〈 ′ ⟦ → ⟧ 픵, ⟧ 픵 = ∧ 픷 = 픶 ( , ∧ ) ⟧ 픷∈dom

픞 ⟦ ⟦ ⟦ ( 〉 ⟦ ∧ 휙 ) ~ ⟦ 픞 ⊆ 픵 dom ∈ 픶 : 픵 ⊆ 픵 ∩ 픞 휙 ∧ 픞 ∈ 푥 픵 ∈ 픶 ) 픵 ∈ 픶 ( ⟦ 픵 퐻 = ⟧ 픵 = 픶 ) ) ( ⟧ 픞 ⋁ = ⟧ ) ′ 픞 ∧ ⟧ ( ) ⟧ dom ⟦ ) . 픷∈dom

⟧ ⋀ = 픵 ∈ 픷 ⋁ ≤ ) ( ′ ( ( ⟧ 픵 ( 픵∈dom (픞 ⋁ = 픵 픞 )〉 픞 픷∈dom = ) ) ) ) ⟧ dom ∈ 픵 : } 픶∈dom ( . Weneed the following two claims: 픞 ⟦ and and 푥) = 픶 ∧ ∧ ) 픵 ( ( ′ ⟦ 픞 ( Pow 픞 픷 = 픶 픵 ⊆ ( ) 픷 ( ( 픞 ) ⟦ ) ( Pow ⟧ 픷 = 픶 ∧ 픞 ) ( ∧ ⟧ ) ⟦ ⟧ ( 픞 ( ) 픷 = 픶 픵 ) ⟦ ⟦ = ) ( ≤ } 픞 ⊆ 픵 픵 ⊆ 픵 ∩ 픞 ∧ 픞 ⊆ 픵 ⟧ 픞 . Then . Then for ∧ ⟦ )( ⟦ ∧ 휙 ∧ 픞 ∈ 픶 ⟦ 픵 ∈ 픶 ⟧ 픵 ⟦ 휙 ) ⟧ 픵 ∈ 픷 ∧ ( = 픵 → ⟦ ) 픵 ∈ 픶 ⟧ ⟧ ⟦ , ⟦

⟧ 픞 ⊆ 픵 ∧ 픞 ⊆ 픵 ) 푉 ∈ 픶 ( ⟦ 픶 ⟧ 픵 = 픶 ) ) ⟧ ⟧ ⟧ . ′

) ( ⟧ for 퐻 1. = ⟧ ) ) ,

∈ 픵 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

Using bothUsing claims, we can finally validate other the direction ofthe powerset axiom:

Let Collection schema tion oftion the previous axioms, this setis awell-defined member of By collection in Obviously, this is shown byinduction: Let induction Set validityThe of the collection schema follows. But thenBut ⟦ 픞 ⊆ 픵 ⟦ 푉 ∈ 픲 푥∈픲 푦 휙 ∃푦. 픲. ∈ ∀푥 ⟧ = ( 퐻 ⟦ ) 푦 픞 ∈ 푦 → 픵 ∈ 푦 ∀푦. and and = 푉 ( 휙 , we can set ,푦 푥, ⟦ ⟦ ⋀ = ⋀ = ⋀ = ( 푥 Pow ∈ 푥 → 픞 ⊆ 푥 ∀푥. 픵 ,푦 푥, ′ 픵∈dom 픵∈dom 픵∈dom 픞 ⊆ ) ⟧ ) ⋀ = beany formula. For each ⟧ ( ( ( 픵∈dom Pow = 픲 픲 픲 ) ) ) ( ⟧ ( ( 픲 = 훼 ⋀ = 픲 픲 ( 픵 ( ( ( 픲 ) 픵 픵 픶∈푉 ⋀ ≥ ( ) ) ) ⟦ ( 픞 → ⋃ 픲 ∀푥 ∀푥 ⋁ → ( ⋁ → )( 픵∈푉 ( { ( 퐻 ⟦ 훼 픶∈푉 픵 픵 ) 픶∈푉 픶∈푉 푦∈ ∃푦 ′ ) ( ⋁ 픵 ( ( ) 퐻 dom ∈ 픵 : ⟦ ∀푦 ∈ 푥 휙 푥 ∈ ∀푦 ( → ) 픵 ∈ 픶 ( ≤ 훼 훼 픞 ( 퐻 ( ( 퐻 퐻 픵 ) ⟦ ) ⟦ ) ) ⟦ ⟧ 픵 = 픵 → 픞 ⊆ 픵 ⟦ ⟦ Col 푦 휙 ∃푦. 휙 픵 푉 ∈ 픞 Col 휙 ⋀ = ′ ⟧ ( ( Pow ∈ ,픶 픵, 휙 픵∈푉 → ,픶 픵, 휙 ( ~ ( ( 푥,푦 ( ( 픲 ) 푦 ⟦ ( 푥,푦 ,푦 픵, ( ) 퐻 ⟧ 퐻 픞 ∈ 픶 ) 105 ) ⟧ 푉 ∈ 픵 ) ) } ) ( ) ⋁ = ) ( and suppose and that for all 휙 → and and ( ) ( 픲 ⟦ 픞 ⟧ 픲 ) Pow ∈ 픵 → 픞 ⊆ 픵 ⋀ ≤ 픶∈푉 ) ) )( 휙 . ⟧ ~ ⟧ ′ ( (픲 ⋀ = ( ) 픵∈dom . ⟧ 픶 퐻 훼

Col 푥 ( ( 퐻 픵 ⋀ ≤ ) ) ∧ ,푦 픵, 픵∈dom ) , there is, there aleast ) ∧ )⟧ ⟦ ⟦ 픶∈dom 휙 휙 픵 → 픞 ⊆ 픵 ⟦ ) ( ( ( 휙 ≤ ⟧) 푥,푦 픲 ,픶 픵, ) ( ( ( 픲 ⟦ ,픶 픵, ) = ) ( 휙 ( 푉 ) 픲 픵 픲 ⟧ ′ ( ( ⟦ ( ) ) ) . 퐻 픶 픵 ( 푥∈픲 푦∈ ∃푦 픲. ∈ ∀푥

⟧ ) ( ) = 픵 ) ) ( ′ by Lemma 6.26. Furthermore, 픵 ⟧ ′ ) ⋁ → ( 픞 ) . Pow ∈ {

픶 ) 〈 ⋁ → ⟧ ) 픶∈푉 ,1 픶, ) → 훼 픶∈푉 훼 픵 ( 〉 such that such 퐻 훼 < 훽 ⟦ 푉 ∈ 픶 : ( ) 픞 ∈ 픶 ( ⟦ 픞 퐻 휙 ) ) ⟦ ⟧ Col ( 휙 ) ,픶 픵, and and ⟧ ( 훼 1. = ( ) ,픶 픵, 휙 퐻 ) ( = ) ⟧ 푥,푦 } ) )

. As in the valida- 푉 ∈ 픶 ⟦ ⟧ ) 푦∈픵 ∈ ∀푦 ) ( 픲 ) 휙 . 훽 ( 퐻 WS WS ( ) ′ ,푦 푥, ,

( 픞 ∈ 푦 2019 ) ⟧ .

/20 ) ∎ ⟧

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. On the other hand, suppose, 2.holds for all Proof { ordered pair of As As we have seen, when validating the axiom of infinity, 6.3.3 induction Bar of Independence and Semantics Topological hn aiaig h aim f ar w hv vrfe ta te nenl ar of pair internal the that verified have we pair, of axiom the validating When 6.3.4 Ⓜ This showsThis the model the product of hence, the orderthe isdecidable in th mirroring modelthe inside definite is numbers natural the orderon the that saying lemma

,픳 픲, ⟦ ⟦ ∀푥 (∀푦 ∈ 푥 휙 푥 ∈ (∀푦 ∀푥 푚 Lemma6. ̂ 1. 2. } : 1.and 2. shownAre bysimultaneous induction: Suppose that 1. holds for all (

푛 = 퐻 ) ⟦ ⟦ Internalsetof natural numbers InternalCross product , where , ̂ ⟦ 푛 푛 ̂ ̂ ∀푥 ∀푥 ⟧ 푚 = 푚 ∈ = 푉 ,픟∈푉 ∈ 픟 픞, ( ( ̂ 픵∈dom ̂ ∀푦 ∈ 푥 휙 푥 ∈ ∀푦 31 퐻 ⟧ ) ⟧ ⋀ (we also say that internally, that say also (we : For all ,픳 픲, ( = dom = ≤ ⋀ = 푦 ( ) { { 푛 ̂ ( is given as 픵∈푉 1, ,else. 0, ⟦ 퐻 1, 0, ) 휙 → [ 휙 ( ) 푛 ̂ ⟦ { can begiven as ( ( ( ( ,픳 픲, 푚 ̂ 퐻 픞 푦 if 픵 if if ,푚∈휔 ∈ 푚 푛, ) ) ( ) ) 푛<푚, < 푛 ⟦ 푛=푚, = 푛 푛≠푚. ≠ 푛 ⟧ 푛 ∈ 푥 퐈퐙퐅 ⟦ 푦∈픵 휙 ∈ ∀푦 } → . 휙 → ∀푥 (∀푦 ∈ 푥 휙 푥 ∈ (∀푦 ∀푥 )

( ̂ 퐻 ⟧ ⟦ ⟧ ) : ≤ ∈ 픵 ) = ( ( = 푥 ⟦ ,픳 픲,

픵∈dom ∀푥 (∀푦 ∈ 푥 휙 푥 ∈ (∀푦 ∀푥 , we have ) 푚 ̂ { )⟧ ( ,픳 픲, ⋁ ⟧] ) 푦 ( WS WS ⋀ ≤ 퐻 ) ∧ ( } ) 푛 ̂ 휙 → and and 픶∈dom 픵∈dom ) = ( [ 2019 푛 ̂ 푦 ⋀ { ( ) ( { 픵 푚 < 푘 픵 ,픲 픲, 휙 → { ) ( /20 ) ( ,픳 픲, 푚 ̂ ∧ 픞 ⟧ 휔̂ ( ) ) ⟦ [ 푦 } ⟦

∧

푚 휙 ̂ } ( ( 푚 is ) ~ ̂ 퐻 ( 푥 ⟦ and ( ( 퐻 휙 → the set of natural numbers). We will need one simple one need will We numbers). natural of set the ) ∀푦 ∈ 픞 픞 휙 ∈ ∀푦 ) 픶 픵 = , ) 106 ⟧ ) { ) ( ,픳 픲, ⟧ 픲 ⋀ ≤ → 픵 ⟧] ) 푛 < 푙 ( ⋀ ≤ 푥 픞∈푉 ⟦ = } ~ = ) ∈ 픵 픶∈dom (퐻) ⟧ 휔̂ { ( ⋁⟦ 푘∈푛 ,픳 픲, ∧ ( 퐻 . Then } plays the role of the set of natural numbers in 푦 푛 ̂ ) (퐻) ⟦ ⟦ ) ⟧] 휙 ∀푦 ∈ 픞 픞 휙 ∈ ∀푦 } ⟧ ( 푚 ( 픞 ̂ . In this section, we will show that the cross ( = 퐻 ) ≤ 픞 ( ) 픞 푘 = ) ( ⋀⟦ 푘∈푛 ⟦ ⟧ ( 픳 ∀푦 ∈ 픞 픞 휙 ∈ ∀푦 픶 ̂ ) = ) ⟧ 1 = 푘 ̂ → ⟦ ( = ∀푥 휙 ∀푥 푦 ∈ ) ⟦ { . It is then clear that the internal the that clear then is It . 0, 0, ⟧ 1, 1, 휙 푚 ̂ ( ( ⟧ ( 픶 if if 푦 푥 ∧ ) 푚∉푛. ∉ 푚 푚∈푛, ∈ 푚 ) ⟧ ) ⋀⟦ 푙∈푚 ⟧ ) 휙 → .

