Large Sets in Constructive Set Theory
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Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a. -
The Logic of Brouwer and Heyting
THE LOGIC OF BROUWER AND HEYTING Joan Rand Moschovakis Intuitionistic logic consists of the principles of reasoning which were used in- formally by L. E. J. Brouwer, formalized by A. Heyting (also partially by V. Glivenko), interpreted by A. Kolmogorov, and studied by G. Gentzen and K. G¨odel during the first third of the twentieth century. Formally, intuitionistic first- order predicate logic is a proper subsystem of classical logic, obtained by replacing the law of excluded middle A ∨¬A by ¬A ⊃ (A ⊃ B); it has infinitely many dis- tinct consistent axiomatic extensions, each necessarily contained in classical logic (which is axiomatically complete). However, intuitionistic second-order logic is inconsistent with classical second-order logic. In terms of expressibility, the intu- itionistic logic and language are richer than the classical; as G¨odel and Gentzen showed, classical first-order arithmetic can be faithfully translated into the nega- tive fragment of intuitionistic arithmetic, establishing proof-theoretical equivalence and clarifying the distinction between the classical and constructive consequences of mathematical axioms. The logic of Brouwer and Heyting is effective. The conclusion of an intuitionistic derivation holds with the same degree of constructivity as the premises. Any proof of a disjunction of two statements can be effectively transformed into a proof of one of the disjuncts, while any proof of an existential statement contains an effec- tive prescription for finding a witness. The negation of a statement is interpreted as asserting that the statement is not merely false but absurd, i.e., leads to a contradiction. Brouwer objected to the general law of excluded middle as claim- ing a priori that every mathematical problem has a solution, and to the general law ¬¬A ⊃ A of double negation as asserting that every consistent mathematical statement holds. -
Sets in Homotopy Type Theory
Predicative topos Sets in Homotopy type theory Bas Spitters Aarhus Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos About me I PhD thesis on constructive analysis I Connecting Bishop's pointwise mathematics w/topos theory (w/Coquand) I Formalization of effective real analysis in Coq O'Connor's PhD part EU ForMath project I Topos theory and quantum theory I Univalent foundations as a combination of the strands co-author of the book and the Coq library I guarded homotopy type theory: applications to CS Bas Spitters Aarhus Sets in Homotopy type theory Most of the presentation is based on the book and Sets in HoTT (with Rijke). CC-BY-SA Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Bas Spitters Aarhus Sets in Homotopy type theory Formalization of discrete mathematics: four color theorem, Feit Thompson, ... computational interpretation was crucial. Can this be extended to non-discrete types? Predicative topos Challenges pre-HoTT: Sets as Types no quotients (setoids), no unique choice (in Coq), .. -
The Iterative Conception of Set
The Iterative Conception of Set Thomas Forster Centre for Mathematical Sciences Wilberforce Road Cambridge, CB3 0WB, U.K. September 4, 2009 Contents 1 The Cumulative Hierarchy 2 2 The Two-Constructor case 5 2.1 Set Equality in the Two-Constructor Case . 6 3 More Wands 9 3.1 Second-order categoricity .................... 9 3.2 Equality . 10 3.3 Restricted quantifiers . 10 3.4 Forcing . 11 4 Objections 11 4.1 Sets Constituted by their Members . 12 4.2 The End of Time . 12 4.2.1 What is it an argument against? . 13 4.3 How Many Wands? . 13 5 Church-Oswald models 14 5.1 The Significance of the Church-Oswald Interpretation . 16 5.2 Forti-Honsell Antifoundation . 16 6 Envoi: Why considering the two-wand construction might be helpful 17 1 Abstract The two expressions “The cumulative hierarchy” and “The iterative con- ception of sets” are usually taken to be synonymous. However the second is more general than the first, in that there are recursive procedures that generate some illfounded sets in addition to wellfounded sets. The inter- esting question is whether or not the arguments in favour of the more restrictive version—the cumulative hierarchy—were all along arguments for the more general version. The phrase “The iterative conception of sets” conjures up a picture of a particular set-theoretic universe—the cumulative hierarchy—and the constant conjunction of phrase-with-picture is so reliable that people tend to think that the cumulative hierarchy is all there is to the iterative conception of sets: if you conceive sets iteratively then the result is the cumulative hierarchy. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