= ,픳∈푉 ∈ 픳 픲, ⟦ 푙 ̂ ( 푦∈픞 휙 픞. ∈ ∀푦

∈ 픞 푛 < 푘 ) 푛 ̂ ⟧ ⟧ ∧ = ⟦ , then ( ∀푦 ∈ 픞 픞 휙 ∈ ∀푦 퐻 { 0, 0, 1, 1, ) ( i gvn by given is 푦 if if ) e fact thatfact e ⟧ 푚≠푛. ≠ 푚 푚=푛, = 푚 ,

( 푦 ) ⟧

∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. On theOn other hand, for So, let Proof Proof Ⓜ Theory Set Constructive of Analysis and Comparison Ⓜ It is easy to see that internally, 6.3.5 mimic the definition of definitionthe mimic

diately get that get diately

the lemmathe implies that hence avoiding taking internal collection and separation. The special case that is important for us is the internal cross product cross internal the is us for important is that case special The has asimplerhas internal representative, namely As As Lemma 6. Lemma 6. 픴 : Weknow that all theorems of : wasarbitrary, this shows the lemma.

푉 ∈ 픴 ⟦ Internalsetof finite sequences of naturalnumbers ∈ 픴 ⟦ 휔 ̂ cp 푛 ( 33 32 퐻 ∈ ( ) ,픟 픞, , then : : For 푉 Se 푉 ( 퐻 ( ) p 퐻 ⟦ ) ⟧ ∃푛푦=휔 푛픶=휔 = 픶 ∃푛 ) ,픟∈푉 ∈ 픟 픞, 휔 ⊨ ⋁ = ⊨ ⋁ = 푉 ∈ 픶 cp 픶∈dom 푉 휔 픲∈dom 픳∈dom { ⋁ = <휔 휔 ( 휔 ( 푛 푉 퐻 ,픟 픞, 픶∈푉 푛 in ) ( ( 휔 ( ̂ = 휔 ∈ 푛 : 퐻 퐻 Co 퐻 휔 × 휔 = 휔 × 휔 ⊨ (

푛 ( ( ) 휔 퐈퐙퐅 cp ) )

( 픟 픞 ⟧ Un 퐻 휔 , the , the just defined ⊨ ∀푤 ∀푤 ⊨ ) ̂ ) l = ( ) 푦=휔 [ → 픞,픟 [ 푛 픞 ⟦ , using the axioms of separation, union and collection. We thus imme-thus We collection.and separation, unionofaxioms the using , { ( ( 푛픶=휔 = ∃푛 픶 ) 휔 = 〈( ⟦ 픲 Se ) } 푛 [ ∈ 픶 ) ,픳 픲, cp = ( ( p 퐈퐙퐅 픟 ∧ 휔 ∈ 푤 푛 ∃푛푦=휔 ( Se ) ) . To define the set of finite sequences of natural numbers, we may { ,픟 픞, ) ( 휔 ( hold hold inside p ̂ 퐻 픳 ⟧ 푛 ∃푛 푦=휔 ∃푛 ) ) cp )( ⋁ = ̂ 푛 픞 , 휔 ∈ 푛 : ∧ ⟧ 푛 픶 ( ( 픶∈푉 ⟦ ,픟 픞, ) ∧ ( 픲 휔 . ̂ ( Co ∧ ) ~ ,픳 푢, ⟦ <휔 휔 cp ( 푛 휔 픟 ∧ ) 퐻 ⟦ ̂ l ( 107 } ) 푤 = 픶 푦=휔 푥∈픞∃ = 푤 픟 ∈ ∃푦 픞 ∈ ∃푥 ↔ ( 푛 . ) ⟧ Co [ ,픟 픞, ( Se 픶 = 픴 = = 픳 푉 l )〉 푛 p 푦=휔 ) ( ~ ( ⟦ 퐻 gives theus internal cross-product. ∃푛 푦=휔 ∃푛 dom ∈ 픲 : ⟧] 휔 푛픶=휔 = ∃푛 픶 ⟧] ) ⟧] , hence is suffices to verify that ) 푛 ≥ ⋁ = )) ( = 휔 ⟦ . In this section, we want to prove that provethat to want section, we this In . 픲∈dom 픳∈dom ⟦( 푛 ) ∃푛 휔 ∃푛 ) ( 푥∈픞∃ = 픴 픟 ∈ 픞 ∃푦 ∈ ∃푥

Co ( 푛 ̂ 픞 ⟧ ( ( l 푛 ) 픟 픞 푦=휔 ) ) ⋁ → dom ∈ 픳 , [ 휔 = 픞 푛∈휔 ( cp 푛 픲 ( ( ) 푛 ,푦 푥, 휔 ( ⟧ ⟦ 픟 ∧ ,휔 휔, ) 휔 ∧ ̂ ) ) 푛 ( ( ) ( ⟦ 픟 픳 . 픶 휔 )

픶 = ̂ ) ) ) . Since . } 푛 ∧ ∧ ,

휔 = ( ⟧ ⟦ ⟦ ,푦 푥, ( 휔 = ̂ ,픳 픲, ̂ 푛

푛 ) cp ⟧ 픶 = )⟧ ) ( ( 1. = 퐻

,휔 휔, ) ⟧] 픴 =

= WS WS ) 휔 × 휔 = ⟧] 2019 ̂ 휔 /20 <휔 ∎ ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. we immediately get where Recall, that the domain of domain the that Recall, Ⓜ induction Bar of Independence and Semantics Topological We want to find a representative for the internal function space function internal the forrepresentative a find to want We 6.3.6 want torestrict our attention to those where the underlying Heyting-algabra has a clopen basis. Note that Proof subset of and and So let

On theOn other hand, let Proposition 6.34 푈 ⟦( : In : In the light ofthe last lemma, it suffices toshow that

( ( =⋁ = 푈 ( 픵 ( Internalfunctionspace ,푎 0, ( ) ,푎 0, 푥 = 풫 ) 푎 0 0 ( 0 → 훼 is single-valued ≡ ∀푚 ∈ 휔 ∀푛 휔 ∈ ∀푚 ≡ single-valued is 훼 sttl≡∀ 푛∈휔. ∈ ∃푛 휔 ∈ ∀푚 ≡ total is 훼 ⟦ ,…,푎 ) 휔 × 휔 ) 푤∈퐿 푤 ∈ 픵 퐿. ∈ ∃푤 … , 푛∈휔 … , ⟦ 푛−1 휔 ∈ 픵 ̂ ( ̂ ( ,푎 1, − 푛 ,푎 1, − 푛 ∈휔 ) ̂ = . Hence, if we repeat this construction inside ≥ : ⟦ 푉 <휔 푉 = 픵 ⟦ ⟦ ( 푛픶=휔 = 픶 ∃푛 ( 퐻 dom ∈ 픵 푤∈퐿. ∈ ∃푤 퐻 ⟧ ) 푛−1 ) 푛−1 = ( ⊨ 휔 ⊨ ( ⟧ = 푈 ,푎 0, , for ⟦ ( ) ) 푤∈퐿 푤 ∈ 픵 퐿. ∈ ∃푤 ̂ 휔 ) ) 픴 <휔 휔 푈 ∈ 0 ( dom ∈ ( Un 휔 푛 ) 푈 픵 ( 픴 = … , ⟧ in one in of these domains. ,푎 0, = ) = ⟧ ( , then → { ( WS WS Un ( ⋁ = Se 휔 0 ,푎 1, − 푛 휔 푉 ⟦ ( ) 푛 푛픶=휔 = 픶 ∃푛 ) 퐻 ∈ 훽 픵∈dom ( p … , 휔 ̂ ̂ ( 퐻 . However, another representation of 휔 ∈ 푛 : 2019 훼 is total ∧훼 is is single-valued is total ∧훼 훼 ( Se <휔 ) ,푛 푚, ̂ ( 휔 ⊨ ⟧ ,푎 1, − 푛 p ) 1 ⋁ → /20 ( ∃푛푦=휔 , then 푛−1 푛 , ̂ 휔×휔 푈 ̂ ) ) <휔 ̂ } 푎 2

훼, ∈ ) 0 푛 ) ~ ⟦ is given by the union of the domains of domains the of union the by given is ,…,푎 ( 휔 ∈ , where )⟧ ⟧ 푤∈퐿 푤 ∈ 픵 퐿. ∈ ∃푤 = 푛∈휔 푛−1 푛 108 1. =

푛−1 ⋁ → ( Un Co ) 푎 ( ∈휔 )

( 0 ( l ~ { 푦=휔 ,…,푎 ,푛 푚, 푤 ∈ 휔 ⟦ ⟦ 푛∈휔 훽 is total is 훽 = 픵 푛 푛−1 ̂ 휔 ∈ 푛 : 푛 ⟧ 1 ( ( ) ( ≥ 휔 Pow ∈휔 ⟧ ( ∧ 훼 ∈ ) ,푎 0, ∧ ⟦( ⟦ )) 푉 = 픵 ⟦( 휔 ( ( } ⟧ and hence ( 0 휔 × 휔 ,푎 0, ) ∧ 퐻 휔 ̂ ( ) .

( ) ,푎 0, dom … , . Externally, this space is defined asdefined is space this Externally, . ⟦ ( i.e. if i.e.if we set ,푛 푚, 훽 is single-valued is 훽 ( 0 ,푎 0, ) ( 0 … , ,푎 1, − 푛 ) ) ( 2 ) … , 픴 0 ) , ̂ (

) 휔 ,푎 1, − 푛 훼 ∈ … , ̂ 휔 ( ) 푉 ,푎 1, − 푛 is the whole set 휔 ( ( ) will be useful in the case 푛−1 퐻 ,푎 1, − 푛 푛 → ) 휔 ⊨ ) 푛−1 )⟧ 1 푛−1

<휔 푛 = ⟧ ) 푛−1 ) 1 = ) = 퐿 휔 ∈ ) 휔 = 2 ) 픵 = ). )⟧ ̂ : ̂

푛 { <휔 휔 ⟧ ⟧ 퐻 1. = ̂ ) 푛 1. = 휔×휔

̂ 휔 ∈ 푛 :

. We ∎ ∎ }

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ⟦ ⟦ Proof Using this,Using Ⓜ Ⓜ Theory Set Constructive of Analysis and Comparison Ⓜ

{ there issome Lemma6.36, the hypothesis Proof

tive of tive of internal calculating the following values: We will show the rather technical proof of And And finally, 퐻 ∈ 훽 ( ( ,푛 푚, ,푛 푚, ̂ Lemma 6.37 Lemma Lemma6. 6.35 Theorem 3. 1. 2. ⟦ : UsingLemma 6. : For the sake of readability of this proof, let us write us let proof, this of readability of sake the For :

훼 is total is 훼 ) ) 휔×휔 ̂ ⟦ ⟦( ⟦ 훼 ∈ = 훼 훼 is single-valued 훼 훼 is total ,푛 푚, ( : ,푛 푚, ̂ ⟧ ⟦ ⟦ 훼 is single-valued is 훼 훽 is total is 훽 = 36 ) 훽 ⟧ 훼 ∈ ℎ,훼 ⋁ ) 휔 : For all : Let : = ⟧ ⟧ 픵∈dom 휔 : Let : 훼 = ⟦ ⟧ = 픉 ∈ , i.e. 푚∈휔∃ 휔. ∈ ∃푛 휔 ∈ ∀푚 훼 ⋁ ⋀ = 훼( = 퐻 ∈ ℎ ⋁ ⋀ ⟧ ( such that such ( 푚 퐻

푚 ( 31 훼 ∧ 푉 ,푛 푚, ̂ 퐻 ∈ 훼 ) be a complete Heyting algebra with clopen basis. Then Then basis. clopen with algebra Heyting complete a be ( [ ⟦ ⟧ ( , wehave for 1., ,푛 푚, ̂ 훼 퐻 푛 훽 is single-valued is 훽 be clopen and and clopen be ⋀ = = ⋀ = = 푛 ) ( 훼 픵 ) ⊨ ( 푚,푛 ⟦ 푛 ) ≤ ℎ ⋀ ) ) ( ⟧ 휔×휔 푚∈휔̂ ∈ ∀푛 ∀푚 1 . 푚 ̂ ) ∧ ̂ 푛 ,푛 푚, ( ( ,푛 = , for all 1 휔 = 픉 ( 1 ⟦ 2 푚 ≠푛 ≤ ℎ ,푛 푚, ̂ ,푛 ⟦ ( [ ⟦ , the following hold 2 ,푛 푚, 훼 is total is 훼 ̂ (훼 ) ∀푚 ∈ 휔 ∀푛 휔 ∈ ∀푚 2 [(⟦ ) ( ( ¬ ,푛 푚, , ⟦ ( )

) 휔 ( 훽 = 훼 ( ) ,푛∈휔 ∈ 푛 푚, ,푛 푚, . ,푛 푚, 훼

) ̂ ̂ 픵 = . ( ) 1 ( 퐻 ∈ 훼 푛 , 훼 ∈ ,푛 푚, ⟧ ̂ 1 1 ⟧ ⟧] ⟦ ℎ,훼 ) ) 2 ∧ 픴 = 픉 1 = ) 훼 ∈ ⟧ 1 = 휔̂∈ ( ⟦ ⟧ 1 훼 ∧ 푛 , , 훼 is single-valued is 훼 = . ) ~ 휔×휔

⋁ ̂ ) } 2 ⟧ and ⟦ 푙,푘∈휔 ( 훼 ∧ 109 휔 ( 휔 ∈ ∧ 푚∈휔 푛∈휔̂. 휔̂ ∈ ∈ ∃푛 ∀푚 ( ( ,푛 푚, such that such ⟧ ̂ ⟦ ( ( ,푛 푚, ( 1 = ( ̂ [ 픉 ,푛 푚, ,푛 푚, 훼 ~ ̂ ̂ 2 ( ( ( 훽 ) in a sequence of several lemmas. We start with ( ( 1 ̂ ) ,푘 푙, ) ,푛 푚, 2 ) 2 0 → ) ) ) 1 = ∧ 훼 ∈ ) 훼 훼 ∈ ) ) ≤ ℎ ) ( 1

,푛 푚, ) for , ∧ ⟧ ] ⟧) ∧ 훼 ∈ reads ⟦ (훼 ¬ ⋀ = ( ⟦ ( ( ,푛 푚, 훼 is total is 훼 → ) ̂ ,푛 푚, ̂ ,푛 푚, ̂ 푛 instead of instead dom ∈ 훽 1 푚 ⟦ ,푛 ( 푛 2 2 ) ,푛 푚, ) ) 1 = 훼 ∈ 훼 ∈ 푛 ≠ ( 2 ⟧ ̂ ,푘 푙, ) ⟧ ( ) ( ∧ 2 픉 ( 훼 ∈ ⟧ ⋁ ⋀ = 푛 → ⟦ ,푛 푚, ) ) 훼 ] ̂ 훼 is single-valued is 훼 ⟧] i aohr representa- another is , 픉 ( 푚 ) ( , defined as defined , 1 ,푛 푚, ̂ 훼 = 1 푛 → 푛 ≠ ) ) 푛 ( 훼 ∧ ) 1 ( ⟦ 2 ) ,푛 푚, ̂ ( ) . In the light of light the In . 푛 ≠ ,푛 푚, ̂ ⟧ ( ( ,푛 푚, ̂ 2 ) ) ) WS WS ) dom ⟧ ∧ 훼 ∈ 2 ⟧ ) 2019 ) . Then . ( ⟧ ) 픉 .

) /20 ∎ =

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological We define the element For totality, we have for each

Let us Let us showfirst that oe that Note that that For hence Proof shownHaving this lemma, theproofTheorem of 6.35 follows easily: e aiy e that see easily We 훽 ( 훽 ⋁ ,푛 푚, 푛 푛 ¬ ≤ ℎ : Clearly, for ⟦ 1 픴 ∈ 훽 푛 , dom ∈ 훽 ) ( and and ,푛 푚, 2 훽 0 ≠ ⟦ ( ( 훽 is single-valuedis 훽 훼 ,푛 푚, ) 휔 훽 ≤ ℎ ( 훽 = ⟧ and and ,푛 푚, ( ⋁ = 픉 1 ℎ∨ℎ=1. = ℎ ∨ ¬ℎ = ⟦ ) ( ) 훽 훼∈dom 훽 = 훼 1 픉 ∈ 훽 ( ,0 푚, . Finally, again invoking Lemma 6.36, 훽 ∧ 푛 ( ) ,푛 푚, ,0 푚, 1 훼 ≥ 훼 ∧ ( 푛 ≠ ( dom ∈ 훽 ) ,푛 푚, ⟦ ,푛 푚, 훽 ( ) 훽 is total is 훽 ⟧ , we have, we ( 훽 ⋁ ∨ ) 픴 ℎ,훼 ,푛 푚, 훼 → 휔 ⋀ = 2 훽 ∧ 푛>0 , ) ) 2 훼 ⋁ ⋀ ≤ ℎ [ ( 훽 = ) 픴 훽 ≤ ℎ ∧ 푚,푛 ⟧ ( 2 ( = ,푛 푚, 휔 ,푛 푚, ) 푚 1 = (

( ) ( 픉 ) as ,푛 푚, ⋀ = ( , i.e. , 훽 훽 [ ⟧ 훼 푚 = ≤ 훼 훼 ) ( ( ) ∧ , i.e. ) mas that means ) ( ( ( ,0 푚, ,푛 푚, [ [( 푚,푛 which shows that ∧ ,푛 푚, ,푛 푚, = ⟦ WS WS ( ( ) 푛 훽 is single-valued is 훽 훼 ∧ ℎ ,푛 푚, 훼 ⟦ = [ ) ( 훼 = 훽 ( ⟦ ) [ ( ) ,0 푚, ,0 푚, 2019 1 훼 ) 훽 is total is 훽 훼 ( = ( 훼 = ( ) ( 훼 ) ( → ,푛 푚, 훼 ,푛 푚, ℎ ∧ ,0 푚, and and ( ( ( ) ) ,0 푚, 훼 /20 ⟦ ,푛 푚, ⟧] ( ℎ ∧ 훼 ∧ ( ( ,푛 푚, ) ) ,푛 푚, ,0 푚, ) )

⋁ = ∧ 훽 ¬ ⋀ ∧ 훽 ) ~ ℎ ∧ 훽 → ) 1 ( ( ⟧ ( 푛 ( 훼∈퐻 ℎ ∧ ) ,푛 푚, ,푛 푚, ) ∧ 훼 ,푛 푚, ∧ 1 ) ) 110 훼 ∧ 푚 ∧ ℎ ℎ ∧ ≠푛 ) ( ( 훽 ∈ ℎ ∧ ⟦ ( ,푛 푚, 훼 ) ¬ℎ ∨ 휔×휔 ,푛 푚, 훽 is single-valued 훽 ̂ 2 ) 1 ( ℎ∨⋁ ∨ ¬ℎ ∨ ( ⟧ ) ) ≤ ℎ ) ,푛 푚, ,푛 푚, ) ~ ⟧] 훼 ≤ ℎ ∧ ∧ ℎ ∧ 훽 ∧ [⟦ ( 2 ¬ℎ, ∨ ) 훼 ) ] ⟦ ⋀ ∧ 훼 is total is 훼 ] 훽 = 훽 ( ⟦ ∧ ℎ ∧ ) 2 ] ⋀ ∧ ( ( ,푛 푚, 훽 = 훼 ) 푚,푛 ,푛 푚, ℎ ∧ ∨ ( 푛>0 ) ( 훼

푚,푛 ) [ 0 = ( ℎ∧ℎ ∧ ¬ℎ ) ( = ) 1 for [ ,푛 푚, ,푛 푚, ( ⟧ 2 ] 훽 ) ) 훼 ) [ ⟧ ∨ ( 1. = ( 훼 ∧ for 훽 ⟧ . 푛>0. > 푛 ( 훼 0 = ,푛 푚, [ ,푛 푚, ( ∧ ) ( ) ℎ∧ ¬ℎ ,푛 푚, ,푛 푚, for all all for ( ℎ ∧ ] ⟦ ⟧

푛 ,푛 푚, for 훼 is single-valued is 훼 0. = 0 ∨ 0 = ) 1 1 = ) ) → )

ℎ ∧ 푛 ≠ 1 ( ) 훼 → 2 훼 푛 . ⟦ ) 훼 ∧ 1 ( ( ) ) 2 ,푛 푚, ,푛 푚, ,푛 푚, . 푛 ≠ . Indeed, for ℎ∨( 훼 ⋁ ∧ (ℎ ∨ ¬ℎ =

( ( ,푛 푚, ,푛 푚, ) 2 ) Hence, . . We will use the fact the use will We . ℎ ∧ 훼 ∈

) 2 ] ) .

) ) ⟧] ] 0, = ℎ ∧ ⟧ ∧ 훼 ≤ ℎ 0 > 푛 ⟦ 푛 훼 = 훽

( , ( ,푛 푚, ,푛 푚, ⟧] ) ) ) → ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. The proofsThe in this and the next two chapters is taken from[ an observation about the basic open sets open basic the about observation an image we mark Using thisUsing theorem, we will show theusing topology thisIn chapter, we will show that 6.4 Theory Set Constructive of Analysis and Comparison

Putting these things together, Using Lemma 6. pen elements, ⟦ 픴 = 픉 ⟦

픉 ∈ 훼 Independence of Bar induction 휔 ⟧ ⟧ ⋀ = ⋁ = 훼∈dom

훽∈dom ⋁ ⋀ = 푈 37 퐵 푢,휆 ′ 훼∈dom , and writing for , we have ( 픉 in red in and ( 픉 ) [ ) [ 픴 픉 ( 휔 픉 ( ) ( 훽 [⟦ 훼 ) ) 훼 is total is 훼 ∧ → ⟦ 푈 훽 = 훼 ⟦ 〈 픉 ∈ 훼

〉 휇 , 퐻 ∈ 훼 Cons ⟧ in green, in where ⟧] ∧ ⟧] ⋁ ≥ ⟦ 휔×휔 훼 is single-valued is 훼 ⋀ ∧ ( ̂ ℎ∈퐵 퐙퐅 푈 훽∈dom 푢,휆 ) , the element ′ ⟦ : For simplicity, consider the finite path finite the consider simplicity, For : Cons ⇒ ~ 훽 = 훼 ( 픉 111 ) [ 픉 ℎ,훼 〈 ( 풯 (

~ 훽 퐙 ¬ + 퐈퐙퐅 〉 on ⟧ isthe empty sequence and ) ℎ ⋁ ≥ → ⟧ ⟦ ℕ 59 ℎ∈퐵 → 훼 is total is 훼 휆 ⟦ ∗ 픴 ∈ 훽

] fromExample 6.17 that 푢 . ⟦ BI ′ 픉 ∈ 훼 In order to applyTheorem 6.35, we need

D = ) 휔 .

⟦ ⟧ ⟧] 휇 ⟧] 훼 is totalis 훼 ∧ …

⋀ ∧ ⟦

훼 is single-valued is 훼 훽∈dom …

⟧ ( ∧ 픉 ) ⟦ ⟦ 훼 is single-valued is 훼 픴 ∈ 훽 = 휇 … … = 푢 〈

휔

,,,,,… 2,3,0,0,0, ⟧ ⟧

as join of clo- 1. = 1 ∧ 1 = 〈 2,2 WS WS 〉 . In the In . 2019 ⟧ 〉 .

. /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. We already know that Since this union is disjoint, for each disjoint,for is unionthis Since Each Each oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological Ⓜ = 푈 Hence, we may write where allwhere three sets are clopendisjoint. and as before. Again, we can write: Proof Proof ⋃ Notice that 푘 ≠ 푘 Proposition 6.38 formalizeus this argument: Note that 푥∉푈 Lemma 6. ⋂ : : Let : 푢,휆 푂 푛 ⟦ 픉 ∈ 훼 푛<푚 훼 , bydisjointness. Hence, 푢 ∪푈 is | 푚 〈 푢

푈 〉 훼 cl ,휇 푢 = be of length of be 푢,휆 means that means for each 푈 ( { open: This works almost the same as before: Set :푣⊇푥 ⊇ 푣 푣: ( 39 푢,휆 ̂ , ,푘 푛, ̂ 푡 푈 : Let ⟧ and and 〈 : The basic sets

⋀ = 푛 〉 휇 , ) ( ) ̂ 푛,푘 ℕ ∈ 푡 and and the grey area are disjoint and their union is ⋀ = isan open neighborhood of 푈 } . Since . ℕ 〈

) ( 푡 〉 ̂ ∈dom 푛,푘 ∗ ,휇 is contained in the second conjunct. For each as the following asthe disjoint of union open sets: 푛 푢 are disjoint and all points innot ∗ , i.e. , ) 푡 푛<푚 , ∈dom ( 푛 픉 ∈ 훼 ( ) 훼 { ≠푘 :푣⊇푥 ⊇ 푣 푣: | 푚 = 푢 ( ) 훼 푈 ℕ ℕ 푛, [ ) and 훼 푢,휆 ∗ [ ∗

¬훼 ( 〈 푈 = WS WS 푈 = ( 푢 are clopen and so are sets the 푛 ̂ ,푘 푛, 0 } , there is a uniquea is there , ( 푢 , … , 휔 ∈ 푚 ( 푈 = 2019 푢,휆 푢,휆 ̂ ,푘 푛, ) ) 푈 ∪ 푈 ∪ → ⋃ 푥,푥∗0 /20 푛−1 ) 푘 • ⏟ )] . Then there is some ⟦ 〈 〈

̅

( 훼 ∧ 〉 〉 〉 ̂ ~ , where where , ,휇

,휇 ,푘 푛, . We define the infinite path path infinite the define We . ( 푛<푚

⋂ ( ∪ ∪ 푡 ̂ 112 ,푘 푛,

. Let )

푥∉푈 푥∉푈 푢 ∈

훼 )

( ) ̂ ~ 푢,휆 푢,휆 ⋃ ⋃ (

푢 0 ̅ 푡 ℕ = ,푘 푛, 푘 ̂ 푡

∪푈 ⟧] ∪푈 푂 = 푛 be the bethe sequence 푢 푈

such that such 〈 ⋀ ∧ 〈

푛 〈

푢,휆 〉

∗ 〉 ,휇 0 ∗ 푢 = 휆 ,,,… 0,0,0, ,휇 ) . (

̂ 푛,푘 { ) { :푣⊇푥 ⊇ 푣 푣:

:푣⊇푥 ⊇ 푣 푣: . 푈 ∪

) ∈dom 푢

〈 푡

푂

〉 ,휇

푢 휔 ∈ 〉 ̅ ℕ , this shows that shows this , 훼 ∈ 푡 , then

lie to the lie left of it. ( ℕ = 푚 < 푛 푢 ∗ } } 푡 . The grey area may be written as , , <휔

)

[ 푢 ∗ ( such that such ( 푡 ∖ ( { 푘 ( , ,푘 푛, :푣⊇푢 ⊇ 푣 푣: ̂ = 휇 ( 푛 { 훼 ̂ ,푘 푛, :푣⊇푢 ⊇ 푣 푣: ) ( 푛<푚 푛 ( ,푘 푛, ) 〈 ̂ ) 푢 ) ) . In particular, the settheparticular, In . , then 0 → 푛 푢 , … , } ∈ 푡 ) } 푈 = ⟦ 푈 ) for ( 푢,휆 ¬훼 ⊆ ̂ ,푘 푛, ⟦ 푛−1 훼 푢,휆 is clopen. Let clopen. is ℕ ∈ 푢 | 푚 ) and and let ,,,… 1,0,0, + 훼 ∈ ( 푢 = ( ̂ ,푘 푛, ̂ ∗ ⟧] . 푡 ⟧ ) . ) 휇 for be ∎ 〉 .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ⋃ Proposition 6.38 formalizeus this argument: Note that The secondThe antecedent is basically the thatfact the oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison We can now show that the special case of for each Finally, the third antecedent: For

ℕ { in valid not is

For the first antecedent, let immediately getthat We will show that all the antecedents evaluate to ℕ ∈ 푣 ⟦ ∗ 푥∉푈 ̂∈픅∨푢 픅 푢̂ ∉ ∨ 픅 푢̂ ∈ . As inLemma. As 6.39, let 푢,휆 ∗ ∪푈 푢 ⊇ 푣 : ∈ 푡 푚 < 푛 〈

푈 〉 ,휇 ⟦ 푢,휆 { 훼 :푣⊇푥 ⊇ 푣 푣: | , } 푚 , which shows, which that the first conjunct is an open neighborhood of . Note that for 푈 ⟧ 〈 푢 = ): Let):

푉 〉 픅 = 휇 , ( ̂ 풯 푡 and and the grey area are disjoint and their union is ) } ⟧ ⟦ . Since Since . Here . ( ℕ ∈ 푢 = ⊆ 〈 푂 ∩ 푢̂

〉 ) ⟦ 푛∈휔 ⋃ ⋃ 픅 ∈ 푘∈휔 훼 휔. ∈ ∃푘 ¬픅 ∪ 푢 푢 [ 푡 푡 픉 ∈ 훼 훼∈휔 ∈ ∀훼 푢∈휔 ∈ ∀푢 푢∈휔 ∈ ∀푢 ∗ = ⟦ 푢∈휔 휔 ∈ { , then , then ⟧ 휔 ∈ 푢 ̂∈픅 푢̂ ∈ 훼 ⊆ 푈 ∈ 푡 :푣⊇푥 ⊇ 푣 푣: 픅 푂 = ⟦ ( 훼 <휔 푢̂ i gvn by given is <휔 . For each | ℕ ∈ 푢 ) 푚

〈 <휔 <휔 휔 [⟦ |

픅 = besuch that <휔 ⟧ 〉 푘 푛∈휔 훼 휔. ∈ ∃푛 . 푢 = 훼 ℕ = = (

픅 ∈ } ( | , wehave that 푘∈휔푢∗ 푢 휔 ∈ ∀푘 ̂ 푛 픅 ∉ 푢 ∨ 픅 ∈ 푢 푈 = ∗ ( 푣∈휔 ( 푡 , wehave 푢̂ ( 푢 = ⋃ ∗ ⟧ ,푘 푛, ⟧ ) ̂ BI ∖ ∩ . <휔

푥,푥∗0 Int ∪ ℕ ∈ 푡 { ⟧ D ⟦ 푛 ℕ ∈ 푣 [ 푢 픅 ∈ ̂

∩ 픅 ) ̅ | dom 푡 ) 푛 ~ , ( where where ⟦ ( ∗ ∈ 푡 푢̂ ⋀ ⊆ 픅 ∈ 푢 ℕ ∧ 픅 ∈ let 〈 113 ℕ ) 푘 ∗ ∗ ( ( 푂 ∩ ⊇ 푣 : ̂ ∗ 〉 푛,푘 픅 픅 ∖ ⟧ ⟦ while the consequent has truth value 푢 ) 픅 ∈ 푢 → 픅 ∈ length = 푚 훼 푢 ⟦ ) ⊆ s are clopen ( ) ∧ 푡 ̂=푢̂푣̂ = | ~ ⟧] ∈dom ( dom = 0 ̅ 푚 ( 푛 푢∈휔 푢̂ 〈 ) = =

≠푘 ⋃ 푢 = ) 〉 ) } ⟦ 〈 ( <휔 ̂ ⟧] 훼 ,,,… 0,0,0, ℕ = 푂 = 푛∈휔 푢∈휔 ∈ ∃푢 휔. ∈ ∃푛 푡 )

( [ [⟦ ⟧ 픅 = 픴 ¬훼 . Then 훼 푢 ( ∗ <휔 푡 | ℕ ∖ Int ∪ ) ( ) 푚 ( ] ( ) 1 + 푢̂ ̂ ,푘 푛, 〉 → 푢 = and and , this shows that that shows this , ℕ ) ∗ ∗ 푂 = ∅. = ( 〈 . Then clearly, . The grey area may be written as ) ℕ

⟧ 〉 )] ∗ 픅, ∈ ∩ 푢 픅 , 푂 ∖

. <휔 ⟦

( 픅 ∈ 푢 푢̂ 푢

( ) ) 훼 푂 = 푡 푂 = | aswell. 푛 ⟧] 픅 ∈ 푢 ∧ 푢 = 푢 푢 , where where , 푂 ∈ 푡 ∪ 푈 푢,휆 ( ℕ is clopen. Let clopen. is ∗ 푢 ∅ for each 푂 ∖ . Indeed, we WS WS 푂 ) 푢 ⟧ 푢 ) 2019 ℕ = ℕ = ∈ 푢 /20 ∗ ∗ ∎ . ∖

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. h udryn te) W dfn te olwn tplg on topology following the define We tree). underlying the 푇 through Ⓜ In this section we will show that 6.5 induction Bar of Independence and Semantics Topological Lemma6.41 following the results: two star We thing. same the to down come formulations these all that following the in show will We Letfollowingthe way: in topologizefanany to wantwe First, 6.5.1 We will start with discussing some equivalent formulations ofthe fan theorem. We thus have sequences of natural numbers in such a way that way a such in numbers natural of sequences Theorem 6.11 and the topology a basis of a topology on Proof This concludesThis the proof of independence of where the equalitythewhere before lastthesince holds, any open set containing

ing versionsing of the fan theorem: <휔 Theorem6.40

oflength : We can think of the underlying tree of tree underlying the of think can We :

⟦ Compatibility with the Fan theorem ⟦ 푘∈휔 ∗ 푢 휔. ∈ ∀푘 Fan Fan theoremequivalentand formulations 푘∈휔 ∗ 푢 휔. ∈ ∀푘 푇 , starting at the root. For each each For root. the at starting , The spaceThe ( spaceThe ( : Each fan 훼∈2 ∈ ∀훼 휙 휔 ∈ ∃푛 푇 ∈ ∀훼 푛 , wedefine ̂ ̂ : Cons 〈 푘 〈 휔 푘 〉 ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 〉 Int = n ( Int = ̂∈픅 푢̂ ∈ → 픅 ∈ 푇 푇 푇 픅 ∈ ( 2 is compact. is 퐙퐅 푇 is homeomorphic toasubfan of is compact. is . By ) ⟧ ( Φ { Cons ⇒ ⋀ = 푘∈ℕ ⋂ 푢 ( ( 푇 푢 } 훼 ( 푘∈휔 2 ) ∪ 훼 | we denote the important case of the binary tree. We consider the follow- 풯 ℕ as finitethe sequence 푛 퐈퐙퐅 | ( on ) 푛 ∗ ⟦ ℕ ) ⟧ ) ∖ ∗ 푢 (

WS WS ) ∗ → is compatible with the fan theorem. To be more precise, we will use 퐈퐙퐅 = ̂ { ℕ ∖ → ℕ ∈ 푣 ( ⟦ 〈 ∗ 2019 { 푘 ∃푚 ∈ 휔 ∀훼 ∈ 푇 ∃푛 < 푚 휙 푚 < ∃푛 푇 ∈ ∀훼 휔 ∈ ∃푚 from Exampleto show: 6.17 푘∈휔 ∗ 푢 휔. ∈ ∀푘 ( ℕ ∈ 푣 퐅 + 푇 ∈ 푢 〉 푚∈휔∀ 2 ∈ ∀훼 휔 ∈ ∃푚 픅 ∈ /20 ∗ ) | . ∗ 푢 ⊇ 푣

BI ∗ 푇

<휔 ⟧ | ~ as subtree of subtree as D 푢 ⊇ 푣 ⋀ = . , let , ̂ 114 푘∈휔 ∗ 푢 〈 푘 〈 } 푈 〉 ⟦ 푘 ) 푢 ~ ∗ 푢 〈 ) 픅 ∈ 〉 푘 } ̂ 휔 = ℕ = n (ℕ Int = ) 〉 푇 ∃ .휙 푚. < ∃푛 2 〈 푇 ∈ { ⟧ 푘 . . :훼⊇푢 ⊇ 훼 푇: ∈ 훼 〉 → ∗ 휔 픅 ∈ ∖ <휔 푇 ⟦ 푇 휔 Let : { ̂∈픅 푢̂ ∈ be a fan (thought of as infinite pathsofinfinite as of(thought fan a be ℕ ∈ 푣 , where the nodes are labelled by finite by labelled are nodes the where , ( ∗ 푢 → ⟧ 훼 ∗ n ( Int = | ( 푛 ∖ 푇 훼 ) ⟧ ∗ ) 푘∈ℕ | ⋃ <휔 | 푢 〈 푛

= 푢 ⊇ 푣 } 푖 ) must alsocontain must some . Then . 〉 dnt te e o fnt paths finite of set the denote ) ⟦ { 푘∈ℕ ⋂ 푇 ∈

ℕ ∈ 푣 ̂∈픅 푢̂ ∈ } <휔 푂 픅 = 푢∗ = ℬ ∗ for all all for ⟧ | 〈 푘 ∗ 푢 ⊇ 푣 → 〉 ( ) 푢 { ⟦ 푈 ) ̂∈픅 푢̂ ∈ 푢 = 푘 < 푖 푇 ∈ 푢 : ⟦ 〈 ̂∈픅 푢̂ ∈ 푘 ⟧ 〉 } . Given Given . ℕ = ) <휔 (퐅 (퐅 (퐅 (퐅 ⟧ } . ∗ ∗ 푢

2 푇 푇 2 forms 퐜 퐜 . t with t

) ) ) )

〈 ∈ 푢 푘 〉 . Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. neighborhood max of = 푚 Then Then Proof o te otniy of continuity the For Lemma6.42 Theorem6. We can now establish equivalence of differentthe of versions the fan theorem: ping of fans, i.e. fans, of ping oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison Proof borhood of

and and

opcns, hr are there compactness, max = 푚 훼 퐅 define the formula Proposition 6.44 each each 푚 Hence, there is some thereis Hence, 푇 푖 } | Φ 푛 covers 퐅 → 푖 { ( 푛 ∑ : For each path : 훼 푇 ∈ 훼 훼 = 훼 :푇→푇 → 푇 Φ: with 퐅 ( 푖=0 푛−1 푇 ) 퐜 푇 푖 퐜 , wemayassume that : We showopensubcover:everysetscoverfinitehasthat We of basicLet a : ) | { 퐅 → 푇 ∈ 훼 : 푛 푛 훽 푇 , 푖 푖 푖 , weconclude 휙 43 훼 푈 . 푘 < 푖 : : Every subfan 푛 + ( disjoint from 훼 푇 : For each fan 훼 2 | : Suppose, : 푉 푛 is defined is by | } . 푛 of . Now, for any open neighborhood open any for Now, . 풪 ∈ : Let Φ ) } the set the [ Fr each For . 휙 푇 Φ for some 푇 ∈ 훼 onfinite sequences as ] ( 휔 ∈ 푚 푇 is a fan as well. Clearly, well. as fan a is 훼 ′ ) be a subfan of Φ 훼 such that for all for that such 훼∈푇 푛∈휔 휙 휔. ∈ ∃푛 푇. ∈ ∀훼 휙 not in 0 푈 −1 훼 , … , 푇 푇 ( 훼 such that for all all forthat such 푇 훼 | . Wecan write ′ te at that fact the 푛 Φ , ofafan | 푇 ∈ 훼 푈 푚 < 푛 to 훼∈푇 푛∈휔 휙 ∈ 훼 휔. ∈ ∃푛 푇. ∈ ∀훼 ⇔ 훼∈푇 푢∈푇 ∈ ∃푢 푇. ∈ ∀훼 ⇔ 훼∈푇 ∃푈 푇. ∈ ∀훼 퐅 푛 ( is of the isof form 푖 푇 훼 푘−1 ) ,…,0 … 0, 푢 ⏟ 푇

many . and 풪 ) (

′ 0 , there , there is some initial segment . By the premise and the fact the and premise the By . = ) tee s some is there , +1 푇 ∈

. As

⋃ 퐅 푇 ,…,1 … 1, , 푇 푇 푛∈휔 퐜 isclosed. 푢 ⏟ and and

. Then many ( are equivalent over 푈 ∈ 훼

1 ) 푉 ∈ 훽 ( Φ +1

푇 ∖ 푇 휙 훼 푢 푇 푇 ∈ 훼 ( 푛 | ( 훼 ,…,0 … 0, , .훼∈푈 ∈ 훼 풪. ∈ 푛 푈 1 i fntl bacig s rca: o each For crucial: is branching finitely is 푢 훼 푢 ⏟ ~ 퐙 퐅 ⊢ 퐈퐙퐅 Φ | 푛 , … , ) | Φ many ) ( 푛 푛 <휔 , . We construct an open cover open an construct We .

2 ′ is injective. Given injective. is ( , we have shown that the finite set ) 푈 ≡ Φ as the asthe union of all such neighborhoods. ) 훼 115 . By our assumptionBy our . structuretheonof , there is someis there , +1

(훼 . | −1 푛 푘 < 푖 푘−1 ) 푈 ,…,1, … 1, , 푢 . Then for all ( 푢 | ⏟ 훽 of ~ 푇 풪 ∈ ( dom many ( 훼 푢 )

3 휔 ∈ 퐅 → sc that such )

| 푈 ∈ 훼 +1

푛 . Since . Since ( (again, assume (again, 푢 ) 푇 . sc that such ) ,…,∗ … ∗, , …

퐈퐙퐅 . Such a Such . ′ 휙 ∧ 푢 = . 푢 . 푈 ∈ 훼 ( ⏟ 휔 ∈ 푛 푛−1 훼 many 풪

푈 ∈ 훽 | covers 푛 푈 ∈ 훼 푇 ∈ 훼

) +1 푇 ∉ ( 푉 훼 푢 with (푈 | . can be given as given be can ) 푛

훼 ) 푛 , 훼 |

and an open neighborhood open an and 푛 훼 풪 . Hence 푖 푇 | , wehave that 푖 푛 푈 = 푈 | is indeed an open cover. Bycover. open an indeed is 푛 , 푖 휙 푖 Since . ) ( 푖<푘 훼 풪 ℬ ⊆ 풪 | for 훼 푛 covers 푈 | ) 푛 . This meansthatfor This . 훼 { ), we have to find a find to have we ), 휙 푈 | 푇 푇 푛 with 푢 ( isan open neigh- : Add for each for Add : 훼 :length 풪: ∈ 푖 푈 | Φ 푇 푇 푛 푇 Φ 푖 ( , ⋃ ) ( s el Let well. as 훽 푛 훼 Φ hls and holds, WS WS ) | 푇 = 풪 푚 let , is a map- is 푈 ∈ ) , where , 2019 ( . 푢 . We. 훽 ) 푖 /20 ∎ ∎ ∎ < = 푈 훼

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. This meansthat This As As fied, fied, Hence, that conclude to 6.46 Corollary apply to want would We equivalent to the fan theorem (Theorem 2.4). This implies that implies This 2.4). (Theorem theorem fan the to equivalent However, atthis point, all we can sayis the following weaker version: Proof Ⓜ We are now ready to our proof ofcompatibility: 6.5.2 induction Bar of Independence and Semantics Topological Ⓜ To infer the generaltheinfer Toform of compact in Corollary holds. Then, if Then, holds. Corollary Corollary6.46 Proof pact as well. By Lemma 6.41, Lemma By well. as pact sition 6.20. fore Theorem by compactnessof by Proof Corollary 6.48 Proposition 6.47 푡 wasarbitrary, this concludes the proof. 퐅 : Let : : Thedirectionright : fromto Lettrivial.isleft

푇 퐅 ′ 푇 Proof compatibilityProof of . 퐅 →

ℬ ⊆ 풪 6.45 퐙퐅 푇 퐜 . ). Hence,). there issome finite subcover : Over : 푇 퐙 퐅 ⊢ 퐈퐙퐅 and let and { ′ is a closed subset of a compact space, hence compact itself. This shows This itself. compact hence space, compact a of subset closed a is :푡∈ 푡 풪: ∈ 푂 : 푉 푇 : (풯) 2 푉 퐈퐙퐅 and Proposition 6.21 it is compact itself. We conclude We compactthat itself. is it 6.21 Proposition and ∈ 푡 ( 푇 풯 퐜 ∀ 푇 ∈ (∀훼 ⊨ ℕ ∈ 푡 frec a 퐅 ↔ for each fan 푇 ) , all , all formulationsthe ofthe fan theorem are equivalent. ⟦ 훼∈푇 ∈ ∀훼 푇 ⊨ 푂 1 ⟦ ̂ 퐅 푂 2 푂 ∧ … ∧ 풪 ∈ ∗ ̂ 2 푇 is compact and and , let us argueinsidelet us , 풪 ∈ is homeomorphic to a subfan a to homeomorphic is 2 ̂ ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 2 ̂ ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 ⟧} ∈ 푡 WS WS 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 is an openancoverof is ⟦ ∈ 훼 ⇔ ∈ 푡 2019 훼 2 ∀훼̂ ∈ 푛 . 풪 ∈ 2 퐜 ( /20 . 훼 ( 훼 | 푂∈풪 ̂ ⋃ 푛

⟧ 푡∈ | 휔 ~ ) 푛 ∧ ⋃ ⟦ , then in particular, the premise of the weak the of premise the particular, in then , ∃푂 ) 푂 ⟦ ̂ ⟦ ∃ 훼∈푇 ∈ ∀훼 휔 ∈ (∃푚 → ) 116 ̂ ∈풪 ̂∈푂 훼̂ ∈ 훼∈2 ∈ ∀훼 ̂ 푇 푉 2 ̂ 2 ∃푛 < 푚 휙 푚 < ∃푛 ⟧ 풪 ∈ ( be compactand be 풯 푂 푂 ) 1 ~ ̂ . ̂ : Suppose, theSuppose, weaker form: of

푂 … , 훼 푂 훼̂ ∈ 푂 ∧ 2 휔

휔 퐅 . ̂ 2 ( . Classically,. 푛 푂 ∈ 훼 풪 ∈ (and hence (and of ̂ ⟧ ( 푇 ̂ , i.e. for each 훼 2 ⟧ ′ | of

푇 휔 푛 1 2 ) . This shows. This is externally compact (i.e. provably (i.e. compact externally is .훼∈푂 ∈ 훼 ∨. … ∨ . 푇

̂ 2 2 푇 . By Lemma 6.42, Lemma By . ∃푛 < 푚 휙 푚 < ∃푛 any fan. Weshow fan.that any König’s lemma 퐅 푇 for all trees all for 푇 ∈ 훼 푛 ) 푇 ( ⟧ 2 훼 . is compact by Propo-by compact is

, | 푛 ) 퐅 ) 2 . . 푇 ′ 2.1 holds and isand 2.1holds 푇

of the previousthe of ) ′ 퐅 is closed and closed is holds in holds 푇 풄 ′

퐅 and there- and 2 푇 ′ is satis- is is com- is 푉 ( 풯 ∎ ∎ ∎ ) .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Proof where where This This mula mula Ⓜ thisIn section, we will show that the WeakContinuity Principle 6.6 Theory Set Constructive of Analysis and Comparison Ⓜ Lemma 6.50

for all for all some this that show will We set 푔 푢 This showsThis that 1 푛 ̃ is defined as . Wecan therefore conclude Proposition 6. Theorem 6.49 푔 = ℎ 4. 1. 2. 3.

푓 : Let

휙 푢 Compatibility with the Weak Continuity Principle is continuous: Let continuous: is 푈 ∈ 푥 픞 푛 ( :ℕ ℎ: :ℕ 푓: 푔 푔 ⟦ ℎ 푥 훼∈푇 ∈ ∀훼 1 2 2 푇 ∈ is defined recursively by 1 ℕ : ℕ : length 푥 , … , 푔 ∘ ∗ ∗ 휆 ̃ ∗ ∗ 2 : There are continuous functions <휔 푈 → → ,푣 1 −1 → 푈 → , we thathave { 2 such that such 푛 :푣⊇푢 ⊇ 푣 푣: 푔 ( { 푓 ∘ ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 휆,푤 푉 ) :푣⊇푢 ⊇ 푣 푣: 푢 : 1 휆,푤 51 with all free variables shown and (풯) Cons ( ) with with . 푣 : Let 푛 = homeomorphism with ) 퐅 ⊨ 푣 ∗ 푢 = ≤ ≤ ≤ ≤ ( } . Wedefine 푚 퐙퐅 푓 푓 with with } ℎ 2 :퐻→퐺 → 퐻 ℎ: ⟦ ⟦ ⟦ ⟦ homeomorphism with ( ( ′ works for works 푓 | 훼 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 (

푣 푣 { ) 훼 (and thus(and 푣:푣⊇푢 ( | ) ) 푛 푣 Cons( ⇒ | 푓 푛 = 푢, 푢, = 푢 = and and is clearly continuous and ) 푢 = 푓 ) ( 푈 ∈ 휙 | ⟧ 푥 } { : 푣:푣⊇푢 ) ( ≤ 0 { be a homomorphism of complete Heyting-algebras. Then for each for- 푛 훼 :푣⊇푢 ⊇ 푣 푣: 푢 , … , 푈 ∈ 휆,푤 . But But . ℎ | 픞 ⟦ 푓 푛 ℎ ( 훼∈푇 ∈ ∀훼 퐈퐙퐅

푇 } as follows: For any ) 푉 . Without loss of generality, of loss Without . ⟦ 휆,푤 2 . Formally, the proof looksthis: like = = 휙 (풯) as well: Let well: as 푛−1 ̂ ( 푇 퐖퐂퐍) + { . id 2 픞 2 2 2 2 〈 } <휔 ∃푛 < 푚 휙 푚 < ∃푛 1 ∃푛 < 푚 휙 푚 < ∃푛 ∃ 푇 ∈ ∃푢 ∃ 푢∈푇 ∈ ∃푢 푚 < ∃푛 ̂ 퐅 ⊨ 픟 { 푣 , 푔 푣:푣⊇푢 픞 , … , 2 푈 → ℎ 2 ∃푛 ∈ 휔 휙 휔 ∈ ∃푛 ℎ , 푛 푇 = ( 푇 푣 , … , 〈 (

휆,푤 〉 for any treefor any ~ } ̂ 픞 푛 ) . 푔 ̂ 2 (

2 <휔 ) 1 푚 homeomorphism and 푤 = 픟 117 ⟧ ( length )

) 〈 ( and hence and ) 푇 ∈ 훼

( ( 픞 〉 훼 = 푢 = 〉 훼 훼 ) , 1 dom ∈ 픟 : ( | | 픞 , … , 푢 = 훼 ~ ⟦ 푛 푛 ( 휙 | ̂ ) ) 푣 푛 2 ℕ ∈ 푣 2 ⟧ ⟧ 푛 ) (

) . We know that for each for that know We . −1 | , . 픞

푚 ⟧ ( 푇 푛 1 ℎ 푔 훼 = 푢 , , length if ≤ ) 픞 , … , ∃ .휙 푚. < (∃푛 ∧ 푉 ∈ 2

and concludes theTheorem proof of 6.40. is defined similarly. Finally, for ∗ ( ⟦ 휙 , weset 픞 푚∈휔∀ 푇 ∈ ∀훼 휔 ∈ ∃푚 if must hold true for one of the of one for true hold must ) ( 퐖퐂퐍 length 퐻 푛 ℎ } | 푛 푢 ⊇ 푤 . ) )⟧ , wehave that 휙 ∧ ,

is compatible with ℎ ( ( ( ( . Let . 푢 푣 푣 푢 ( | ) ) 푛 ) 푢 푛 < ) 푛. ≥ | 푤 = )⟧ 푛 ̂ 휆 ) ̃ 2 )⟧

∗ 푢 = ∃ .휙 푚. < ∃푛 .

푚 < 푛 〈 휆 푛 휆 , , there is some is there , 퐈퐙퐅 ( 푛+1 훼 | : 푛 WS WS … , ) ℎ ⟧ 푢 , we can 〉 2019 푛 . Then . s, say s, /20 ∎

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oooia eatc n needneo a nuto induction Bar of Independence and Semantics Topological Ⓜ That showThat the desired equality of functions. tions: As any continuous function continuous any As tions: ℕ → 휔 × 휔 Proposition 6.52 Remember, that in our topology on topology our in that Remember, set of allset of On On the other hand, let and functionsand ⟦ In In regard with this proposition and Proposition 6.51, we may ask how bras of the topological spacestopological the of bras shown by simultaneous induction. Proof Proof Hence, for each for Hence, fines amap fines nuity only for these sets. We have is continuous: Note that the sets the that Note continuous: is (픞 휙 Lemma6. : We thehave chain of equivalences : This is shown by a straightforward induction on the structure of structure the on induction straightforward a by shown is This : 1 푓 −1 픞 , … , 휔 × 휔 ∈ 훼 ∗ . Clearly both constructions are inverses ofone another and we provedhave 훼̃: ℕ 53 푛 푓 ̂ −1 ℕ → 휔 × 휔 : Let ∗ ) ℕ ∈ 푡 : The assignment ⟧ 휔 → . Weare interested how the elements of such that such for all 픉 ∈ 훼 ∗ 휔 :ℕ 휂: and each and . If we equipweIf . ∗ an ∗ . 휔 → d 푓 :ℕ 푓: −1 휔 푛 푉 be continuous, then . We can thus apply Proposition 6.51 to get that that get to 6.51 Proposition apply thus can We . 푛,푚 훼̃ ↦ 훼 , there is a unique a is there , ∗ ℕ WS WS 훼 = = = 훼̃ 푛 ℕ → ℕ 휔 ∗ −1 , = ℕ → ∗ 휔 2019 , the function space function the , [ with the withtreeusual topology the fromsection,last see thatwe 푉 { is a 1-1 correspondence between continuous functions ∗ 푛,푚 휔 ∈ 훼 푓 ⇔ 훼̃⇔ 훼̃ 훼 ∈ 푡 ⇔ 푓 ∈ 푡 ⇔ continuous. Then ∗ ⋃ /20 gives rise to the homomorphism of complete Heyting-alge- complete of homomorphism the to rise gives 푚 • ] = ( (

푓 훼 푡 휔 푓 −1 ~ ( ) { { { ( 훼 : 푛 푚) (푛, :훼̃ 푇: ∈ 푡 :푡∈훼 ∈ 푡 푇: ∈ 푡 :훼̃ 푇: ∈ 푡 ( 푡 훼 ∈ ( −1 ̂ 푓 118 푡 ) ( −1 ( )( ) ,푚 푛, ̂ 푛 [ ( 훼 ( ( 푚 푚 ) 푚 ( ( 휂̃ ( ~ 푚 = ,푛 푚, ̂ ) ̂ ,푛 푚, ) ) ( defined as such that such ) ,푛 푚, ̂ ( ( 푛 = ℕ = ) 푛, = 푡 푡 .

) )( ) 픉 ) } 휔 푉 ∈ ) form a subbasis, so we need to check conti- check to need we so subbasis, a form ) ( 푛

) 훼̃ look like thisunder mapping: ∗

( )] . 휔 )

푓 ,푚 푛, ̂ 푛,푚 , as given by given as , −1

푚 = ̃∘푓 훼̃ ∘ = 훼 ∈ 푡 } ) 휂̃

)} } (

,푚 푛,

훼̃ ( behaves under continuous func- 푛 푚) (푛, . ̂ ) 휂 = 휙 픉 , the cases for atomic for cases the , from Theorem 6.35, is the is 6.35, Theorem from ) −1 . Hence, externally, Hence, . [ 푉 푚,푛 푓 −1 ] defines a function ( ⟦ 휙 ( 픞 1 픞 , … , ℕ ∗ 푛 휔 → 휙 훼 ) ⟧ de- are ) ∎ ∎ = 휔 훼̃

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. Then Then clearly Set Let Possibly redefining a contradiction. such thatsuch obtain some We define Hence, for all all for Hence, Proof 휔 ∈ 휉 as continuous function 푈 As the interior of a set is the largest open set contained in it, this means that there is some is there that means this it, in contained set open largest the is set a of interior the As Claim oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

Suppose, towards acontradiction, that for some Theorem6.49 of Proof With thesepreparations, we arenow ready to prove main the result this of section: 푢,휅 푠 ( with 푖 푛 : As As : 휔 ∗ 푡 = 푖 : There are 푡 ) such that such 푚 푛 푖∈휔 int ∉ 푡 ∈ 휌 푈 ∈ 푡 :ℕ 휂: be a sequence of natural numbers such that each each that such numbers natural of sequence a be 푖 〈 ⟦ ( 푖 푠 휙 푡 ⟦ 〉 푖 푚 푛 and 휉 ( ) ,푛∈휔 ∈ 푛 푚, ∗ 푤 | 훽 ( 푚 푡 = 푈 ∈ 푚 휔 → 푈 푚 푛 ⋂ ∉ 푡 푡 푡 ,휆 푢,휅 푛 , 훽 = 푚 푚 푛 푛 훿∈픉 푚 푉 푖 푛 푢,휅 휂 푖 ) : We will show thatshow Wewill : and and 푖 휔 ∉ 푈 ∈ ⊈ and and ⟦ 푚 푛 ⟧ ⊆ = 푚 ⟦

푛 but but innot the intersection. Hence there are ∃푦, 푏 ∈ 휔 휔 ∈ 푏 ∃푦, by ⟦ 휉 by , = ⟦ 휉 | ⟦ 휉 ̃ { | ∉ 푡 푤 푚 훼∈휔 ∈ ∀훼 ℕ : ℕ ∈ 푣 휙 푚 |

⟦ ∗ 푡 푚 푚 ⟧ 휌 휙 ( 훽 ,휆 훿 = ∗ 휂 | ℕ = ⟦ 푚 ( 휂 = 푛 푉 푚 푚 〈 푛 휔 → 훿∈휔 ∈ ∀훿 훽 푖 푚 , wecan assume that and and 푉 : 푚 푛 , 푛 ∗ | 푠 ⊇ 푣 : n ( int = 〉 푚 ∗ 푚 푛 휔 [ 푛 , 푖 ⟦ ) . If . If 훿∈휔 ∈ ∀훿 푈 ∉ 휔 휉 푦 훿∈휔 ∈ ∀훿 휔 ∈ 푏 ∃푦, | ⟧ 푈 → 휙 → 푚 ) | , . By this continuity, for every

푡 ⟧ 푚 훽 ̃ ⟧ 푡 푚 푛 휔 푤 푚 ℕ ∩ 푚 푛 푛 → 휂 = 푤 휉 . 푖 푚 ( 휔 ∈ ( werein } 휉∈휔 푖 ,푛 훿, ⋂ ,휆 푡 B Lma 6. Lemma By . ,휆 | 휔 ⟦ )( ∗ 푚 푚 푖 푉 휙 푚 푛 휉 . are homeomorphisms. 휔 = such that such such that such for every 푘 ( ( ) 훿 = | ⟦ 풯 ) | 푚 휂 ⟧ ∃푦, 푏 ∈ 휔 휔 ∈ 푏 ∃푦, 푚 ⟦ ) ) 푚 푛 = ⟧ 휙 , for each for , 퐖퐂퐍 ⊨ | 푛 , 훿 = ℕ = ( 푚 ⟦ { ~ 훽 휙 휉 휂 ) 푚 푈 푛 휙 → ( 푚 ⟧ 푛 ( 119 | 휔 푡 푢,휅 훽 푛 , ∗ 푚 ⟦ = ( )( and 푡

푚 푛 휉 [ 푡 ) 훿∈휔 ∈ ∀훿 we have , where where , )( 휙 → 푘 | ( ( 푛 , [ ⟧ 푚 ℕ ,푛 훿, ) 훿∈휔 ∈ ∀훿 50 ~ 푘 ∩ , , ,푛 푚, ) ∗ 휂 = ) ⟧ , there are continuous functions continuous are there , 푚 ( 푛 , , ⟦ ∖ , then ) ,푛 훿, 휉 ⟧ ⟦ for for ∉ | we can use a form of the axiom of choice to choice of axiom the of form a use can we 푚 . As in Proposition 6.52, we may interpret may we 6.52, Proposition in As . 푛 휔 휉 푚 풯 | ⟦ | 휉 휔 ) 푈 ∈ 푥 푘≥푚. ≥ 푘 푚 푘<푚, < 푘 푚 휔 ∈ 푛 푈 휙 훽 = ]⟧ is again the topology from Example 6. Example topologyfromthe again is 휉 . | ⟧ 푢,휅 ℕ ∈ 푚 푚 ( 휂 = .

휂 | ℕ = 푚 푚 푚 훿 = 휂 푛 푛 ⊆ 푤 푚 푛 | 푛 , 푚 appears in it infinitely many times. many infinitely it in appears 푛 훿 = 푚 푚

⟦ ∗ | 픉 ∈ ̅ ( ) ,휆 ⟧ there exists a basic open 푚 : We simply define 훼∈휔 ∈ ∀훼 푚 ⟧ 푚 = | ⟧ .

푚 ) we have ) such that such ⟦ 휙 → ∪ 휙 → 휙 ⟦ ( 휙 훽 휔 ( 푚 ( ( 푛 ∃푥 ∈ 휔 휙 휔 ∈ ∃푥 ,푛 훿, ,푛 훿, 휂 푛 , 푚 푛 휉 ) 푛 , ) ) ( ⟧ ]⟧ ]⟧ 푥 ) ∩ )( ⟧ ).

⟦ 푚

휂 ( 푚 푛 ,푥 훼, ) 훽 ̃ 휌 푚 푛 휉 = 푖 훽 = ℕ : 푈 ∈ 푡 as ) WS WS ⟧ ( ∗ 푈 , but 푚 푛 푡 푤 푈 → )( ⟧ 푢,휅 2019 푚 푚 ,휆 and 푤 ) 푚 . /20 17. 푖 ,휆 ∎ ⊆ 휉 푖

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. common and widely accepted formalization of mathematics, namely There are two main ideas behind the axiomatic set theories levels levels of constructive demand. As discussed in section 2.4, the rather minimalistic requirement for towards existing everyday mathematical practice. Second, the two theories are tailored to meet differ NuPRL andNuPRL Coq have been given in the past (see, [ for example, Idr Agda, like implementations Various assistants. proof automated and languagesprogramming in Being explicitly based justification thatjustification motivationThe behind this is indeed so, is given in only chapter4 via the metamathematical tool of realizability. that it does not go beyond the realm of intuitionistic logic, i.e. ocuin Conclusion Martin- In this thesis we have analyzed and compared three different kinds of set theories: Brouweria Conclusion developed a proof-theoretical analysis of the notion of predicativity (see [ (see predicativity of notion the of analysis proof-theoretical a developed

have have been a thesis on its own. Instead we have justified interpretation of is continuousis at neighborhood 휂 notion ofsets. e hc that check We 휂 Since Since ovre to converge ⟦ ⟦ As As 휙 휙 is well defined, as the the as defined, well is ( 푥 ( ( ) 휂̃, 휂 휂 푡 푖 푖 푛 푛 휂 = 푈 ∈ 푡 푖 푖 푛 , 푖 푛 휌 = Löf’s settheory 푖 푖 푛 )⟧ 푖 )⟧ 푖 ( 푢,휅 , where , where ( 휌 , acontradiction to how the 푠 푖 푖 푡 ( ) ⊆ 푥 ad since and 푈 ∈ 푈 = 푈 ) ⟦ 휕 ) of 퐂퐙퐅 퐂퐙퐅 ⟦ 훼∈휔 ∈ ∀훼 and and ( 휙 ⋃ 푛 푡 ( 푖 푖∈휔 is indeed fully predicative, however, is more intricate. Schütte and FefermanandhaveSchütte intricate. more predicative, however,isindeedfully is into into sc that such ̃ ∘휌 휂̃ ∘ 휆,푡 푛 = on 휌 퐂퐙퐅 does the job: Indeed for Indeed job: the does 퐌퐋 푖 푉 푉 ( 휔 paradigms from programming, 푖 . 푥 푖 푖 −1 ) 퐌퐋 푛 푥∈휔 휙 휔. ∈ ∃푥 . are mutually disjoint. To show that it is continuous it suffices to show that itthat show to suffices it continuousis it that show To disjoint.mutually are ) and the and constructive axiomatic set theories = tofurther restrict apas niiey ay ie in times many infinitely appears 푛 , 푈 ∈

cl – 푖 )⟧ a theory that is considered to give a constructively clear and well-justified 휂 ( 휂 푤 ( ⋃

( 푖 푥 ,휆 (em 65) and 6.53) (Lemma 푥 푖∈휔 ) 푖 ) 휂 | . Hence, | 푖 ( ( 푘 ,푥 훼, 푠 푉 휂 = ) 푖 휂 = ) 푡 = 푚 ∖ 푛 ) 푖 푛 ⟧ ( 푖 wereconstructed. { ⋃ ( for 푡 휂 휉 휌 ) 푖∈휔 ( 푖 | 푛 푖 푘 ( 퐈퐙퐅 푠 푖 푥

휂̃ = 훼 ) ~ ( 휉 = 푉 , ) 휌 푖 )| istoend up with atheory that is predicative.Giving a 푖 120 푡 ≠ 푥 = ( 푖 ( 푠 푡 휉 = ) { , there is an ) ) 푡 | , } ~ 푘 , 퐂퐙퐅 (LemmaGiven6.22): 푡 ( for all all for 푈 ∈ 푥 푖 푛 휌 if if 퐌퐋 푖 푖 푠∈푉 ∈ 푠 푠∉ 푠 ( 퐈퐙퐅 ∈ in chapter 5, by giving a meaning-preserving 푥 ) seems to be the best suited for applications ⟦ 퐙 퐋퐄퐌 ⊬ 퐈퐙퐅 휆,푡 )| ̃ ∘휌 휂̃ ∘ and and ⋃ 푖∈휔 푈 ∈ 푥 푖 푖 mas that means 2 ( 휔 ∈ 푛 , 휉 = 푛 ], ], [ 푉 푖 푖 −1 퐂퐙퐅 ) 푖 33 푖∈휔 ( .

Let . 푡 퐙퐅퐂 휂 = with ], ], [ ) 퐈퐙퐅 : First, they are based on the most | tee s some is there , 푖 . The ultimate confirmation that . 39 푖

푛 . Hence, their usage is oriented 푖 and and 휆 23 ⟧ ], [ ], be the path path the be ∈ 푡 푉 ∈ 푥 w cnld, that conclude, we , ]), but presenting it would it presenting but ]), 푘 BO , we needtowefindopenan , 퐂퐙퐅 ⟦ 휙 ] 푖 ). ( fr some for

휂̃, 푛 . ) ⟧ . The 휔 ∈ 푖 ∗ 푡 〈 n n set theory, 푘 ≥ 푖 ,000 … 0,0,0, 푘, WS WS 푠 with 푖 ∗ 푡 = 2019 Thus . 퐈퐙퐅 푡 푖 푛 푠 ent /20 푖 푖 〈 is, is ∎ 푖 〉 ∈ ∈ 〉 .

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. full full our realizability structure for 퐑퐄퐀 oms of countable choice, dependent choice theories different ways to interpret its meaning. While meaning. its interpret to ways different choice of axiom classical The constructive in principles choice of that is marginally only touched we that topic Another ing, not purely set theoretical result, we have shown in chapter 6, that 6, chapter in shown have we result, theoretical set purely not ing, tion 2.4.2 are actually underivable statements in oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison A A further interesting question is which mathematical loss we have when passing form where witnesses can beconstructed. of case the in least At shown, relying shown,

principle and principle in in a different way in universe of all sets pendent from Brouwerian analysis basedon for true holds same the that is,thesis this in with dealt compactification is alwaysnot guaranteed in choice, of axiom the ing many results from classical topology relying on the set- property both that was 4 chapterof result Another by more elaborate metamathematical tools, that the stronger existence property does not hold for hold not does property existence stronger the that tools, metamathematical elaborate more by result, showing that showing result, true intrue of bar induction and and tions. Brouwer’s set theory goes far beyond what we have sketched in sketched have we what beyond far goes theory set Brouwer’s tions. co to differentfundamentallyapproachhis and continuum the ofjustification Brouwer’s un to enough theory, set Brouwerian of concepts basic some discussed have we 2.2, section In à-vis à-vis in concepts ordinalslike and cardinals. Itwould beinteresting to seehow his notions differ from their vis- We have shown in chapter 6, that Brouwerian analysis is, in principle, compatible to compatibleprinciple, in is, analysisBrouwerian that 6, chapter in shown have We 퐙퐅퐂 퐂퐙퐅 (for (for an overview, see [ 퐂퐙퐅 퐈퐙퐅

(see (see [ (section 2.4.2), it is an easy theorem in theorem easy an is it 2.4.2), (section 퐈퐙퐅 – a feature that it certainly expected from constructive theories. Anyways, it has been shown been has it Anyways, theories. constructive from expected certainly it that feature a in section in 2.4.3. The implications are far-reaching: The and and and and 25 ¬BI on ], ], [ 퐙퐅퐂 BI 퐂퐙퐅 퐷 the result that result the 퐷 63 푉

has been proved by Ščedrov in in Ščedrov by proved been has

. are not sets any more. The mathematical discipline of topology has to be dealt with turns turns out to be to strong an assumption for Brouwer’s proof of the fan theorem. Th 퐂퐙퐅 퐈퐙퐅 . Since . ]). ]). This may raise doubts for the aptness of like Tychonoff’s theorem. It turns out however, that existence of the Stone the of existence that however, out turns It theorem. Tychonoff’s like 퐂퐙퐅

. Instead of topological spaces one passes to point-free topologies, thus preserv- may be equiconsistently extended with the fan theorem, Brouwer’s continuity Brouwer’s theorem, fan the with extended equiconsistently be may this doubt is cleared up by the aforementioned interpretation of interpretation aforementioned the by up cleared is doubt this 53 퐋퐄퐌 퐈퐙퐅 ]). ]). Actually, interpretation of 퐀퐂 is not derivablenotin is plays very different roles in the set theories at hand. This is due to the to due is This hand. theoriesat set the in roles differentvery plays and and 퐂퐙퐅 퐂퐙퐅 is compatible with compatible is from section 3.3 works for 퐂퐙퐅 퐃퐂 퐈퐙퐅 퐂퐙퐅 , the representation axiom and the regular extension axiom and and ~ 퐌퐋 퐀퐂 퐈퐙퐅 . (see, for example[ 121 is a weak counterexample and hence extends hence and counterexample weak a is

퐈퐙퐅 (section 5.1.9). Further choice principles are the axi- the are principles choiceFurther 5.1.9). (section . For example, 퐈퐙퐅 [ 59 ~ possess the disjunction- and numerical existencenumericaldisjunction-and the possess , it follows that all weakcounterexamples it followsallthatsec-of , ]. Another result of this paper that we have not have we that paper this of result Another ]. BI 퐷 퐆퐔퐏 ch 퐂퐙퐅 in place ofplace in aracter of topological spaces and especially into into , that the powerset axiom does not hold not does axiom powerset the that , 퐈퐙퐅 푉 퐙 Foundation ⊬ 퐈퐙퐅 훼 and and 퐌퐋 19 s in thes in hierarchical structure ofthe 퐙 퐑퐄퐀 + 퐈퐙퐅 ¬BI ], ], [ works as well for 퐂퐙퐅 퐙 BI ⊬ 퐈퐙퐅 20 this thesis: In [ In thesis: this 퐷 ] making Bar induction inde-induction Bar making as constructive set theories. ).

and and 퐷 . As for As . . As a more interest- 퐙 퐑퐄퐀 + 퐂퐙퐅 퐈퐙퐅 퐙퐅퐂 7 ntinuous func-ntinuous . The principle The . 퐙 + 퐂퐙퐅 ], he develops he ], to the weaker 퐂퐙퐅 set theories. set 퐂퐙퐅 WS WS , we have we , derstand , too (see 퐃퐂 2019 in in 퐈퐙퐅 -Cech , and 퐌퐋 퐈퐙퐅 /20 to is ,

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. ocuin Conclusion principles we discussed chapterin 4. [ 1 ] , [ 55 ]). Hence both extensions also enjoy the metamathematical properties and compatibilities with compatibilities and properties metamathematical the enjoy also extensions both Hence ]). ~ 122 ~ WS WS 2019 /20

Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. References Theory Set Constructive of Analysis and Comparison

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Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. oprsnadAayi fCntutv e hoy Theory Set Constructive of Analysis and Comparison

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Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek. References

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Die approbierte gedruckte Originalversion dieser Diplomarbeit ist an der TU Wien Bibliothek verfügbar. The approved original version of this thesis is available in print at TU Wien Bibliothek.

Pair, axiom of axiom Pair, Pow(픞) ordinal ordinal open numerical existence property numerical ℕ ℕ MR MP I Intuitionism Intuitionism Infinity, axiom of Infinity, axiom Heyting-algebra axiom Foundation Extensionality axiom existence property property existence Empty set,of Empty axiom EI Disjunction property Disjunction DET DET CR CP Constructivism Collection schema Collection Col closed sets closed clopen sets clopen class choice sequencechoice Brouwer Brouwer Bounded separationschema Bounded Boolean algebra Boolean BI BHK-interpretation BHK-interpretation basis (of topology) basis a + { Index Index Theory Set Constructive of Analysis and Comparison

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