A Taste of Set Theory for Philosophers
Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics. -
Weak Measure Extension Axioms * 1 Introduction
Weak Measure Extension Axioms Joan E Hart Union College Schenectady NY USA and Kenneth Kunen University of Wisconsin Madison WI USA hartjunionedu and kunenmathwiscedu February Abstract We consider axioms asserting that Leb esgue measure on the real line may b e extended to measure a few new nonmeasurable sets Strong versions of such axioms such as realvalued measurabilityin volve large cardinals but weak versions do not We discuss weak versions which are sucienttoprovevarious combinatorial results such as the nonexistence of Ramsey ultralters the existence of ccc spaces whose pro duct is not ccc and the existence of S and L spaces We also prove an absoluteness theorem stating that assuming our ax iom every sentence of an appropriate logical form which is forced to b e true in the random real extension of the universe is in fact already true Intro duction In this pap er a measure on a set X is a countably additive measure whose domain the measurable sets is some algebra of subsets of X The authors were supp orted by NSF Grant CCR The rst author is grateful for supp ort from the University of Wisconsin during the preparation of this pap er Both authors are grateful to the referee for a numb er of useful comments INTRODUCTION We are primarily interested in nite measures although most of our results extend to nite measures in the obvious way By the Axiom of Choice whichwealways assume there are subsets of which are not Leb esgue measurable In an attempt to measure them it is reasonable to p ostulate measure extension axioms -
Faculty Document 2436 Madison 7 October 2013
University of Wisconsin Faculty Document 2436 Madison 7 October 2013 MEMORIAL RESOLUTION OF THE FACULTY OF THE UNIVERSITY OF WISCONSIN-MADISON ON THE DEATH OF PROFESSOR EMERITA MARY ELLEN RUDIN Mary Ellen Rudin, Hilldale professor emerita of mathematics, died peacefully at home in Madison on March 18, 2013. Mary Ellen was born in Hillsboro, Texas, on December 7, 1924. She spent most of her pre-college years in Leakey, another small Texas town. In 1941, she went off to college at the University of Texas in Austin, and she met the noted topologist R.L. Moore on her first day on campus, since he was assisting in advising incoming students. He recognized her talent immediately and steered her into the math program, which she successfully completed in 1944, and she then went directly into graduate school at Austin, receiving her PhD under Moore’s supervision in 1949. After teaching at Duke University and the University of Rochester, she joined the faculty of the University of Wisconsin-Madison as a lecturer in 1959 when her husband Walter came here. Walter Rudin died on May 20, 2010. Mary Ellen became a full professor in 1971 and professor emerita in 1991. She also held two named chairs: she was appointed Grace Chisholm Young Professor in 1981 and Hilldale Professor in 1988. She received numerous honors throughout her career. She was a fellow of the American Academy of Arts and Sciences and was a member of the Hungarian Academy of Sciences, and she received honorary doctor of science degrees from the University of North Carolina, the University of the South, Kenyon College, and Cedar Crest College. -
Arxiv:1901.02074V1 [Math.LO] 4 Jan 2019 a Xoskona Lrecrias Ih Eal Ogv Entv a Definitive a Give T to of Able Basis Be the Might Cardinals” [17]
GENERIC LARGE CARDINALS AS AXIOMS MONROE ESKEW Abstract. We argue against Foreman’s proposal to settle the continuum hy- pothesis and other classical independent questions via the adoption of generic large cardinal axioms. Shortly after proving that the set of all real numbers has a strictly larger car- dinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert’s famous list presented in 1900 [19]. G¨odel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but G¨odel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. G¨odel remained hopeful, however, that new mathemati- cal axioms known as “large cardinals” might be able to give a definitive answer on CH [17]. The independence of CH from ZFC was finally solved by Cohen’s invention of the method of forcing [2]. Cohen’s method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. -
Constructive Set Theory
CST Book draft Peter Aczel and Michael Rathjen CST Book Draft 2 August 19, 2010 Contents 1 Introduction 7 2 Intuitionistic Logic 9 2.1 Constructivism . 9 2.2 The Brouwer-Heyting-Kolmogorov interpretation . 11 2.3 Counterexamples . 13 2.4 Natural Deductions . 14 2.5 A Hilbert-style system for intuitionistic logic . 17 2.6 Kripke semantics . 19 2.7 Exercises . 21 3 Some Axiom Systems 23 3.1 Classical Set Theory . 23 3.2 Intuitionistic Set Theory . 24 3.3 Basic Constructive Set Theory . 25 3.4 Elementary Constructive Set Theory . 26 3.5 Constructive Zermelo Fraenkel, CZF ................ 26 3.6 On notations for axiom systems. 27 3.7 Class Notation . 27 3.8 Russell's paradox . 28 4 Basic Set constructions in BCST 31 4.1 Ordered Pairs . 31 4.2 More class notation . 32 4.3 The Union-Replacement Scheme . 35 4.4 Exercises . 37 5 From Function Spaces to Powerset 41 5.1 Subset Collection and Exponentiation . 41 5.2 Appendix: Binary Refinement . 44 5.3 Exercises . 45 3 CST Book Draft 6 The Natural Numbers 47 6.1 Some approaches to the natural numbers . 47 6.1.1 Dedekind's characterization of the natural numbers . 47 6.1.2 The Zermelo and von Neumann natural numbers . 48 6.1.3 Lawv`ere'scharacterization of the natural numbers . 48 6.1.4 The Strong Infinity Axiom . 48 6.1.5 Some possible additional axioms concerning ! . 49 6.2 DP-structures and DP-models . 50 6.3 The von Neumann natural numbers in ECST . 51 6.3.1 The DP-model N! ..................... -
Indiscernible Pairs of Countable Sets of Reals at a Given Projective Level
Indiscernible pairs of countable sets of reals at a given projective level Vladimir Kanovei∗ Vassily Lyubetsky† January 1, 2020 Abstract 1 Using an invariant modification of Jensen’s “minimal Π2 singleton” forcing, we define a model of ZFC, in which, for a given n ≥ 2, there exists 1 an Πn unordered pair of non-OD (hence, OD-indiscernible) countable sets 1 of reals, but there is no Σn unordered pairs of this kind. Any two reals x1 6= x2 are discernible by a simple formula ϕ(x) := x < r for a suitable rational r. Therefore, the lowest (type-theoretic) level of sets where one may hope to find indiscernible elements, is the level of sets of reals. And in- deed, identifying the informal notion of definability with the ordinal definability (OD), one finds indiscernible sets of reals in appropriate generic models. Example 1. If reals a 6= b in 2ω form a Cohen-generic pair over L, then ω the constructibility degrees [a]L = {x ∈ 2 : L[x] = L[a]} and [b]L are OD- indiscernible disjoint sets of reals in L[a, b], by rather straightforward forcing arguments, see [2, Theorem 3.1] and a similar argument in [3, Theorem 2.5]. Example 2. As observed in [5], if reals a 6= b in 2ω form a Sacks-generic pair over L, then the constructibility degrees [a]L and [b]L still are OD-indiscernible disjoint sets in L[a, b], with the additional advantage that the unordered pair arXiv:1912.12962v1 [math.LO] 30 Dec 2019 {[a]L, [b]L } is an OD set in L[a, b] because [a]L, [b]L are the only two minimal degrees in L[a, b]. -
The Realm of Ordinal Analysis
The Realm of Ordinal Analysis Michael Rathjen Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, United Kingdom Denn die Pioniere der Mathematik hatten sich von gewissen Grundlagen brauchbare Vorstellungen gemacht, aus denen sich Schl¨usse,Rechnungsarten, Resultate ergaben, deren bem¨achtigten sich die Physiker, um neue Ergebnisse zu erhalten, und endlich kamen die Techniker, nahmen oft bloß die Resultate, setzten neue Rechnungen darauf und es entstanden Maschinen. Und pl¨otzlich, nachdem alles in sch¨onste Existenz gebracht war, kamen die Mathematiker - jene, die ganz innen herumgr¨ubeln, - darauf, daß etwas in den Grundlagen der ganzen Sache absolut nicht in Ordnung zu bringen sei; tats¨achlich, sie sahen zuunterst nach und fanden, daß das ganze Geb¨audein der Luft stehe. Aber die Maschinen liefen! Man muß daraufhin annehmen, daß unser Dasein bleicher Spuk ist; wir leben es, aber eigentlich nur auf Grund eines Irrtums, ohne den es nicht entstanden w¨are. ROBERT MUSIL: Der mathematische Mensch (1913) 1 Introduction A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their ‘rank’ or ‘complexity’ in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of ‘proof the- oretic ordinals’ to theories, gauging their ‘consistency strength’ and ‘computational power’. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen’s head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories.