Axiomatic Set Teory P.D.Welch

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Axiomatic Set Teory P.D.Welch.

August 16, 2020 Contents

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1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11

2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28

2.4.2 H  - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36

3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48

ii iii

3.5.1 Incompleteness and Consistency Arguments 50

4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. 58 4.5 Ordinal Definable sets and HOD 60 4.6 Criteria for Inner Models 63 4.6.1 Further examples of inner models 64 4.7 The Suslin Problem 66

Appendix ALogical Matters 71 A.1 The formal languages - syntax 71 A.2 Semantics 73 A.3 AGeneralised Recursion Theorem 75 symbols 80 index 82 iv Chapter 1

Axioms and Formal Systems

1.1 Introduction

1 Te great German mathematician David Hilbert (1862-1943) in his address to the second International 2 Congress of Mathematicians in Paris 1900 placed before the audience a list of the 23 mathematical prob- 3 lems he considered the most relevant, the most urgent, for the new century to solve. Hilbert had been a 4 defender of Cantor’s seminal work on on infnite sets, and listed the Continuum Hypothesis as one of the 5 great unsolved questions of the day. He accordingly placed this question at the head of his list. Te hy- 6 pothesis is easy to state, and understandable to anyone with the most modest of mathematical education: 7 if A is a subset of the real line continuum , then there is a bijection of A either with the set of natural 8 numbers, or with all of . Phrased in the terminology that Cantor introduced following his discovery 9 of the uncountability of the reals and his subsequent work on cardinality the hypothesis becomes: for 10 any such A, if A is not countable then it has the cardinality of itself. CH thus asserts that there is no 11 cardinality that is intermediate between that of and that of . If the cardinality of is designated  12 (or  and that of the frst uncountable cardinal as (or ) then CH is ofen written as “ ”  13 (or  ) the point being here is that the real continuum can be identifed with the class of infnite 14 binaryℵℵ sequences)  and the latter’s cardinality is   . ℵ = = ℵ 15 Sometimes called the Continuum Problem, Cantor wrestled with this question for the rest of his ca- 16 reer, without fnding a solution. However, in this quest he also founded the subject of Descriptive Set 17 Teory that seeks to prove results about sets of reals, or functions thereon, according to the complexity 18 of their description. Such hierarchical bodies of sets were to become very infuential in the Russian school 19 of analysts (Suslin,Lusin,Novikoff) and the French (Lebesgue,Borel,Baire). Te notion of a hi- 20 erarchy built up by considering complexity of defnition of course also invites methods of mathematical 21 logic. Descriptive Set Teory has fgured greatly in modern set theory, and there is a substantial body of 22 results on the defnable continuum where one tries to establish CH type results not for the whole contin- 23 uum but just for “defnable parts” thereof. Cantor was able to show that closed subsets of satisfed CH: 24 they were either countable or could be seen to contain a subset which was of cardinality the continuum. 25 Tis allows one to say that then countable unions of closed sets also satisfed the CH. Cantor hoped to 26 be able to prove CH for increasingly complicated sets of real numbers, and somehow exhaust all subsets 27 in this way. Te analysts listed above made great strides in this new feld and were able to show that any

1 2 Axioms and Formal Systems

28 analytic set satisfed CH. (At the same time they were producing results indicating that such sets were 29 very “regular”: they were all Lebesgue measurable, had a categorical property defned by Baire and many 30 other such properties. Borel in particular defned a hierarchy of sets now named afer him, which gave 31 very real substance to Cantor’s eforts to build up a hiearchy of increasing complexity from simple sets.) 32 However it was clear that although the study of such sets was rewarded with a regular picture of their 33 properties, this was far from proving anything about all sets. We now know that Cantor was trying to 34 prove the impossible: the mathematical tools available to him at his day would later be seen to be for- 35 malisable in Zermelo-Fraenkel set theory with the Axiom of Choice , AC (an axiom system abbreviated 36 as ZFC). Within this theory it was shown (by (Cohen (1934-2007)) that CH is strictly unprovable. If he 37 had taken the opposite tack and had thought the CH false, and had attempted to produce a set Aneither 38 of cardinality that of nor of he would have been equally stuck: by a result of Godel¨ within ZFC it 39 turns out that CH is strictly unprovable. 40 It is the aim of this course to give a proof of this latter result of Godel.¨ Te method he used was ¬ 41 to look at the cumulative hierarchy V (which we may take to be the universe of sets of mathematical 42 discourse) in which all the ZF axioms are seen to be true, and to carve out a special transitive subclass 43 -theclassofconstructible sets, abbreviated by the letter L. Tis L was a proper transitive class of sets (it 44 contains all ordinals) and it was shown by Godel¨ (i) Tat any axiom from ZF was seen to hold in L; 45 moreover (ii) Both AC and CH held in L. Tis establishes the unprovability of CH from the ZF axioms: 46 L is a structure in which any axioms of ZF used in a purported proof of CH were true, and in which 47 CH was true. However a proof of CH from that axiom set would contradict¬ the fact that rules of frst 48 order logic are sound, that is truth preserving. ¬ ¬ 49 In modern terms we should say that Godel¨ constructed the frst inner model of set theory: that is, 50 a transitive class W containing all ordinals, and in which each axiom of ZFC can be shown to be true. 51 Such models generalising Godel’s¨ construction are much studied by contemporary set theorists, so we 52 are in fact as interested in the construction as much as (or even more so now) than the actual result. 53 It is a perhaps a curious fact that such inner models invariably validate the CH but most set theorists 54 do not see that fact alone as giving much evidence for a solution to the problem: the inner model L and 55 those generalising it are built very carefully with much attention to detail as to how sets appear in their 56 construction. Set theorists on the whole tend to feel that there is no reason that these procedures exhaust 57 all the sets of mathematical discourse: we are building a very smooth, detailed object, but why should 58 that imply that V is L? Or indeed any other of the later generations of models generalising it? 59 However it is one of facts we shall have to show about L that in one sense it is “self-constructing”: the 60 construction of L is a mathematical one; it therefore is done within the axiom system of ZF; but (we shall 61 assert) L itself satisfes all such axioms; ergo we may run the construction of the constructible hierarchy 62 within the model L itself (afer all it is a universe satisfying all ZF axioms). It will be seen that this process 63 activated in L picks up all of L itself: in short, the statement “V L” is valid in L. Te conclusion to be 64 drawn from this is that from the axioms of ZF we cannot prove that there are sets that lie outside L. It 65 is thus consistent with ZF that V L is true! If V L is true,= then there are many consequences for 66 mathematics: the study of L is now highly developed and many consequences for analysis, algebra,... have 67 been shown to hold in L whose proof= either remains= elusive, or else is downright unprovable without 68 assuming some additional axioms. It is a corollary to the consistency of V L with ZF, that we cannot 69 use this method of constructing an inner model to fnd one in which CH holds: if it is consistent that = ¬ Preliminaries: axioms and formal systems. 3

70 V L then it is consistent that L is the only inner model there is, so no construction using the axioms 71 of ZF alone can possibly produce an inner model of CH. = 72 We are thus lef still in the state of ignorance that Hilbert protested was not the lot of mathematicians 1 ¬ 73 as regards the CH. Cohen’sproof that CH is not provable from the ZFC axioms does not proceed by using 74 inner models (we have mentioned reasons why it cannot) but by constructing models of the axioms in 75 a boolean valued logic: statements there do not have straightforward true/false truth values. In Cohen’s 76 models, when constructed aright, all axioms of ZF (and sentences provable from them in frst order 77 logic) receive the topmost truth value “1”,and contradictions , receive the bottom value “0”. Cohen 78 constructed such a model in which CH received a “non-0” truth value in the Boolean algebra, value p 79 say. Consequently CH is not provable from ZF, else the Boolean¬ ∧ model would have to assign the non- 80 zero p to the inconsistent statement¬ CH CH and such is not possible in these models. Tis literally 81 taken, says absolutely nothing about sets in the universe V since the model is a sub-universe of V with a 82 non-classical interpretation. It speaks only∧ ¬ about what can or cannot be proven in frst order logic from 2 83 the axioms of ZFC. 84 Tere are many results in set theory, in particular in axiomatic systems that enhance ZFC with some  85 “strong axiom of infnity” that indicate that the CH is actually false (that   ofen occurs in such 86 cases). At present this can only be taken as some kind of quasi-empirical evidenceℵ and so is a source of 87 much discussion. = ℵ 88 Prerequisites: Cohen’s proof is beyond the scope of this course, but we shall do Godel’s¨ construction 89 of L in detail. Tis will involve extending the basic results on ordinal and cardinal numbers and their 90 arithmetic; we shall have recourse to schemes of ordinal and -recursion. Te reader is assumed familiar P 91 with a development of these topics, as well as with the notion and basic properties of transitive sets. 92 Although Godel¨ gave a presentation of the constructible hierarchy using a functional hierarchy, with 93 almost all logic eliminated, (mainly as a way of presenting his results to “straight” mathematicians) we 94 shall be going the traditional route of defning a “Defnability” operator using all the syntactic resources 95 of a formal language and the methods of modern logic. Formal derivability T will always mean 96 that is derivable from the axioms T in one, or any, system of classical frst order calculus familiar to 97 the reader. L ⊢ 98 Acknowledgements: these notes are heavily indebted to a number of sources: in particular to Ronald 99 B. Jensen : Modelle der Mengenlehre (Springer Lecture Notes in Maths, vol 37,1967), and his subsequent 100 lecture notes. 101

102 1.2 Preliminaries: axioms and formal systems.

103 We introduce the formal frst order language ,andseehowwecanuseclass terms expressed in it. We 104 then give a formulation of the Zermelo-Fraenkel axioms themselves. L 1Or any mathematical problem “You can fnd [the solution to any mathematical problem] by pure reason, for in mathematics there is no ignorabimus” Hilbert, Lecture delivered to the 2nd International Congress of Mathematicians, Paris 1900. 2Tis is only one way of interpreting Cohen’s forcing technique. See Kunen [4] Set Teory: An Introduction to Independence Proofs 4 Axioms and Formal Systems

Hilbert in 1900

105 1.2.1 The formal language of ZF set theory; terms

106 ZF set theory is formulated in a formal frst order language of predicate logic with axioms for equality. 107 Te components of that language ˙ are: P 108 (i) set variables; v, v,...,vn,... (for n L = L P 109 (ii) two binary predicate symbols: ˙, ˙ P ) 110 (iii) logical connectives: , = 111 (iv) brackets: , ∨ ¬ 112 (v) an existential quantifer: . ( ) D 113 Te formulae of are defned inductively in a way familiar for any frst order language. We assume 114 the reader has seen this done for his or herself and do not repeat this here. We assume also that the notion 115 of free variable (FVblL and subformula of a formula as inductively defned over the collection of 116 all formulae is also familiar. We shall use the notation y x for the formulae with the free variable 117 occurrences of the variable( )) x replaced by the variable y. A formula with no free variables is called a 118 closed formula or a sentence. It is sometimes convenient to( augment/ ) the language with other predicate

119 symbols A A, A,...; if this is done we denote the appropriate language by A. L 120 We use the binary predicate symbol as a relation to be interpreted as membership: “v v”will ⃗ = P L ⃗ P 121 be interpreted as “v is a member of v” etc. We ofen use other letters also to stand in for variables vk: 3 122 typically x, y, z, and recalling the convention from ST: , , for ordinals etc. etc. . It is so convenient 123 to adopt these conventions that we do so immediately even when we write out our basic axioms. Note 124 that in our statement of the Extensionality Axiom Ax  we also abbreviate “ v ”asusualby“ v ”. D k @ k 125 Ax0 (Extensionality) ¬ ¬ 126 x y z z x z y x y . @ @ @ P P 3Formally speaking the symbols( ( x, y, ↔, ,...are) not↔ part= of ):theyhavethestatusofmetavariables in the metalanguage;the latter is the language we use to talk about . Te metalanguage consists of English with a liberal admixture of such metavariables and other symbols as and when we require them. Some of ourL metatheoretical arguments require some simple arithmetic, as when we prove something about formulaeL or terms by induction. Tese arguments can all be done with primitive recursive arithmetic. Preliminaries: axioms and formal systems. 5

127 Tis single axiom expresses the fact that identity of sets is based solely on membership questions 128 about the two sets. 129 We have seen that collections, or “classes” based on unguarded specifcations within the language 130 can lead to trouble; recall Russell’s Paradox: R df x x x was a class that could not be considered R L 131 to be a set. Likewise V df x x x is not a set. Such collections we called “proper classes”. It 132 might be thought that this mode of introducing= collections{ ∣ or} classes is fraught with potential danger, 133 and although we successfully= used{ ∣ these= ideas} in ST perhaps it would be safer to do without them? In fact 134 such methods of specifying collections is so useful that instead of being wary of them, we shall embrace 135 them full heartedly whilst keeping them at a safe distance from our formal language .

L 136 Definition 1.1 (i) A class term is a symbol string of the form x where x is one of the variables vk 137 and is a formula of our language. { ∣ } 138 (ii) A term t is either a variable or a class term. 139 (iii) Te free variables of a term t are given by: 140 FVbl t df FVbl x if t x ; FVbl x x if x is a variable.

141 We allow( ) = terms to( be)/{ substituted} = { for∣ variables} in( ) atomic= { } formulae x y and x y,andforfree P 142 variables in general in formulae of .Wethusmaywrite t x for the formula with instances of x 143 replaced by t. Just as for substitutions of variables in ordinary formulae in f=rst order predicate logic, we 144 only allow substitutions of terms t intoL formulae where free( / variables) of t do not become unintention- 145 ally bound by quantifers of . Substitutions can always be efected afer a suitable change of the bound 146 variables of . A term t with FVbl t is called a closed term. A term of the form x is not part of our language : it is to be understood purely as an abbreviation. Likewise t x is not part of( our) = language∅ if t is a class term. We understand these abbreviations as follows: { ∣ } L ( / ) y x is y x ; P x z is y y x y z @ z {x ∣ } is (y /y )z y x @ P P {x ∣ } = {z ∣ } is y(y ( /x) ←→ ( y/ z)) P D x= { ∣ }z is y(y z ←→y x{ ∣ }) { ∣ } { ∣ P } D ( =P { ∣ } ∧ ( / )) 147 Although class terms appear{ on∣ both} sides of the( above,∧ this= { in∣ fact}) gives a precise recursive way 148 of translating a “generalised formula” containing class terms into one that does not. Note that a simple 149 consequence of the above is that for any x we have x y y x . Note in particular that the fourth P 150 line ensures that if we write “s t” for terms s, t then s must be a set. P = { ∣ } 151 We now name certain terms and defne some operations on terms. Again these are metatheoretical 152 operations: we are talking about our language , and talking about, or manipulating terms,ispartofthat 153 meta-talk. L

154 Definition 1.2 (i) V df x x x ; df x x x ; 155 (ii) s t df x x s x t Ď @ =P { ∣ =P } ∅ = { ∣ ≠ } 156 (iii) s t df x x s x t ;s t df x x s x t ; = ( P %→ P ) P P 157 s df x x s ;s t df x x s x t ∪ = { R∣ ∨ } ∩P = {R ∣ ∧ } ¬ = { ∣ } / = { ∣ ∧ } 6 Axioms and Formal Systems

158 (iv) s x y y s x y ; s x y y s x y df D P P df @ P P 159 (v) t,...,tn df x x t x t x tn = { ∣ ( ∧ )} = { ∣ ( %→ )} 160 (vi) ⋃x, y df x , x, y (the ordered⋂ pair set) { } = { ∣ = ∨ = ∨⋯∨ = } 161 (vii) x, x,...,xn df x,...,xn  , xn (the ordered n-tuple) 162 ⟨ ⟩ = {{ } { }} (viii) x z df u, v u x v z− (the Cartesian product of x, z) ⟨  ˆ n⟩ = ⟨⟨nP P ⟩ ⟩ 163 t df t t; t t t; =ˆ {⟨ ⟩∣ ˆ ∧ } 164 (ix) x df y y+ x (the “power class” of x. = Ď= 165 At theP( moment) = { the∣ above} objects just have the status of syntactic names of certain terms, but we are 166 going to adopt axioms that will assert that the classes defned are in fact sets. Indeed we shall say “x is a 167 set” “x V”. In (viii) we have introduced a useful syntactic device: instead of writing df P 168 x z y u v u x v z y u, v ˆ D D P P 169 we have⇐⇒ placed the constructed term u, v to the lef of the . In general we introduce this abbreviation: 170 we let t = { ∣ z u( z t∧ (whose∧ = ⟨ notation⟩)} is probably more easily understood through the df D 171 example above, here u is a list of variables⟨ ⟩ containing all those∣ free in t and ). 172 { ∣ } = { ∣ ⃗( = ∧ )} ⃗ 173 Ax1 (Empty Set Axiom) V. P 174 Ax2 (Pairing Axiom) x, y V. P 175 Ax3 (Union Axiom) x∅ V. { P } 176 Lemma 1.3 t V y⋃y t . P D 177 Proof: (Actually 1.←→3 is a theorem( = ) scheme: for each term there is a lemma corresponding to the defnition 178 of the term t. By our rules on translation 1.2, t V y y t x x y x y y t y P D D 179 y y y t . Q.E.D. D ) ↔ ( = ∧ ( = )( / )) ↔ ( = ∧ ( = 180 L))emma↔ 1.4( Ax= 0)-3 prove: x y V; x ,...,x V. P  n P 181 Proof: By Ax2 x, y V and then by Ax3 x, y V.And x, y x y (by Ax0). Repeated P ∪ { } P 182 application of Ax0-3 shows x ,...,x V (Exercise). Q.E.D.  n P 183 Tere now follow{ } a sequence of defnitions⋃ of{ basic} notions which⋃{ we} have= ∪ already seen in ST. { }

184 Definition 1.5 Let r be a term. (i) r is a relation df r V V n Ď ˆ 185 (ii) r is an n-ary relation df r V . Ď ⇐⇒ 186 We write in (i) xry or rxy instead of x, y r and in (ii) rx x instead of x ,...,x r. ⇐⇒ P  n  n P

187 Definition 1.6 If r, s are relations and u⟨ a term⟩ we set: ⋯ ⟨ ⟩ 188 (i) dom r x y xry ; ran r y x xry ; feld r dom r ran r . df D df D df 189 (ii) r u df x, y xry x u . ( ) = { ∣ ( )} P ( ) = { ∣ ( )} ( ) = ( ) ∪ ( ) 190 (iii) r“u df y x x u xry . ↾ = {⟨ D ⟩∣ P ∧ } 191 (iv) r df y, x xry . = { ∣ ( ∧ } 192 (v) r − s df x, z y xry ysz . = {⟨ ⟩∣ D } 193 We may de○ f=ne the{⟨ unicity⟩∣ quanti( ∧fer: )} Preliminaries: axioms and formal systems. 7

194 Definition 1.7 !x z z x . D df D 195 We now defne familiar⇐⇒ functional({ } = concepts.{ ∣ })

196 Definition 1.8 Let f be a relation. 197 (i) f is a function (Fun f df x, y, z fxy fxz y z (we write f(x)=y). @ n 198 (ii) f is an n-ary function df fisafunction dom f V ( )) ⇐⇒ ( ∧ %→Ď = ) 199 (we write f x,...,xn y instead of f x,...,xn y). ⇐⇒ ∧ ( ) 200 (iii) f a b df Fun f dom f a ran f b. ( ) = (⟨  Ď⟩) = 201 (iv) f a   b df f a b Fun f (“ f is an injection or (1-1)”). ∶ %→ ⇐⇒ ( ) ∧ ( ) = ∧ ( ) 202 (v) f a onto b df f a b ran f − b (“ f is onto”). ∶ %→ ( − ) ⇐⇒ ∶ %→ ∧ ( ) 203 (vi) f a b df f a   b f a onto b (“ f is a bijection”). ∶ %→ ⇐⇒ ∶ %→ ∧ ( ) = a ( − ) 204 Definition∶ ←→1.9⇐⇒(i) b df∶ %→f f a ∧ b∶ the%→ class of all functions from a to b. (ii) Let f be a function such that ran f . Ten the generalised cartesian product is = { ∣ ∶ %→R } f h Fun h ∅dom h( )dom f x dom f h x f x . df @ P P 205 Note that f∏consists= of{ choice∣ ( functions) ∧ for( ran) = f : each( ) ∧h “chooses”( an)( element( ) from( ))} each appropriate 206 set. ∏ ( )

207 1.2.2 The Zermelo-Fraenkel Axioms

208 Te axioms of ZFC (Zermelo-Fraenkel with Choice) then are the following: 209 Ax0 (Extensionality) x y z z x z y x y @ @ @ P P 210 Ax1 (Empty Set) V P 211 Ax2 (Pairing Axiom) x, y( (V ↔ ) ↔ = ) P 212 Ax3 (Union Axiom)∅ x V P 213 Ax4 (Foundation Scheme){ }For every term a: a x x a x a D P 214 Ax5 (Separation Scheme)⋃ For every term a: x a V P 215 Ax6 (Replacement Scheme) For every term f : ≠Fun∅ %→f ( f “x ∧V.∩ = ∅) P 216 Ax7 (Infnity Axiom) x x y y x ∩y y x D P @ P P 217 Ax8 (PowerSet Axiom) x V ( ) %→ P 218 Ax9 (Axiom of Choice) Fun(∅ f ∧ dom( f %→V ∪ { ran} f ) f . P( ) P R 219 Note 1.10 (i) ZF comprises Ax0-8(; Sometimes) ∧ ( ) Ax6 is∧∅ replaced( by:) %→ ∏ ≠ ∅ 220 Ax6’ (Collection Scheme) For every term r: xr“x w t u w v t u, v r . @ @ D @ P D P P 221 Te Axiom of Choice is equivalent over ZF to the Wellordering Principle: 222 Ax9’(Wellordering Principle) x r Rel r x, r ≠is∅ a wellordering%→ ( . (⟨ ⟩ ) @ D 223 Tere are two useful subsystems. ZF with Replacement dropped is called Z for Zermelo. ZF is 224 Ax0-5,6’, 7; ZFC is ZF with Ax9’. ( ( ) ∧ ⟨ ⟩ ) − 225 (ii) ZF is an− infnite− list of axioms: Ax4,5,6, (and 6’) are schemes: there is one axiom for each formula 226 defning the mentioned terms a and f (or r in 6’). We shall later prove that it cannot be replaced by a 227 fnite list with the same consequences. 8 Axioms and Formal Systems

228 (iii) Te statements difer in their formulation from ST: Foundation was there stated just for sets, and 229 was a single axiom; Separation was given its synonym “Comprehension”: and was stated as follows: 230 (Te set of elements of a set z satisfying some formula, form a set.) For each formula v, ...vn  (with free variables amongst those shown), z w ...+ w y x x y x z z, x, w ,...,w . @( @  @ )nD @ P P  n 231 Te formulation above shows how powerful( and↔ succinct∧ a[ formulation we]) have if we allow ourselves 232 to use terms. Likewise Replacement there had a much longer (but equivalent) formulation. 233 (iv) Te axioms are of diferent kinds: one group asserts that simple operations on sets leads to further 234 sets (such as Union, Pairing). Another group consists of set existence axioms (Empty Set, Separation). 235 Others are of “delimiting size” nature: the power class x may be thought to be a large incoherent 236 collection of all subsets of x. Te power set axiom claims that this is not a large collection but merely 237 another set. Te Replacement Scheme assures us that functionsP( ) however defned cannot create a non-set 238 from a set. It thus also in efect delimits size. Tis axiom is due to Fraenkel. Te term ‘Replacement’ 239 comes from the idea that if one has a set, and a method (or function) for replacing each member of that 240 set by a diferent set, then the resulting object is also a set. Zermelo’s achievement was to recognize (a) 241 the utlility, if not the necessity, of formulating a formal set of axioms for the new subject of set theory 242 - which he then enunciated; (b) that the Separation scheme was a method to avoid paradoxes of the 243 Russell/Burali-Forti kind. Zermelo essentially wrote down the system Z although Separation was given 244 a second order formulation. Later Skolem gave the familiar frst order formulation equivalent to the 245 above. Again the Axiom of Choice asserts the existence of a rather specialised set: a choice function for a 246 collection of sets. In ST we adopted the axiom that “Every set can be wellordered” for AC (on pedagogical 247 grounds). We saw there that this principle was equivalent to the existence of choice functions. 248 (v) One may ask simply: Are these right axioms? Tere are indeed other formulations of set theory, 249 some involving class terms more directly as further objects. Our point of view is that the V hierarchy 250 comprises all that is needed for mathematics, further we have a somewhat less developed intuition about 251 what such “objects” these free-standing class terms could possible be: if they are attempts to continue 252 the V-hierarchy even further, by using the power class operation “just one more time” this would seem 253 to miss the point. Since we have no need for classes as some other kind of separate entities of a diferent 254 sort, we avoid them. 255 One formulation of set theory (which Godel¨ used - and is named von Neumann-Godel-Bernays)¨ 256 does however include class variables in the object language but disallows quantifcation over classes: it 257 can be shown that this system is conservative over ZFC: that is, it proves no more theorems about sets 258 than ZFC itself, and so is treated by set theorists virtually as a harmless variant of ZFC. 259 We use a frst order formulation of set theory (meaning that quantifers x , x quantify only over  D @ 260 our objects of interest, namely sets. A second order formulation ZF is possible, where, as in any second 261 order language, we are allowed quantifers such as P, P that range over predicates P of sets. Tere are D @ 262 two points that could be made here. Firstly, as a predicate P is extensionally a collection of sets itself, 263 even to understand the meaning of a second order sentence involving say a quantifer P is to already @ 264 claim an understanding about the universe V.AnditisV itself that we are trying to understand in the 265 frst place. As in all areas of mathematics, frst order formulations of theories are the most successful: we 266 may not know of a frst order sentence whether it is true or not, but we do know precisely what it means Transfnite Recursion 9

267 for it to be true. Secondly the tools of mathematical logic are the most useful in the setting of frst order  268 logic. Te deductive system associated to ZF lacks a Completeness Teorem, and hence Compactness  269 and Lowenheim-Skolem¨ Teorems fail. In ZF it is possible to argue that since the only possible models  270 of ZF are V itself and possible initial segments of V of the form V (as Zermelo demonstrated), then  271 ZF shows that, e.g. CH has a defnite truth value: namely that obtained by inspecting that level of the 272 V-hierarchy where all subsets of live: V . However as to what that truth value is, we have no idea. 273  Hence we are no further forward! Indeed second+ order logic and ZF seems not to give us any tangible 274 information about the universe of sets that we do not obtain from the frst order formulations of ZFC 275 and its extensions. 276 (vi) Some formulations or viewpoints concerning the mathematical hierarchy of sets take as the base 277 of that hierarchy not the empty set (“V ”) but rather a collection of “atoms” or base objects: thus 278 instead we take V U U where U is this collection of Urelemente and we build our hierarchy by 279 iterating the power set operation from this= point∅ onwards. Tis may be of presentational beneft, but, at 280 least if U is a set (meaning[ ] = that it has a cardinality), then this is of limited foundational interest to the 4 281 pure set theorist. Te reason being, that, if U say, then we may build an isomorphic copy of V U 282 inside V, by starting with some sized set or structure which is an appropriate copy of U. Hence again 283 to study V is to study all such universes V ∣U∣ ,= and we may limit our discussion to universes of “pure[ ] 284 sets” without additional atomic elements. [ ]

285 1.3 Transfinite Recursion

286 We recall the defnitions of transitive set.

287 Definition 1.11 xistransitive (Trans x ) if z x z x . @ P Ď 288 We have the following scheme of -induction: P( ) ( ) Lemma 1.12 (scheme of -induction) For any formula : P x y x y x x x . @ @ P @

289 Tis principle was used in the proof[ of: ( ) → ( )] → ( )

290 Theorem 1.13 (Transfnite Recursion along ) P If G is a term and G V V V then there is a term F giving F V V ˆ ∶ → xF x G x, F x . ∶ → ˚ @

291 Moreover the term defnes a unique( function,) in( that) = if F( is any↾ other) term satisfying then, xF x ˚ @ 292 F x . ′ ′ ( ) ( ) = 4 ( T)is is not to say that models with atoms are without utility: formulations of ZF with atoms, “ZFA”, are of great use for studying universes in which the Axiom of Choice fails. Te point being made is that we cannot get any additional knowledge about foundational questions by using them. 10 Axioms and Formal Systems

293 Note: (i) Usually one speaks instead of G, F being defned by formulae G , F etc., but we have 294 replaced that with talk about terms. In the proof of Teorem 1.13 we, in efect, saw how to build up from 295 the formula G the formula F. Tis is in essence a Teorem Scheme: it is one theorem for each term G. 296 Te ‘canonical’ procedure for building the formula F given G now becomes a method for building a 297 canonical term defning F from one defning G. 298 (ii) Ofen one frst proves a transfnite recursion theorem along On: as the ordering relation amongst 299 ordinals is the -relation, we can view the latter theorem as simply a special case of Teorem 1.13.From P 300 these we proved the existence of functions giving the arithmetical operations on ordinals, and their basic 301 properties. It is ofen useful to have the notion of a wellfounded relation in general:

302 Definition 1.14 If R is relation on a class A then we say R is wellfounded if for any z, if z A then 303 there is y z A which is R-minimal (that is x z A xRy ). P @ P ∩ ≠ ∅ 304 An important example of a defnition by transfnite recursion along is that of the transitive closure ∩ ∩ (¬ ) P 305 operation TC.

Definition 1.15 TC is that class term given by Teorem 1.13 satisfying

x TC x x TC y y x @ P 306 Exercise 1.1 In ST TC was given an alternative( ) (but= equivalent)∪ ⋃{ ( de)f∣nition,} and was shown to satisfy the defnition 307 of TC above. Rework this by showing, using the above defnition, that: (i) x y TC x TC y . (ii) Show P Ď 308 that TC x is the smallest transitive set t satisfying x t. [Hint: Use -recursion.] (Tus if Trans t x Ď P  n  Ď 309 t TC x t.) Moreover Trans x TC x x. (iii) Defne by recursion%→ on( ): x ( )x; x n Ď n 310 x(;tc) x x n . Show that tc x TC x . ( ) ∧+ %→ ( ) ( ) ←→ ( ) = ⋃ = ⋃ = 311 D⋃(efinition⋃ ) ( 1.)16= ⋃For{⋃ x( )On∣ ,

313 Definition 1.17 (Te rank function ) Terankfunctionisde( ) = ⋃ fned by transfnite recursion on : P x sup y  y x . P ( ) = { ( ) + ∣ } 314 Exercise 1.2 Show that: (i) the relation xRy x TC y is wellfounded; (ii) x x TC x ; P @ 315 (iii) Trans x “x On . P ←→ ( ) ( ( ) = ( ( ))) 316 Definition( ) 1%→.18 (Te Cumulative Hierarchy) TeV function is defned by transfnite recursion on On 317 as : V x x .

318 In ST= { we∣ de(f)ned< the} V hierarchy by iterating the power set operation. Te previous defnition 319 does not use AxPower and together with the next exercise shows that we can defne the latter hierarchy 320 without it.

321 Exercise 1.3 Defne by recursion R , R  R and for Lim( , R R . Show by transfnite 322 induction that for any On that R V . + P = ∅ = P( ) ) = ⋃ < = Relativisation of terms and formulae 11

323 1.4 Relativisation of terms and formulae

324 We may classify concepts according to the syntactic complexity of their defnitions. Accordingly we then 325 frst classify formulae of our language as follows. 326 Bounded quantifers: vi v j , vi v j abbreviate: vi vi v j and vi vi v j @ P LD P @ P D P 327 respectively. We allow terms for v j here: x a is x x a etc. @ P @ P ( → ) ( ∧ ) 328 Te Levy hierarchy: We stratify formulae according to their complexity by counting alternations of ( %→ ) 329 quantifers. We frst defne the ∆-formulae of inductively: 330 (i) vi v j and vi v j are ∆. ) P L 331 (ii) If , are ∆, then so are and . = 332 (iii) If is ∆ so is vi v j . D P ¬ ( ∨ ) 333 Having defned ∆ as those without unbounded quantifers, we then proceed, frst setting Σ Π 334 ∆: = = 335 (i) If is Π then v v is Σ . n  D i D in n 336 (ii) If is Σn then is Πn. − ⋯ 337 One should note that if a formula is classifed as Σn then it is logically equivalent to a Σm formula ¬ 338 (or to a Πm-formula) for any m n, by the trivial process of adding dummy quantifers at the front. 339 Of particular interest are existential formulae: those that are Σ : x with ∆ . Such assert a simple  D  340 set existence statement, and universal≥ formulae: these are Π : x whose truth requires, prima facie,  @ 341 an inspection over all sets (although in practice we shall see that by the Downward Lowenheim Skolem 342 Teorem, we may sometimes restrict that apparent unbounded search). Occasionally, for T a fnite set 343 of formulae, we write T for the single formula that is their conjunction. 344 Some terms will be seen to be defnite in that they defne the same object in whatever world the 345 defnition takes place. ⋀⋀Tis may sound obscure at the moment, but one can perhaps see that the defnition 346 of the empty set provides a defnite object which is “constant” across possible universes where it might 347 be defned; likewise given any structure U with sets x, y as members and in which the Pairing Axiom 348 holds, then the term t u u x u∅ y defnes the same object in U as in any other structure 349 satisfying these conditions. Tis is in contradistinction to a term such as t y y x which defnes Ď 350 the power set of a set x: although= { ∣ the= de∨fning= } formula “v v ” is extremely simple, which subsets of x  Ď  351 get picked up when we apply the defnition, depends on which structure U we= { apply∣ the} defnition in. It 352 is thus not a defnite term. We shall need investigate this and give a criterion for when terms are defnite. 353 Tis leads on to the important notion of absoluteness. 354 We shall be interested in looking at models M, E of ZFC Φ for various statements Φ. For this to 355 be really meaningful we shall want that certain terms and notions defned by certain formulae that are 356 interpreted in the model M, E mean the same⟨ thing⟩ as when+ that term or formula is applied in V, : P 357 this is the notion of “absoluteness”. Certain (simple) objects, such as , and the like, are defned by the 358 same syntactic terms evaluated⟨ in⟩ V or in M. It is possible to think about models where the interpretation⟨ ⟩ 359 of the symbol is something other than the usual set membership relation.∅ Such models are called non- P 360 standard models, and do not feature highly in this course (or in the wider development of set theory). 361 We shall be most interested in transitive sets or classes W and where E is taken to be the genuine set 362 membership relation . Such an W, is called a transitive -model. However terms can have diferent P P P 363 interpretations even when considered in V and in a standard transitive model W, .Wefrst have to ⟨ ⟩ P ⟨ ⟩ 12 Axioms and Formal Systems

364 say what it means for an axiom or sentence to “hold” or “to be interpreted” in such a structure. We 365 build up a defnition by recursion on the structure of by straightforwardly restricting quantifers to the 366 new putative “universe” W.

367 Definition 1.19 Let W be a term. We defne by recursion on complexity of formulae of the relativi- W 368 sation of to W, : W W L 369 (i) x y df x y ; x y df x y ; P W WP 370 (ii) df ; ( ) W= ( W) (W = ) = ( = ) 371 (iii) df ; (¬ ) W= ¬( ) W 372 (iv) x df x W if x is not in FVbl W ; otherwise this is undefned. (D ∨ ) = D ( P ∨ ) W 373 Notice that( we) can= always ensure that is defned( ) by replacing the bound variables in by others 374 diferent from those of W. We tacitly that this has always been done when discussing relativising formu- W W 375 lae. It is immediate that, e.g. x ( ) x W . We shall be thinking of class terms W as being @ @ P 376 potential - structures - meaning that we shall be thinking of them potentially as models W, .We P W P 377 shall read as “ holds in( W”or“) ←→holds relativised to W”. Te following theorem (with Γ 378 says the theorems of predicate calculus in are valid in non-empty -structures W, .Weusethe⟨ ⟩ P P 379 shorthand( that) if Γ is a fnite set of formulae, then Γ is the single formula that is the conjunction= ∅ of) 380 those in Γ. L ⟨ ⟩ ⋀⋀ 381 Theorem 1.20 Let Γ be a fnite set of sentences in and W a transitive non-empty term; assume 382 that if x is a list of all the variables occurring in Γ then x FVbl W . W W 383 If Γ then ∪Γ{ } . L ⃗ ∪ { } ⃗ ∩ ( ) = ∅ 384 Proof: By⊢ induction(⋀⋀ on) the%→ length of the derivation of from Γ. Q.E.D. 385 386 Tis is just as it should be: roughly, it is a form of Soundness: if we can prove that is derivable from 387 a set of axioms true in a structure, then should be true in that structure.

W 388 Lemma 1.21 Let W be a transitive class term, Ten AxExt .

389 Proof: Te Axiom of Extensionality relativised to(W is: ) W 390 x y z z x z y x y @ @ @ P P W 391 x W y W z z x z y x y @ P @ P @ P P W W 392 ( x ( W( y W↔ z W) →z =x )) z y x y @ P @ P @ P P P 393 ↔ x W y W( z( W z↔ x )z →y = x) y @ P @ P @ P P P 394 Since↔ W is transitive,( if x, y (W then↔x, y )W→. Hence( = if) z)z x y y x then z W z x y y x . P Ď D P D P P 395 Hence↔ the of the( last equivalence( ↔ is true!) → = ) Q.E.D. 396 ( / ∪ / ) ( / ∪ / ) 397 Te next→ concern is how to relativise a formula that contains class terms. It should turn out that if we 398 have such a formula we should be able to frst relativise the terms it contains to W (Def.1.22) and then 399 substitute the results into the relativised formula of .

W W 400 Definition 1.22 Let t x be a class term; theLrelativisation of t to W, is: t x W . df P = { ∣ } = { ∣ } Relativisation of terms and formulae 13

W W W W 401 Example (i) V x x x x W x x . Since x x is just x x, this renders W P 402 V V W W. W W W 403 Example (ii) x= { ∣ =z } y =z { y x ∣ ( = )z }W (y z= )y x = z W y W D P P P D P P P D P 404 W z= y ∩ x = z W y W z y x . P P P D P P P 405 Notice that if( additionally⋃ ) = ({W∣ is a( transitive)}) term, (=i.e.{ defnes∣ a( transitive( class))) then} =x{ W ∣ x P Ď 406 W;( moreover) y} =y{ x ∣ y W(. Hence )}z W y W z y x z y z y x @ P W Ď P D P P P D P P 407 and so in this case x x. Tis demonstrates that is an absolute operation for transitive%→ classes 408 and the process of( relativisation%→ yields) the same{ set. We∣ shall be particularly( interested)} = { ∣ in( such absolute)} 409 operators and similarly(⋃ ) absolute= ⋃ properties for transitive classes.⋃

Lemma 1.23 Let t,...,tn and W be terms, with W transitive, let x,...,xn be in ; then assuming y FVbl t,...,tn : Ě ( ) L ⃗ ( ( )) y W t ,...,t W W tW ,...,tW . @ P  n  n

410 Remark: Te lemma is about⃗ syntax,( formulae( and) terms.←→ T(e xi’s are (meta-)variables)) (“meta” because 411 they are standing in for some ofcial variables vi ,...vin ). In this context the notation is supposed 412 to mean that each of the terms t,...,tn is then substituted for the corresponding variable x,...xn. 413 Above we said that we should more properly indicate this by: “ t x,...,tn xn ” but this becomes 414 too cumbersome, and too tedious to do all the time, so we just leave it for the reader to do depending on 415 the context. ( / / ) 416 417 Proof: By induction on the complexity of . Q.E.D.

418 Exercise 1.4 Convince yourself of the truth of the last lemma. [Hint: At least set out the base cases of the induc- W W W 419 tion: suppose is v v and let t x, t z . Ten x t x z x z x z W W P W P P P 420 z W x t . Te other base cases are relatively straightforward, but a little lengthy to write out. P P  421 Te inductive step for non-atomic formulae= = is{ easy∣ } by comparison.]( ) ↔ ( { ∣ }) ↔ ( / ) ↔ { ∣ ∧ } ↔ ( ) W 422 Lemma 1.24 Let W be a transitive term and suppose for any x, y W, x, y W, then AxPair . P P Proof: We need to show x, y V W . First just note that by Def. 1.22: P { } ( ) x, y W ({z W} z ) x z y W z W z x z y x, y . P P

423 By supposition{ we have} = that:{ ∣x(, y= W∨ x=, y) }W= { ∣ = ∨ = } = { } @ P W PW W 424 x, y W x, y V x, y x, y V . @ P P @ P 425 (Telast uses implicitly an atomic formula({ clause} from) ↔1.23) Q.E.D. ↔ ({ } ) ↔ ( ({ } ) 426 Lemma 1.25↔ Let W be a transitive term. W 427 (i) If for any x W, x W then AxUnion ; P P W 428 (ii) If W then Ax . Infnity . P W W 429 (iii) If for any x W⋃ and any term( a x a ) W then AxSeparation ; P W P W W 430 (iv) If for any x W(, and term f) with f being a function, f “x W holds, then AxReplacement . P ∩ ( P ) ( ) 14 Axioms and Formal Systems

W W 431 Proof: (i) By Example (ii) above, because W is assumed transitive, x x. Moreover V z P 432 W z z W. By assumption x W x W. Hence W @ P PW W W 433 x W x W x W x V x x V (⋃; the) latter= ⋃ is AxUnion . = { ∣@ =P } = P @ P PW @ P 434 (iii) We need to show xa x V ⋃ . Suppose y FVbl a . Tis is equivalent (by Lemma 1.23) @ P 435 to, y W:(⋃ ) ↔ (⋃ ) ↔ ( ⋃ ) ( ) @ P W W W ⃗ W 436 x W a x( V∩ ) x W a =x (W) . ⃗ @ P P @ P P 437 But, for any x W, WP(( ∩ ) ) ↔ W (( ∩ ) )W 438 a x z W z a z x z W z a z x . ∶ P WP P P P P 439 As Trans W , x W, so this is a x. By assumption this is indeed in W. Q.E.D. ( ∩ )Ď = { ∣ ( ∧ ) } = { ∣ ∧ } 440 Exercise(1.5 )Show (ii) and (iv) of the last∩ Lemma.

441 Lemma 1.26 Let W be a non-empty transitive term satisfying all the hypotheses of Lemmata 1.24, 1.25. Ten W 442 ZF that is, each axiom of ZF holds inW. − − W 443 (Proof) : We are only lef with the Axioms of the Empty Set and Foundation. But (Check!), and 444 is a member of any non-empty transitive class (why?). For Foundation let a be a term, and suppose W W ∅ = ∅ W 445 that a . Suppose x a W. Now, by Axiom of Foundation (applied in V)asa , let x W P W W 446 be∅ an element of a with x a . Hence a z z a . Q.E.D.  D 447 ( ≠ ∅) ∩ ≠ ∅ 448 Lemma 1.26 is again a theorem∩ = scheme:∅ given( ≠ a∅ class→ term( ∩ for= which∅)) we can prove the assumptions 449 hold for it, (which itself is an infnite list of proofs in ZF if all the assumptions of Lemma 1.25 are verifed) W 450 then the lemma states that for any axiom of ZF then ZF .(Tis can be trivially extended to a 451 fnite list of axioms by taking a simple conjunction− - but it cannot be extended to an infnite list!) Te 452 next lemma gives a sufcient (but not necessary) condition for⊢AxPower to hold in a transitive class term. 453 Te proof is similar⃗ to those above.

W 454 Lemma 1.27 Let W be a transitive term satisfying for any x W, that x W; then AxPower . P P W 455 Consequently if W satisfes this in addition to the hypothesis of the last lemma then ZF ,thatisallof 456 ZF holds in W. P( ) ( ) ( ) 457 We shall see later that we can prove the existence of transitive -models W, ,withW a set, for W P P 458 which ZF , by establishing the existence of transitive sets satisfying precisely the above closure con- W 459 ditions. We− thus shall show for such a W that, assuming ZF,wecanshow ZF⟨ .⟩ However in ZF we W 460 cannot( prove) the existence of sets (transitive or otherwise)W for which ZF −. (We shall see that this 461 leads to a contradiction with Godel’s¨ Second Incompleteness Teorem.) ( ) ( ) 462 Exercise 1.6 Let v ,...,v be any formula. Let g y the least such that x x, y x V x, y if  n D D P 463 such an x exists; let it be  otherwise. Show that g “V V. Deduce that f sup g “V is a welldefned @ P df 464 function. ( ) (⃗) ≈ ( ⃗) → ( ⃗) ( ) = ( ) 465 Exercise 1.7 Let W be the class term . Which axioms of ZFC hold in W, ? Consider the class term On. P 466 Which axioms of ZFC hold in On, ? (NB For the latter On, just is On, .) P {∅} P ⟨ ⟩ 467 Exercise 1.8 Which axioms of⟨ ZFC hold⟩ in V ? ⟨ ⟩ ⟨ <⟩ Relativisation of terms and formulae 15

468 Exercise 1.9 Check, or recheck, the following basic properties of the V using the Defnitions 1.17, 1.18 of and 469 V : (i) Trans V ; in particular show if x V then y x y V y x ; P @ P P 470 (ii) V V ; Ď 471 (iii) V ( ) V ; ( ∧ ( ) < ( )) 472 (iv) If

483 Initial segments of the Universe

484 In this chapter we look at some properties of initial segments of the universe V: typically local properties 485 of singular and regular cardinals, and the classes of sets hereditarily of cardinality less than some . Tese 486 do not depend on the whole universe of sets. We shall see that when studying wellfounded models of 487 our theory, it sufces to concentrate our eforts on models M, where M is a transitive set, rather than P 488 more general N, E . An important application of the Axiom of Replacement is the Montague-Levy 489 Refection Teorem: this says that for any given fnite set of⟨ formulae,⟩ we can prove in our theory that 490 there are arbitrarily⟨ ⟩ large V that correctly ‘refect the truth’ as regards what those formulae say about 491 the sets in V . Cardinals that are simultaneously both fxed points of certain functions and regular are 492 called strongly inaccessible. If such exist then we can fnd models, indeed of the form V , ,ofallthe P 493 ZFC axioms. We discuss these in the last section. ⟨ ⟩

494 2.1 Singular ordinals: cofinality

495 We frst do some basic work on notions of regularity, singularity and cofnality. Tis then leads into the 496 concepts of normal functions and closed and unbounded sets,andstationary sets. From these further large 497 cardinals can be defned, and although we give the briefest of illustrative examples, it is not the intention 498 of the course to go down this route, rich as it is.

499 2.1.1 Cofinality

500 Definition 2.1 Afunctionf is a cofnal map, if sup ran f .

501 In other words the range of f ∶is unbounded%→ in . ( ( )) = 502 Example (i) f given by f n n ; 503 (ii) f given by f n n; 504 ∶ %→ + ( ) = + (iii) g   given by g are all cofnal maps. ∶ %→ ( ) = 505 (iv) Defne the sequence f  ; f n  f n . Let sup f “ . Ten f is cofnal 506 - by construction.∶ %→ ( ) = ( ) 507 (v). Let E be any subset.( ) Suppose= ( its+ order) = type is . We= use the( notation) f for∶ the%→ function that Ď E 508 enumerates E in strictly increasing order. Tus dom fE will be which will necessarily be no greater

17( ) 18 Initial segments of the Universe

509 than .IfE is now unbounded in -thatis , then f will be a cofnal map, @ D P E 510 that is moreover (1-1) and strictly increasing. < ( ) ∶ → 511 Definition 2.2 If A is a set of ordinals, and Lim , then we say that A is unbounded below (or in) if 512 A . @ D P ( ) 513 Definition< < 2.(3 T< e cofnality) of a limit ordinal is the least so that there is a cofnal map f . 514 It is denoted cf . ∶ %→ 515 Taking f as( the) identity map, shows immediately that cf .

516 Definition 2.4 (i) A limit ordinal is singular cf ( .) Otherwise≤ it is called regular. 517 (ii) We set: ←→ ( ) < 518 Reg df is regular ; Card df acardinal ; 519 Sing df Card asingular ; LimCard df On alimitcardinal . = { ∣P } = { ∣ P} 520 Example= { (i) cf ∣ . Te} above example= shows{ that cf∣ ; but it} cannot be strictly less 521 since no function with fnite domain can have unbounded range in . Te same holds for (ii) above 522 cf .and( + ) = is an example of a cardinal with a smaller( + co) f≤nality. It will follow from below 523 + that cf  . ( ) = ℵ = 524 Exercise( 2.1)If=Lim show that for any , cf cf cf .

525 One could defne( ) cofnality for successor> , but( ⋅ then) = it comes( + ) out= always( ) as 1, and this has little utility!

Lemma 2.5 cf . Tus, a regular ordinal must be a cardinal; to rephrase:

( ) ≤ ∣ cf∣ ≤ is regular is regular and a cardinal.

526 Examples: Reg (Hausdorf 1908); Reg, indeed:   P( ) = ←→ ←→  P

527 Lemma 2.6 (H=ausdorff= ℵ 1914) Any Reg. = ℵ P + 528 Proof: Suppose this failed then note that if f with ran f unbounded in ,but , 529 we would have that f - in other words, the+ union of many sets+ of size +. 530 Assuming AC this is impossible+ - this union could∶ have%→ size at most( !) + Q.E.D.< + < 531 = ⋃ ( ) ∣ ∣ < <

532 Tus any  is regular. (Tese are called successor cardinals.) Te frst singular cardinal is , 533 the next is ;also + , Sing. By Hausdorf’s observation above, a singular cardinal is always a +  P 534 ℵ = ℵ ℵ limit cardinal:+ it occurs as a limit point of the cardinal enumeration function: . We shall consider 535 the questionℵ of whetherℵ theℵ converse fails, that is whether there are cardinals that are simultaneously 536 limit cardinals and regular later. ↣ ℵ

537 Lemma 2.7 For any limit ordinal : 538 (i) cf is the least ordinal so that there is a (1-1) strictly increasing cofnal map f ; 539 (ii) cf cf cf ; hence (Hausdorf 1908) cf is regular; 540 (iii) If( f ) is cofnal and strictly increasing, then cf cf . ∶ %→ ( ( )) = ( ) ( ) ∶ %→ ( ) = ( ) Singular ordinals: cofnality 19

Proof: (i) Let f cf be any cofnal map. We defne a g cf of the desired kind from f by recursion on cf : ∶ ( ) %→ ∶ ( ) %→ g  f  ;

541 Note( that) = a)( for)g ( + ) implies= { g( ) + ( )}and b) for any( )Lim→ ( ),= cf{ ( ), g∣ < is} properly 542 defned simply because cf . Tus we have dom g cf . By defnition g is strictly increasing 543 (and moreover is continuous( ) < at limit ordinals( + ) <- see Def. 2.11(ii) below).( ) As it< dominates( ) ( f) it is cofnal 544 into . < ( ) ( ) = ( ) 545 (ii) Let cf cf . Ten cf . However if cf and f , g are chosen so that f 546 cf , g cf are both strictly increasing and cofnal, then their composition g f 547 cofnally, contradicting= ( ( )) the defnition≤ of(cf) . Hence

549 Corollary 2.8 If Lim then cf( cf .

550 Exercise 2.2 Prove (iii) of( lemma) 2.7 and) the= corollary( ) following.

551 Te following gives an alternative characterisation of cofnality for cardinals.

552 Lemma 2.9 For any infnite cardinal cf is the least ordinal so that there is a sequence X 553 with each X X and X . Ď ( ) ⟨ ∣ < ⟩ < 554 Proof: Let be the∧ ∣ least∣ < such ordinal⋃ defned= in the lemma. If cf then for some cofnal function 555 h ,wehave h .So cf . So suppose for a contradiction that cf ,andwe 556 have X , with each X X .Defne f (X ) = .As cf we have ran f < Ď 557 is∶ bounded→ by some = ⋃ . Let(g ) X ≤ X( be) a bijection. Defne G , g

563 For the rest of this section we let Ω denote a regular, uncountable cardinal.

564 Definition 2.10 Let A be a term and suppose A Ω. (i) Ten A is closed if Ω A is unbounded Ď @ 565 in A . P 566 (ii) We say A is c.u.b. in Ω if it is both closed and unbounded in Ω. < ( ∩ %→ ) 567 Note: In clause (i) we deliberately do not require Ω to be in A if the latter is unbounded in Ω. Closure 568 is equivalent to requiring that (ii) : for any x Vifx A then sup x A Ω . (Exercise: Check this P Ď P 569 equivalence.) ′ ∪ { } 20 Initial segments of the Universe

570 Examples (i) Te cofnal maps from the Examples of the last subsection are all closed and cofnal, 571 although the frst three which were maps just from cofnally into their range are rather trivially closed. 572 Te function g in the proof of (iii) of Lemma . was deliberately constructed to have range closed and 573 unbounded in - closure was obtained by taking for limit ordinals , g to be the supremum of g“ . 574 (ii) Te class terms Lim df On a limit ordinal , Card, LimCard, are all c.u.b. in On. P ( )

575 Definition 2.11 (Normal Function).= { Let∣ f Ω Ω . Ten} f is normal if 576 (i) f f ; ∶ %→ 577 (ii) (continuity) Lim f sup f . < %→ ( ) < ( ) 578 Property (ii) says that( f)is%→continuous( ) = . Normal{ ( ) functions∣ < } are quite common: all the ordinal arithmetic 579 operations yield normal functions: A ; M . , E are all normal functions. 580 Te -function which enumerates the cardinals is normal by design. ( ) = + ( ) = ( ) = 581 Exerciseℵ 2.4 Let Reg.Defne by induction on a function f ,byf  ; f  P 582 f and Lim f sup f . Ten check that f is indeed defned for all and that f is 583 normal. Use f to defne≤ a partition of into many disjoint sets< of cardinality ∶by setting%→ D ( )f = ( + ) =. 584 Check( ) + that D D( ) → for( ) = { (;andthat∣ < } D . < ′ ′ = { ( )+ ∣ > } < 585 Enumerating∩ functions= ∅ of≠ sets< of ordinals⋃ are usually= taken as monotonic on their domains, where, 586 if A On we say f enumerates monotonically A if f dom f A is a bijection enumerating the Ď A A A 587 elements of A in increasing order; to spell it out: fA  min A ; f min A f “ if the latter is 588 non-empty; otherwise f is undefned, and so dom f∶ (for) the←→ least such undefned f . ( ) = ( ) ( ) = ( / )

589 Lemma 2.12 (Veblen 1908( )) ( ) = ( ) 590 (i) Let A Ω. Ten A is c.u.b. in Ω if the enumerating function for A, f , is normal with dom f Ď A A 591 Ω; 592 (ii) let f Ω Ω be increasing. Ten f is normal if ran f is c.u.b. in Ω. ( ) =

593 Proof: (i) Let∶ f%→fA. As dom f Ω,andf is (1-1), ran( ) f cannot be bounded in the cardinal 594 Ω.SoA is unbounded in Ω. Te continuity of f translates directly into the closure of A: if Ω and 595 suppose A is= unbounded(⇐) in . Let( ) = Ω be such that f ( enumerates) A ; then we have that 596 Lim and by continuity of f , f sup f “ and so must be in A. < 597 Clearly∩ f is a monotone increasing< function: ↾Ω f f ∩ .AsA is closed, then 598 f will( ) be also continuous: if (Ω)is= a limit then= A f “ is unbounded in sup f “ . So by closure the A P 599 latter(⇒ is) in A and is then f .Notenowthatdom f must< < be Ω%→since( otherwise) < ( ) it is some Ω and 600 f would witness that cf Ω . However Ω was assumed∩ regular. 601 (ii) Similar. See the Exercise( ) below. ( ) < Q.E.D. ( ) ≤ 602 Exercise 2.5 Let f Ω Ω be strictly increasing. Ten f is normal if ran f is c.u.b. in Ω.

∶ %→ ( ) 603 Lemma 2.13 Let C Ω be c.u.b. in Ω.Letf be the enumerating function of C. Ten the class of fxed Ď C 604 points of fC D df Ω fC is c.u.b. in Ω. Hence for any normal function f Ω Ω there 605 is a c.u.b. class of points Ω that are fxed points for f : f . ∶ = { < ∣ ( ) = } ∶ → < ( ) = Singular ordinals: cofnality 21

606 Proof: Again let f f . Let Ω be arbitrary. We fnd a member of D above (this shows that D is C P 607 unbounded in Ω). Defne:  ; n  f n ; sup n n .Notethat Ω. Tis is clear 608 by the assumption of=Ω’s regularity. We claim that D. Let . Ten for some n . + P n 609 Hence f f = . Hence= (f “) = .As({ f is∣ continuous,< }) f ≠ D.Weare n n  Ď P 610 < < < lef only with showing that+D is closed. Let Ω with D unbounded in . Similar to showing the 611 closure of( )

613 Definition 2.14 For any( E ) =On defne E˚ to be the class of limit points (of E:) namely those limit ordinals Ď 614 such that Eisunboundedin .

615 Exercise 2.6 For∩ any E On,showthatE˚ is a closed class, and if E V with cf sup E , then E˚ is c.u.b. Ď P 616 below sup E . ( ( )) > 617 Exercise (2.7) (i) Suppose Ω Reg Let C, D Ω be c.u.b.in Ω.ShowthatC D is c.u.b. in Ω. (ii) Now generalise P Ď 618 this argument: suppose Ω is a regular cardinal. Let Ω. Let C be a sequence of c.u.b.in Ω classes. 619 Show that C is c.u.b. in Ω. ∩ < ⟨ ∣ < ⟩ ⋂ < 620 Remark 2.15 We used the letter Ω isn this subsection rather than a generic mid-alphabet letter such as 621 for a cardinal (our usual convention) since it is possible to construe the results here as also holding when 622 Ω is interpreted as the class term On. To this extent On behaves like a ‘regular cardinal’, and we can 623 interpret many results here as holding about terms a On which are not necessarily sets. One should Ď 624 be a little more careful than we have, when talking about sequences of classes if we allow Ω On.In 625 this case to defne a sequence of classes C with C On, we should speak about a single class Ď 626 term c of ordered pairs , with C On being defned as the class , c . With care= this is Ď P 627 unambiguous and proper. We could do⟨ the∣ same< in⟩ the following exercise, but have chosen not to, and 628 have returned to our assumption⟨ ⟩ that Ω as a regular cardinal. { ∣⟨ ⟩ }

629 Exercise 2.8 (Diagonal Intersections) Let Ω Reg. Let E Ω be a sequence of subsets of Ω.Defne P 630 the diagonal intersection of the sequence to be the set D ∆ E Ω Ω E . Ω df @ P 631 Now suppose that the E are all c.u.b. in Ω. (i) Show that the diagonal⟨ ∣ < intersection⟩ D is c.u.b. in Ω. (ii) Show that = < ⟨ ∣ < ⟩ = { < ∣ < ( )} 632 D Ω E  .

= ⋂ < ( ∪ ( + )) 633 Definition 2.16 Te (beth) function is defned by: 634  ;   ; sup if Lim . ℶ ℶ + 635 Tisℶ normal= function has a rangeℶ which= as alwaysℶ is c.u.b.= { inℶ On.∣ By< the} last lemma( ) it has a c.u.b. in 636 Ω class of fxed points so that .

637 Exercise 2.9 Show that V = ℶ . @ + 638 Exercise 2.10 (i) Check that(∣ the GCH∣ = ℶ (Generalised) Continuum Hypothesis: that   ) implies that @ 639 . (ii) Show that the frst fxed point of the function has cofnality . (iii) Showℵ that for any regular @ + 640 cardinal there is ,afxed point of the function, with cf . [Hint: this( is simple,= ℵ just) consider an 641 enumeration(ℵ = ℶ ) of the fxed points.] ℶ ℶ ( ) = 22 Initial segments of the Universe

642 Definition 2.17 (The c.u.b. filter on , F ) Let be regular; let 643 X F C C is c.u.b. C X . P D Ď Ď > 644 Exercise 2.11 Show that←→ F has the( following properties:∧ ) 645 (i) X F Y X Y F P Ě P 646 (ii) X, Y F X Y F P P 647 (iii) ∧ F%→ @ R 648 (iv) %→X ∩ X F X F @ @ @ P P 649 (v) X< { } X F ∆ X F . @ < ⟨ ∣ @ < ⟩[P ( ) %→ ⋂P< ] < 650 A non-empty⟨ ∣ < collection⟩[ ( F)of%→ subsets of satisfying] (i) and (ii) is called a flter on ; property (iii) 651 states that the flter is non-principal; (iv) states that the flter is -complete;aflter closed under diagonal 652 intersections (see Exercise 2.8) in (v) is called normal. Not listed is the obvious fact about F that it is 653 non-trivial: F.Aflter is called an ultraflter if for every X either X or X is in F. Te existence R Ď 654 of ultraflters on satisfying additionally (iii) and (iv) cannot be proven in ZFC, (they can for ) 655 but is crucial∅ for studying many consistency results in forcing theory, and for/ considering elementary 656 embeddings of the> universe V to transitive subclasses of V. A class of subsets of on which there= is 657 an ultraflter satisfying (i)-(iv) is ofen said, in an equivalent terminology, to have a -valued measure, 658 in which case property (iv) is called “ -additivity”. Sets have value   depending on whether they are 659 out/in the ultraflter. (iii) then translates as “points have measure 0”. /

660 2.1.3 Stationary Sets

661 Definition 2.18 Let E Ω. Ten E is called stationary in Ω if for every C Ω which is c.u.b., then Ď Ď 662 E C .

663 If we were to talk about a class term S Ω being stationary where Ω is allowed to be the class of all ∩ ≠ ∅ Ď 664 the ordinals, we should declare more precisely what this means: it means that for any class term c such 665 that we can prove in ZFC that c is a closed and unbounded class of ordinals, then we can also prove that 666 c S . 667 Stationary subsets of regular cardinals (or subclasses of On exist: any c.u.b. subset of with 668 regular∩ ≠ ∅ is stationary, by Exercise 2.7 (i). (Similarly for subclasses of On). But there are other stationary 669 subsets of regular cardinals. )

670 Exercise 2.12 Let S Ω be stationary and C Ω be c.u.b. Ten S C is stationary. Ď Ď 671 Exercise 2.13 Let S Ω be stationary. Show that S S˚ is stationary. Ď ∩

672 ∩ Example 1 Let . Ten S df  cf and S  df  cf  are two 673 disjoint stationary subsets of : let C be any c.u.b. subset. Let f C be its strictly increasing  Ď   674 = = { < ∣ ( ) = } = { < ∣ ( ) = } enumerating function. Ten f C S and f  C S  . P P ∶ %→ 675 Exercise 2.14 Can you generalise( this) example∩ to larger( regular) cardinals,∩ e.g. n for n , or any regular ?

676 Exercise 2.15 Find S stationary, for n ,withS S but with S . n Ď  n  Ď n n n < > + + 677 Te reason for the nomenclatureℵ comes< from (ii) of the following Lemma.⋂ = ∅ Singular ordinals: cofnality 23

678 Lemma 2.19 (Fodor’s Lemma 1956) Let be a regular cardinal. Te following are equivalent. 679 (i) S is stationary in ; > 680 (ii) For every function f S On which is regressive (that is S  f ), there @ P 681 is a stationary set S Sandafxed  so that S f  Ď ∶ %→ @ P ( > %→ ( ) < ) 682 Proof: Assume (i). If (ii) failed for some regressive function( ( ) = f )then we should be able to defne for 683 every a c.u.b. C with C S f . Let D C be the diagonal P @ P 684 intersection of C . Ten D is c.u.b. in and for any D S, f . But if D S we P P 685 must have< f , which is a contradiction.∩ %→ ( (ii)) implies≠ (i) is= trivial.{ ∣ < ( )} Q.E.D. ⟨ ∣ < ⟩ ∩ ( ) ≮ ∩ 686 Remark: AC was used heavily in picking the C in the above; if one attempts the proof without using ( ) <  687 AC one obtains in (ii) only the conclusion that for some  that f “  is unbounded in .Because 688 one cannot in general pick class terms, if one attempts to prove the Lemma− by considering regressive 689 functions on all of On, rather than just , one again weakens< the conclusion (see the next Exercise).

690 Exercise 2.16 (E) Let f be a function class term with dom f On and f regressive. Show that for some    691 f “  is unbounded in On. [Hint: Suppose the conclusion fails; then defne g sup f “ ;nowfnd  692 closed− under g: g“  .] ( ) = − { } Ď ( ) = { } 693 We could have defned stationary subsets of ordinals with cf . Tis is possible, but notice 694 that it would make no sense to defne the notion of a stationary subset if cf . For, if f 695 is a strictly increasing function cofnal in then ran f is c.u.b. in (; but) > it is easy to defne another c.u.b. 696 in set C (of order type with ran f C so it makes little sense to even( ) = try to defne stationary∶ %→ 697 in this way. ( ) ) ( ) ∩ = ∅ 698 We saw above that  contained two disjoint stationary subsets. In fact far more is true. (Teproof 699 of this theorem is omitted.)

700 Theorem 2.20 (Bloch (1953), Fodor (1966), Solovay (1971)) Let be regular, and let S be station- Ď 701 ary. Ten there is a sequence of many disjoint stationary sets S Sfor (i.e. for S S Ď 702 )withS S . > < < < ∩ = ∅ = ⋃ < 703 Exercise 2.17 (E) (H.Friedman) Let S be stationary. Ten for any there is a closed subset C S ˚ Ď   Ď 704 with ot C . [Hint: Do this by induction on for any stationary S. Tis is trivial for  assuming it 705 is true for (just( ) add on more point S above sup C to C to get C of< order type ). Assume Lim P  706 ( ) = + = + and for we can fnd such C .Notethatforany we can fnd such +C with min C - by considering 707 the stationary S . Let n be chosen with (sup ) ;forany then pick closed+ subsets C S( of) n n n n Ď 708 order type<  and with min C sup C . Ten C S and is closed in S( with) ≥ the exception of the n n  n n n Ď 709 / ⟨ ∣ < ⟩ = point sup n C n . Call a point arrived+ at as a sup of such a sequence of sets C n an “exceptional” point. We have 710 just shown that+ the exceptional( points) are> unbounded( ) in ⋃. But now just note that a limit of exceptional points 711 is also exceptional.(⋃ ) Tat is, they form closed subset of .AsS is stationary there is an exceptional point S.  P 712 Tis can be added to the top of the sequence of points from the sets C n witnessing the exceptionality of ; this 713 sequence then has order type  and is contained in S. ′ 714 Remark: Tis is not the case at higher cardinals, e.g. . Let be the statement “for any X  and any + Ď 715  either X or  X contains a closed set C with ot C ”. Ten ZFC . (⋆) < / ( ) = ⊢/ (⋆) 24 Initial segments of the Universe

716 2.2 Some further cardinal arithmetic

717 We give some further results on cardinal arithmetic.

Definition 2.21 Let be a sequence of cardinal numbers. Let X be a sequence of disjoint sets, with X . (i) Ten we defne the cardinal sum ⟨ ∣ < ⟩ ⟨ ∣ < ⟩ = ∣ ∣ X .

∑< = ∣ ⋃< ∣ 718 (ii) the cardinal product is defned as X

719 Note: (i) as usual these values are independent∏ < of= ∣ the∏ < choices∣ of the X . For the product the require- 720 ment that the sets X be disjoint may be dropped. 721 (ii) If all the for some fxed ,and Card, then and . P Σ < < 722 Exercise 2.18 Show that= ≥ and ∑ . = ⊗ ∏ = < 723 Exercise 2.19 Show that∏ if <  for= (∏ < , then) ∏ < = .

724 Exercise 2.20 Show that distributes over , i.e. that< < , f , f . ≥ < ∑ ≤ ∏ P < < = ( ) 725 Lemma 2.22 If ∏Card and ∑ is a non-decreasing∏ ∑ sequence∑ ∏ of non-zero cardinals, then P 726 sup . ≤ ⟨ ∣ < ⟩ Proof< : We partition< into many disjoint pieces each of size (by using some bijection ). ∏ = ( ) ˆ Let us say then that X . Because the sequence of the is non-decreasing, and each X is unbounded in ,westillhavesup sup say, for each .Nownotethatwemay∶ ↔ < X reorganise the product= ⋃ P = < = < as . X ⎛ P ⎞ ∏< ∏< ∏ 727 But X sup X , hence we have that⎝ ⎠ . P P 728 Conversely . Hence we have equality as desired. Q.E.D. ∏ ≥ = ∏ < ≥ ∏ < = <   <   729 Exercise 2.21 ∏n n ≤ ∏; n =n ;

< < Theorem 2.23∏(Konig’s¨ =Teorem)∏ If = ( ) for then

< < .

∑< < ∏< 730 Proof: Pick X for with X . We shall show that if Y X for are Ď 731 such that Y , that then Y X . Hence we cannot have . Let < 732 P f f Y be< the projection∣ ∣ of= Y on to the ’th coordinate. As Y∏ X , P

735 Exercise 2.22 Deduce Cantor’s Teorem that  from Konig’s¨ Teorem.

< 736 Corollary 2.24 For all ,andforall cf . Hence in particular cf  for any cardinal 737 . ( ) > ( ) > Proof: Let be a sequence of cardinals for with .Itsufces to show that . Let be the fxed sequence with all .ByKonig’s¨ Lemma then < < ∑ < < = .

∑ < ∏ = ( ) = 738 < < Q.E.D.

cf 739 Corollary 2.25 for any cardinal . ( ) 740 Proof: For cf let> be less than so≥ that cf . Ten cf 741 . Q.E.D. < ( ) cf cf = < ( ) 742 ( ) ∑ < ( ) < ( ) 743 We may put some= ∑ of these facts< ∏ to gether to= get some more information about the exponentiation 744 function under GCH. First:

745 Exercise 2.23 If cf then cf .

< ( ) = < = ( )< < 746 Theorem 2.26 Suppose GCH holds⋃ and , ⋃ . Ten takes the following values: 747 (i) if ; 748 (ii) + if cf ; ≥ 749 (iii) + if ≤ cf . ( ) ≤ < cf 750 Proof: (i) follows< ( from)  . (ii)  ; (iii) We use Ex.2.23. 751 .But  + .So ( ) sup + . Q.E.D. 752 = =∣ ∣ + < ≤ ≤ = =+ < < 753 =Without∣⋃ GCH∣ the∣ only∣ ≤ known∣ ∣ = constraints= ∣ ∣ < on the exponentiation≤ ≤ ⊗ function∣ ∣ for= regular cardinals are 754 (a)  and (b)   . For singular the situation is more subtle and a discussion of this 755 involves large cardinals. < < → ≤  756 Exercise 2.24 Prove that Πn n . [Hint: Every subset of can be coded as a function .] ℵ +  757 Exercise 2.25 Assume CHℶbut= not ℶGCH= ℶ.Showthat n n for ℶ n . → ℶ ℵ (ℵ ) = ℵ ≤ < 758 2.3 Transitive Models

759 We have seen how certain assumptions about a transitive set or class term allows us to conclude that a 760 number of the ZF axioms hold, by relativisation to that set or term. When thinking of a term W as a 761 structure, which we more properly write W, ,wesaythat W, is a transitive model,ortransitive P P 762 model if we wish to emphasise the standard interpretation. We saw that in 1.24 and 1.25 that closure P ⟨ ⟩ ⟨ ⟩ 26 Initial segments of the Universe

W 763 under those lists of conditions ensured that ZF . Te following Lemma allows us to create transitive 764 isomorphic copies M, of possibly non-transitive− structures H, . It is known as the “Collapsing P P 765 Lemma” since it collapses any “ -holes” out( of the) structure H, . Te Lemma is much more general P P 766 and in fact a structure⟨ H⟩, R will be isomorphic to a transitive⟨ model⟩ M, provided that R satisfes P 767 two necessary conditions: that it be wellfounded, and that it be⟨ “extensional”.⟩ Te latter simply requires 768 it to be -like. Clearly these⟨ conditions⟩ are necessary, since is itself wellfounded,⟨ ⟩ and for transitive M P M P 769 we always have that AxExt .

( ) 770 Definition 2.27 Given a term t and a relation R on t we say that R is extensional on tif for any u, v P 771 t, u v there is z twithzRu zRv (i.e. z t zRu z t zRv ). P P P

772 Note≠ that is extensional on←→x ¬if Trans x{ but need∣ not} ≠ be{ in general.∣ } P ( ) 773 Lemma 2.28 (Mostowski (1949)-Shepherdson (1951) Te Collapsing Lemma) 774 Let H V. P 775 (i) Suppose that R is wellfounded and extensional on H. Ten there is a unique transitive term M and 776 auniquecollapsing isomorphism H, R M, .   P 777 (ii) Additionally if R x x ,x H, and Trans x , then x id x. P ∶ ⟨ Ď ⟩ %→ ⟨ ⟩ 778 Proof: (i) (1) If exists,↾ then= it is↾ is unique. ( ) ↾ = ↾ 779 Proof: Suppose , M ran are as supposed. Let u, v H. Note if uRv then u v as P P 780 preserves the order relations. Tus for v H: u u H uRv v . = ( ) P P Ď ( ) ( ) 781 However if z v , then z M,asM is transitive. Hence z u for some u H with uRv. P P { ( ) ∣ ∧ } ( ) P 782 Hence u u H uRv v . Tus v u u H uRv . Tus the isomorphism, if it P Ě P 783 exists must take this form.( ) = ( ) { ( ) ∣ ∧ } ( ) ( ) = { ( ) ∣ ∧ } 784 (2) exists. 785 We thus defne by R-recursion: v u u H uRv P ˚ 786 And take M ran .TriviallyTrans M by .(3)-(5)willshowthat is an isomorphism. ( ) = { ( ˚) ∣ ∧ }() 787 (3) is (1-1). = ( ) ( ) ( ) 788 Proof: If not pick t -minimal in M so that there exist u v with t u v .As u v,andR P 789 is extensional, there is some w with wRu wRv. Without loss of generality we assume wRu wRv. 790 (Te argument in the other case is identical.) Ten w ≠u t = v( .Sowemusthavethatfor) = ( ) ≠ P 791 some xRv: x w (as v is the set↔ of¬ all such x ’s). But now if we set s x ,wehave∧ ¬ s t P 792 and x w s and, as wRv, x w. However( this) s contradicts( ) = = the( ) -minimality in the choice P 793 of t. ( ) = ( ) ( ) ( ) = ( ) ( ) = ( ) = ¬ ≠ 794 (4) is onto. 795 Tis is trivial as M is defned to be ran . 796 (5) is an order preserving isomorphism. ( ) 797 We have already that is a bijection. Tis then follows from the defnition at : uRv u ˚ P 798 v .   ( ) → ( ) 799 Tis fnishes (i). For (ii) we now assume that R x x , Trans x and x H. ( ) P Ď 800 (6) x id x. ↾ = ↾ ( ) ↾ = ↾ TeH sets 27

801 Ten for v x we have v x H. Tus becomes, for v x: v u u v .Now,by P Ď Ď ˚ P P 802 -induction on x x we have v x u v u u v v v x v v . P P ˆ @ P @ P @ P 803 ( ) ( ) = { ( ) ∣ } Q.E.D. 804 ↾ [( → ( ) = ) → ( ) = ] → ( ( ) = ) 805 Te resulting structure M is called the ‘collapse’,or better, the ‘transitive collapse’ of H, R . To illus- 806 trate how the Collapsing Lemma works note the following exercise: ⟨ ⟩

807 Exercise 2.26 Let H, R WO. Apply the Collapsing Lemma. What is the outcome? P

808 Note the use in⟨ the⟩ above of a recursion along the wellfounded relation R rather than . More gen- P 809 eralised forms of this argument are possible. We may take any class term t in place of the set H and 810 provided the wellfounded extensional relation R is set-like - meaning for any u t v vRu V, then P P 811 the same argument may be used, and a class term M defned in the same way. { ∣ }

812 Lemma 2.29 (General Mostowski-Shepherdson Collapsing Lemma) Let A be a class term. 813 (i) If R A A be a wellfounded extensional relation which is set-like in the above sense. Ten there Ď ˆ 814 is a unique term M, and unique collapsing isomorphism A, R M, . P 815 (ii) If R then if s is a transitive term with s A, then s id s. P Ď ∶ ⟨ ⟩ %→ ⟨ ⟩ = ↾ = ↾ 816 Exercise 2.27 Show that V can be ‘coded’ as a subset of : that is there is E so that , E V , . [Hint: n Ď P 817 Defne nEm df the “ ” column in the binary expansion of m contains a 1; (thus n nE , ,  ); check 818 there is u satisfying , E u, with Trans u .Showu V .] ⟨ ⟩ ≅ ⟨ ⟩ ←→ P { ∣ } = { } 819 Exercise 2.28 Show⟨ if A⟩,≅ ⟨, B,⟩ are transitive( ) sets, and=f A, B, is an isomorphism, then f id A. P P P P 820 Exercise 2.29 Suppose⟨Trans⟩ ⟨x and⟩ f x is a bijection.∶ De⟨ fne⟩ ≅E⟨ ⟩ by: , E f = f ↾ . Ď ˆ P P 821 Show that , E x, and that the isomorphism is the Mostowski-Shepherdson collapse map. Let g P  ˆ 822 be a further bijection. Ten if E( ) g “E,∶ we↔ can then think of x as coded by a subset⟨ of ,namelyby⟩ ←→ (E.Notethat) ( ) 823 x will have⟨ -many⟩ ≅ ⟨ such⟩ diferent codes− depending on the function f . ∶ ↔ ̃ = ̃ 824 Exercise 2.30 Find an example of an x, which is not extensional. If we nevertheless apply the Mostowski- P 825 Shepherdson Collapse function to it, what happens? ⟨ ⟩

826 2.4 The H sets

827 Te following collects together sets whose transitive closure is of a certain maximal size. Te phrase 828 “hereditarily of [property ]” means that not only must an x have property , but so must all its members, 829 and their members, and .. and so on.

830 Definition 2.30 Let be an infnite cardinal. H df x TC x is the class of sets hereditarily 831 of cardinality less than . = { ∣∣ ( )∣ < } 832 We summarise the properties of these classes. 28 Initial segments of the Universe

833 Lemma 2.31 Let be an infnite cardinal. 834 (i) On H ; Trans H ; 835 (ii) H V and hence H V , H ; ∩Ď = ( )P 836 (iii) y H x y x H ; P Ď P + ( ) = 837 (iv) x, y H x, x, y H ; P ∧ %→ P 838 (v) (AC) regular x x H x H x . %→ ⋃ @{ }P Ď 839 Proof: (i) Exercise; (ii):%→ if x( H we↔ have TC∧ ∣x∣ < ) ; we use Ex.1.2: let “TC x ,andas P 840 Card, we cannot have ; hence x TC x and thus x V . Tus H V , P ∣ ( )∣ < P = ( ) Ď 841 thence H V ;as H ,wehave H . (ii) is completed. P Ď ≤ ( ) = ( ( )) ≤ < + + 842 Exercise 2.31 Prove (i), (iii)-(iv) here. Give( an example) ≥ to show that (v) fails if is singular.

843 (v) Assume x H .AsTrans H x TC x this follows from the defnition of H . P Ď 844 As TC x x TC y y x , it is the union of less than many sets all of cardinality less than . P 845 By AC such(→) a union has itself cardinality( less) ∧ than so( we) are done. Q.E.D.(←) ( ) = ∪ ⋃{ ( ) ∣ } H 846 Lemma 2.32 (AC) regular ZFC .Moreformally:let be a fnite list of axioms from H 847 ZFC . Ten ZFC “ regular − .” %→ − > ∧ %→ ( ) 848 Proof: We appeal⊢ to> Lemma∧ 1.26 once%→ we(⋀⋀ have%→ ) observed that Separation, Replacement, and Choice H H 849 axioms hold relativised to H , the others follow from Lemma 2.31. AxSeparation holds since if a H 850 is any term, and x H then y= a x is a subset of x and hence it satisfes TC y also. Similarly, P H 851 for the Axiom of Collection, if risarelation xr“x then( let s be the function) (defned in V) ∩ @ ∣ ( )∣ < 852 given by sx y r x, y x, y H z ă y r x, z x H y where H , ă WO ( P @ ∧ ≠ ∅) R P 853 for some wellorder ă. Ten letting w H be arbitrary, and applying Replacement (again in V)we P 854 deduce that=s“w↔V(.( However) ∧ s“w ∧H and has¬ ( at most)) ∨ (w many∧ = elements.∅) ⟨ Hence⟩ setting P Ď H 855 t s“w we have t H as required. For AC let f H . In particular dom f H . Assume P P P 856 x dom f f x .ByACwehaveg f . It is an exercise∣ ∣ < to check that any such g satisfes @ P P 857 TC= g TC f hence g H . ( ) ( ) Q.E.D. Ď ( ) ( ) ≠ ∅ P 858 We remark also that the last lemma is false for∏ singular cardinals . 859 ( ) ( )

860 2.4.1 H - the hereditarily finite sets

861 For then H is known as the class of the hereditarily fnite sets - and is so also more usually 862 abbreviated as HF. = 863 Exercise 2.32 Show that V HF. [Hint: For use induction on n to show V HF.For use - Ď n Ď Ě P 864 induction]. = ( ) ( ) HF 865 Lemma 2.33 ZFC Ax . Inf Ax . Inf

866 Proof: See Exercise.( − +¬ ) Q.E.D. TeH sets 29

Andrzej Mostowski 1913-1975

867 Exercise 2.33 Check that HF is closed under all the assumptions of Lemmata 1.24 and 1.25 (except 1.24 (ii)) and HF 868 even the power set operation. Hence ZFC Ax . Inf .

f y 869 Exercise 2.34 (Ackermann 1937) Investigate( − the following) function f HF : f x Σy x  . P ( ) ∶ → ( ) = 870 2.4.2 H  - the hereditarily countable sets

871 TeclassH  is also known as the class of sets hereditarily of countable cardinality, and so also is given 872 the abbreviation of HC. HC and hence we regard the real continuum as a subclass of HC. At Ď 873 least in one crude sense, HC “is” , see the following Exercise. P( ) 874 Exercise 2.35 If x HC then we haveP( TC) x .Defne a wellfounded extensional relation E on so that P 875 , E TC x , . [Hint: We have a bijection f N TC x for some N ; defne nEm f n f m . P  k P 876 ] If we use a recursive pairing bijection p∣ ( )∣ ≤ (for example p k, l  . l  ) we may further ˆ 877 code⟨ ⟩E≅as⟨ a subset( ) ⟩E .Wethushaveefectively∶ coded←→ up( TC) x −as a subset≤ of .] (By using↔ ( further) ( such) Ď 878 coding devices we may take any countable∶ structure←→ with domain in HC(⟨and⟩ code) = it up( as+ a) subset− of . In this 879 sense to study all countable structures is to study all of .) ( )

880 However unlike the case of and HF, we cannotP( identify) HC with any V : V but V  Ě  881 does not contain any countable ordinal .But  HC as can be easily+ determined from its+  Ď 882 defnition. On the other hand V   HC so V HC. ClearlyP( then) HC is   Ď 883 not closed under the power set operation but> we+ do have that all other ZF axioms hold there: ∣ + ∣ = ∣PP( )∣ = > = ∣ ∣ + / HC 884 Lemma 2.34 ZF . − 885 Proof: As HC( H)  this is just a case of Lemma 2.32. Q.E.D.

= 30 Initial segments of the Universe

886 Exercise 2.36 Which axioms of ZF hold in V if Lim ? Find a wellordering A, R V but for which there P 887 is no ordinal V with A, R , ; hence fnd an instance of the Ax.Replacement that fails in V . P + 888 ( ) ⟨ ⟩ [Te latter is a model+ of Z, the axiom system of Zermelo which is ZF with Replacement removed. For almost+ all 889 regions of mathematical discourse,⟨ ⟩V≅ ⟨ is< a⟩ sufciently large “universe” - mathematicians never, or rarely, need 890 sets outside of this set.] +

891 How large is H ? Tis depends again on the power set operation on sets of ordinals. Every element 892 of H can be coded as a subset of . See the next exercise which just mirrors the argument of Ex.2.35. + 1 893 Exercise 2.37 Extend Ex. 2.35 to any H . [Hint: let p now be any pairing bijection p . Assume ˚ ˆ 894 f TC x and put E if f f+ . Ten by the Collapsing Lemma , E TC x , . Let E  P P 895 p “E . Ten any structure with domain in H can be coded by a subset of E .] Deduce∶ that←→ H .  Ď −∶ ←→ ( ) ( ) ( +) ⟨ ⟩ ≅ ⟨ ( ) + ⟩ = 896 We adopt the notation: For , Card, sup Card . ∣ ∣ = ∣P( )∣ P df P < 897 Exercise 2.38 Let Card . Show that H  =. [Hint:{ for ∣ a successor∧ cardinal,< } this is the last Exercise.] P < 898 Exercise 2.39 (Levy) Let h be the class∣ ∣ of= sets x with (i) y TC x y , (ii) x . Show that if @ P 899 Reg, then H h ); fnd an example where this fails if is singular. P ( ) ( )(∣ ∣ < ) ∣ ∣ < = ( 900 2.5 The Montague-Levy Reflection theorem

901 Tis section proves a Refection Teorem, so called because it shows that in ZF we can prove that the 902 fact of any sentence holding in V is refected by an initial portion of the universe: we shall see that V 903 for some . However these arguments are of more interest than just as a means to solving this 904 problem. ↔ 905 We shall be able to prove from this theorem that any fnite collection S of the ZF (or ZFC) axioms can 906 be shown to hold in a transitive set; indeed we shall see that we can always fnd a level of the cumulative V 907 hierarchy, a V , in which S is true: ZF S . Of course we have just seen that all of ZF is true D 908 in any H . If our fnite list contains the Ax.Power then Refection arguments provide a solution.− From 909 this we shall+ be able to see later that ZFCisnot⊢ ( )fnitely axiomatisable: there is no fnite set of axioms S 910 that have the same deductive consequences as those of ZFC.

911 2.5.1 Absoluteness

912 Definition 2.35 Let W Z be class terms. Let with FVbl x . Ď PW P Z Ď 913 (i) is upward absolute for W, Z if x W ; @ P LW Z { } {⃗} 914 (ii) is downward absolute for W, Z if x W ; ⃗ @ P( %→ ) W Z 915 (iii) is absolute for W, Z if both (i) and (ii) hold: x W ⃗ ( @ ←%P ) 916 If Z V then we omit it, and simply say “ is upward absolute for W” etc. If ,..., n is a fnite list ⃗ ( ←→ ) 917 of formulae then we say that ,..., are upward absolute (etc. ) if their conjunction  n %→  n 918 is. = = %→ = ∧⋯∧ 1An Exercise annotated with a indicates that is perhaps harder than usual. An (E) indicates that it is Extra to the course. ˚ Te Montague-Levy Refection theorem 31

919 Definition 2.36 Given classes W Z and a term t we say t is absolute for W, Zif W Ď Z Z W 920 x W t x W t x Z t x t x @ P P P Z Z 921 (Recall⃗ that( asserting(⃗) t x↔ (⃗Z) is to assert∧ (⃗) that= t(⃗x) )is a set of Z. Note we could have defned P 922 ‘upwards’ and ‘downwards’ absoluteness for terms t as well.) A standard example of a term that is not ⃗ ⃗ 923 absolute is given by “the frst( uncountable) cardinal”(t ( ) On is countable . Suppose W V. V W P Ď 924 Certainly t t is defned: it is .Itmaybethatt is defned, and is a cardinal in W.ButV may 925 simply have more onto functions f with dom f and= { ran f ∣ On,thanW has.}) We may thus have W V Ď 926 t t . Another= example is given by t . ( ) = ( ) < = P( ) 927 Definition 2.37 Alistofformulae ,..., n is subformula closed if every subformula of a formula 928 is on the list. %→ = 929 Te following establishes a criterion for when a formula’s truth value is identical in diferent class 930 terms.

931 Lemma 2.38 Let be a subformula closed list. Let W Z be terms. Te following are equivalent: Ď 932 (i) are absolute for W, Z. ; %→ 933 (ii) whenever is of the form x x, y (with FVbl y )itsatisfes the Tarski-Vaught %→ i D j i Ď 934 criterion between W and Z: Z ⃗ Z ⃗ 935 y W x Z x, y ( x )W x, y (. ) { } @ P D P j D P j Z Z W 936 Proof: (i) ⃗ ii :[ Fix y W( and⃗) assume%→ i y ( x⃗) ]Z j x, y . By absoluteness of i, i y , W P DZ P Z 937 so x W j x, y and by absoluteness j, j x, y ,so x W j x, y . D P ⇒ ( ) ⃗ (⃗) ≡ D P( ⃗) (⃗) 938 ii i : By induction on the length of i: we thus assume absoluteness checked for all j on the ( ⃗) ( ⃗) ( ⃗) 939 list for shorter length, in particular for any subformula of i . ( ) ⇒ ( ) 940 i atomic: absolute by defnition. 941 i j k: then i is absolute since both j and k are by inductive hypothesis. 942 i j: similar; ≡ ∨ 943 i x j x, y . So fx y W. ≡ ¬D P ≡ ( ⃗) y W⃗ x W x, y W x Z x, y Z y Z i D P j D P j i 944 Where : the frst and(⃗ last) ↔ equivalence is( just⃗) the↔ defnition of( relativisation;⃗) ↔ (⃗ the) second equivalence 945 - from lef to right uses the absoluteness of from the Ind.Hyp., and the fact that W Z;andfrom j Ď 946 right to lef uses Assumption (ii) and again the absoluteness of j from the Ind. Hyp.; and the last is 947 relativisation again. Q.E.D.

948 Lemma 2.39 Let W be a transitive class term. Ten any ∆-formula is absolute for W.

949 Proof: Let be ∆ and apply the last argument (with the list of together with all it subformulae). 950 Te point here is that Trans W so W knows the full -relationship on its members. As any ∆ -formula P %→  951 only contains bounded quantifers, this is enough to satisfy the criterion of 2.38 when one comes to the 952 induction step x y (where) is ∆ itself, in the induction at the end of the last proof. D P  ≡ 32 Initial segments of the Universe

953 Exercise 2.40 Fill in the details. [Hint: by what has just been said, only the x y step and the last chain D P 954 of equivalences needs to be argued.] ≡ 955 Exercise 2.41 Let W be a transitive class term. Ten (i) any Σ-formula is upwards absolute for W; (ii) any 956 Π-formula is downwards absolute for W.

957 2.5.2 Reflection Theorems

958 We use the last criterion of absoluteness in our Refection Teorems. the frst lemma really contains the 959 essence of the argument.

960 Lemma 2.40 Let Z be a class term, and suppose we have a function F with F Z so that Z Z Z @ P 961 V . Assume (i) Z Z ; Ď 962 (ii) Lim Z Z ; ( ) = ( ) < %→ P 963 (iii) Z On Z . Ten for any ,..., n : ( )P%→ = 964 ZF ⋃ are absolute for Z , Z . ˚ = ⋃ @ D %→ = 965 Note:( ) Formally here⊢ we are> saying( %→ that if we have a term) for Z and a term for the function FZ,and 966 we can prove in ZF that F has properties (i) - (iii), then for any , there is a proof in ZF of .Weare Z ˚ 967 not saying that in ZF “ ) holds”. (Assertions such as the latter we shall see later are false.) @ ˚ %→ 968 Proof: We apply Lemma 2.38 and try and fnd some W Z such that (ii) of the lemma applies.( ) Tis %→ 969 will sufce. By lengthening⊢ the(( list) if need be we shall assume that is subformula closed. For i n we = 970 defne functions Fi On On.If i x j x, y set: D %→ ≤ 971 Gi y  if x Z j x, y ∶D P %→ ≡ ( ⃗) 972 where is least so that x Z j x, y . (⃗) = ¬ ( ⃗) D P 973 Fi sup Gi y y Z . = P ( ⃗) 974 Note that Gi is a well defned function, and consequently so is Fi: Gi“Z V by AxReplacement; ( ) = { (⃗) ∣ ⃗ } P 975 hence Fi sup Gi“Z is then a well defned term. Note also that each Fi is monotonic: 976 Fi Fi .If i is not of the above form, set Fi  everywhere. ( ) = < %→ 977 Claim: Lim i nFi . ( ) ≤ (@) D @ @ ( ) = 978 Proof of Claim: Defne by recursion on :  ; > ( ( ) ∧ < ≤ ( ) < ) 979 k  max k , F k ,...,Fn k ; supk k. = 980 Ten implies that Lim . Hence if then for some k . Hence F + k k  k P i 981 F = .{ + ( ) ( )} = Q.E.D.(Claim) i k k  + 982 Now that< the Claim is proven, then( we) may verify the< Lemma< with such a for Z and Z. ( ) ≤ + 983 ( ) ≤ < Q.E.D.

984 Exercise 2.42 Carry out this fnal verifcation.

985 We may immediately set Z to be V and Z to be V and obtain the immediate corollary:

986 Theorem 2.41 (Montague-Levy) TeRefection Teorem.Let be any fnite list of formulae of . 987 Ten %→ 988 ZF are absolute for V . Q.E.D.L @ D ⊢ > (%→ ) Te Montague-Levy Refection theorem 33

989 As cautioned above, this is a theorem scheme again: it is one theorem of ZF for each choice of . 990 Notice that if in particular are sentences, we may write the conclusion as: V %→ 991 ZF . @ D %→ 992 Moreover if the are axioms of ZF we have that they are true in V. In this case we may write: V %→ %→ 993 ZF ⊢ > . ( ←→ ( ) ) @ D %→ 994 In other words: for any fnite list of ZF we can fnd arbitrarily large so that those axioms hold in %→ 995 V .⊢ We can state> (( something⋀⋀ ) ) stronger:

996 Corollary 2.42 Let T be any set of axioms in extending ZF,and a fnite list of axioms from T. Ten V 997 T . @ D L %→ 998 Proof⊢ : Since> T((extends⋀⋀ %→ ) ZF) T proves the existence of the V hierarchy, and T i for each i from . V 999 Hence T trivially. And T Q.E.D. @ D %→ 1000 At frst blush it might look as if the restriction to fnite lists of is unnecessary.⊢ Why could we %→ %→ %→ 1001 not look at⊢ a⋀⋀ recursive enumeration⊢ of all> axioms( ⋀⋀ of←→ ZF say,(⋀⋀ and) fnd) some V in which they were all i %→ 1002 true? We know from the Godel¨ Second Incompleteness Teorem that there is no way to formalise that 1003 argument within ZF, since it would be tantamount to proving the existence of a model of the ZF axioms, 1004 and hence the consistency of ZF. So what goes wrong? Lemma 2.40 can only work for fnite lists :the 1005 statement “ are absolute for Z , Z” involves a conjunction of the formulae from the list: we cannot %→ 1006 write an infnitely long formula in , so we have no way of even expressing the absoluteness of such an %→ 1007 infnite list. Another paraphrase on this is in the following Exercise. L 1008 Exercise 2.43 Show that for every formula of V 1009 ZF “Tere is a c.u.b. class C On so that C x V x x ” Ď @L ∶ P @ P 1010 [Hint: Te reasoning of Lemma 2.40 pretty much gives the relevant cub class as the closure points of the Fi .] ⃗ ⃗ ⃗ 1011 Remark:⊢ One might think that one could enumerate all the axioms( ( of)ZF↔ ( ,( )),...,)fnd the appropriate classes 1012 C n and take D n C n . Tis appears then to be an intersection of only countable many c.u.b. classes and so V 1013 must be c.u.b. in On? But for any element D we’d have ZF , and we appear to have proven the existence P 1014 of models of ZF -= contradicting⋂ Godel.¨ What is wrong with this reasoning? ( ) 1015 Exercise 2.44 Find a sentence so that if is absolute for V then is a limit ordinal. Repeat the exercise and 1016 fnd so that if is absolute for V then (the ’th infnite cardinal). [Hint: consider the statement: “For 1017 every exists”.] = 1018 As the last exercise shows, if we insist on fnding a V which is absolute for any particular sentence, 1019 then we may need to fnd a very large for this to happen. If we are content to merely fnd asetfor 1020 which a formula is absolute, we can fnd a countable such set. More generally:

1021 Lemma 2.43 Let Z be a term, and be any fnite list of formulae of . Ten 1022 ZFC x Z y x y Z are absolute for y, Z y max , x . @ Ď D Ď Ď%→ L 1023 Proof: We de⊢ fne from[ the term Z the∧ %→ term giving the function∧∣ ∣ ≤F { Z∣ ∣}]V which we shall call 1024 Z . Again assume that is subformula closed. As x is a set, by the AxReplacement G“x V where P 1025 G u the least such that u Z . Ten sup G“x = G“x V(. Call) = this∩ ordinal . By Lemma df %→ P P  1026 2.40 fnd  with absolute for Z , Z.ByACfx a wellorder of Z . Without loss of generality we ( ) = ⋃ > %→ ⊲ 34 Initial segments of the Universe

1027 assume Z .If is of the form x x, y ,...,y (with FVbl y ) we defne a function P i D j  k j i k j 1028 Gi Z Z by the following clauses: ∅ ( ) Z ( ) = {⃗} 1029 h y the -least x Z so that x, y ,...,y if such exists i P j  k j 1030 ∶ %→ otherwise. (⃗) = ⊲ ( ) 1031 We also set hi to be the constant -function in the cases that i is not of the above form, or that = ∅ 1032 i has no free variables. With hi now defned in every case, we look for the least set y closed under ∅ 1033 the hi.Wecanfnd such a y by repeatedly closing under the fnitary functions hi, and obtain a y with 1034 cardinality no greater than max , X (see Exercise). We can then appeal to the criterion in Lemma 1035 2.38, which asserts in this case that is absolute for y, Z .But is absolute for Z , Z,andthusthe 1036 Lemma is proven. { ∣ ∣} Q.E.D. %→ %→

ni 1037 Exercise 2.45 Let x be any set, and fi V V for i be any collection of fnitary functions (meaning n 1038 that n ; show that there is a y x which is closed under each of the f (thus f “ iy y for each i)and i Ě i i Ď 1039 y max , x . [Hint: no need for a formal∶ %→ argument here:< build up a y in many stages y y at each k Ď k  1040 step applying< ) all the fi . ∣ ∣ ≤ { ∣ ∣} + 1041 Te last lemma then] says that, e.g. , if were a fnite list of axioms of ZFC, and x , then y, P 1042 would be a countable structure in which those axioms were true. 1043 Returning to our refection results, we may apply the above to obtain corollaries to Lemma= ∅ 2.43⟨ . ⟩

Corollary 2.44 Let Z be a term, and be any fnite list of formulae of . Ten ZFC x Z Trans x %→w x w are absolute for w, Z w max , x @ Ď D Ď L 1044 Proof: We directly⊢ apply[ the( Mostowski-Shepherdson) %→ [ ∧ %→ Collapsing Lemma∧ to∣ the∣ ≤ set y {appearing∣ ∣}] in the 1045 statement of Lemma 2.43, thereby collapsing it to the transitive w here. As w, y, we have y w P P 1046 v v . Hence are absolute for w, Z. Obviously y w . Q.E.D. 1047 In the special case that Z V and x in the above we may get: ⟨ ⟩ ≅ ⟨ ⟩ (⃗) ↔ ( (⃗)) %→ ∣ ∣ = ∣ ∣ Corollary 2.45 Let T be any= set of axioms= in extending ZFC,and a fnite list from T, then T y Trans y y y%→ . D L 1048 Tus we can fnd for any fnite⊢ set of[ ZFC( axioms) ∧ ∣ a∣ = countable∧ ⋀⋀ (%→ transitive) ] set model in which all those 1049 axioms come out true. Again the fniteness of is necessary. %→ 1050 2.6 Inaccessible Cardinals

1051 We shall encounter in this section an example of a ‘large cardinal’: this is a cardinal whose existence does 1052 not follow from the axioms of ZFC. In general this is because such cardinals allow one to conclude that 1053 there are structures (typically V where is the cardinal number under consideration) in which all the 1054 ZFC axioms are true. If ZFC could prove the existence of such a then this would contradict the Godel¨ 1055 Second Incompleteness Teorem. From these further large cardinals can be defned, and although we 1056 give the briefest of illustrative examples, it is not the intention of the course to go down this route, rich 1057 as it is. Inaccessible Cardinals 35

1058 2.6.1 Inaccessible cardinals

1059 Definition 2.46 Acardinal is a strong limit cardinal,ifforany  . ∣ ∣ > < %→ < 1060 Definition 2.47 Aregularcardinal is 1061 (i) weakly inaccessible if it is a limit cardinal (Hausdorf 1908); 1062 (ii) (Sierpinski-Tarski (1930); Zermelo> (1930)) strongly inaccessible if in addition it is a strong limit 1063 cardinal.

1064 Te idea behind the nomenclature is that an accessible cardinal is one that can be reached from 1065 below by either the successor cardinal operation, or else the power set operation, as per Note (1)that 1066 follows. 1067 Notes (1) Another way of putting this is to say that a cardinal is weakly inaccessible if it is (a) regular 1068 and (b) . It is (strongly) inaccessible if it is both (a) regular and (c) 1069 . + 1070 (2) T

1075 Lℵemma 2.48 (AC) Let Reg. Te following are equivalent: P 1076 (i) is strongly inaccessible; 1077 (ii) V H ; < H 1078 (iii) ZFC ; 1079 (iv) = . ( ) 1080 Proof: i ii . Since Card,wehaveH V (Lemma 2.31(ii)). But x V x V . = ℶ P Ď P D P 1081 By induction on one shows that V : suppose true for : then V V if , V 1082 and as V( ) ⇒ (, then) V  as is strongly inaccessible; if Lim then⇒ V

1091 Exercise 2.46 Verify that is weakly inaccessible if is regular and .

= ℵ 36 Initial segments of the Universe

1092 Exercise 2.47 Does V H imply that is strongly inaccessible?

> ∧ = 1093 Definition 2.49 (Mahlo 1911)Aregularlimitcardinal is called a weakly Mahlo cardinal in case Reg 1094 is stationary below . is called (strongly) Mahlo if it is both weakly Mahlo and strongly inaccessible. ∩

1095 Lemma 2.50 If is weakly Mahlo then in fact is the ’th weakly inaccessible cardinal, and the class 1096 of weakly inaccessible cardinals below is stationary below . Te same sentence is true with ‘strongly’ 1097 replacing ‘weakly’ throughout.

1098 Proof: As Reg is unbounded in , Reg ˚ is c.u.b. below . But such are all limit cardinals. As 1099 Reg is moreover stationary below , D df Reg Reg ˚ is stationary below (see Ex.2.13). 1100 But all members∩ of D are then weakly inaccessible( ∩ ) cardinals. Q.E.D. ∩ = ( ∩ )∩ ( ∩ ) 1101 Exercise 2.48 Let be the least weakly inaccessible cardinal which is itself a limit of weakly inaccessible cardinals 1102 (meaning the weakly inaccessibles below are unbounded in ). Show that is not weakly Mahlo. Tesame 1103 sentence is true with ‘strongly’ replacing ‘weakly’ throughout.

1104 2.6.2 A menagerie of other large cardinals

1105 We briefy consider some other notions of “large cardinal” stronger than Mahlo. (For a full account see 1106 Drake [2], Devlin [1], Jech [3].) We do this to give some favour to the rich structure of even the so-called 1107 small large cardinals. Tey are called ‘small’ because, if they are consistent, then they are consistent with 1108 the statement that “V L” - they can thus potentially be exemplifed in L. Several depend upon the 1109 notion of a homogeneous set for a certain kind of function. = n 1110 Definition 2.51 (i) denotes the set of all n element subsets of . 1111 (ii) denotes the set of all fnite subsets of < [ ] n n 1112 Definition[ ] 2.52 H is homogeneous for f f“ H . Ď df 1113 A homogeneous set is one therefore that every∶ [ n]-tuple%→ there⇐⇒ from∣ gets[ ] sent∣ = by f to the same ordinal n 1114 .Ofen in applications  ,  so we can think of f as partition of into two colours. If 1115 H is homogeneous, then this means that all n-tuples from H are assigned the same colour. For colours 1116 the< same applies. If a longer order= type= { is} specifed on H then the harder it is to[fnd] such homogeneous 1117 sets. Large cardinals can then be specifed by putting requirements on H and so forth as in the next two 1118 defnitions.

 1119 Definition 2.53 Acardinal is weakly compact if for every f  there is a homogeneous subset 1120 H with H unbounded in . Ď ∶ [ ] %→

 1121 Definition 2.54 (Jensen) A cardinal is inefable if for every f  there is a homogeneous 1122 subset H with H stationary in . Ď ∶ [ ] %→ 1123 By themselves the bare defnitions may not mean too much. We give some equivalent formulations. Inaccessible Cardinals 37

1124 Definition 2.55 (i) A tree T, is a wellfounded partial ordering so that for any s T, s T s T P  P  T 1125 s is linearly ordered. 1126 (ii) A branch through a⟨ tree< T⟩ is a maximal linearly ordered set; { ∣ < 1127 } (iii) T s T rank s is the set of elements of the tree of tree-rank or ‘level’ . df P T 1128 A tree thus= looks{ how∣ it sounds.( ) = }

1129 Definition 2.56 Let us say that a cardinal has the tree property if for every tree T , T with 1130 T has a branch of order type . @ = ⟨ < ⟩ 1131

1135 Lemma 2.57 For a cardinal the following are equivalent: 1136 (i) is strongly inaccessible and satisfes the tree property; 1137 (ii) is weakly compact; 1138 (iii) for every A there is a transitive M, and a B, j with j V , , A M, , B an elementary Ď P P 1139 embedding with j id and j . ∶ ⟨ ⟩ %→ ⟨ ⟩ 1140 Tere are many↾ further= ↾ characterisations( ) > of weakly compact. See Jech, Drake. One property of 1141 weakly compact cardinals is that every stationary subset of refects this property below , as in the 1142 following Exercise.

1143 Exercise 2.49 Let be weakly compact. Show that for any stationary subset S , there is so that ˚ Ď 1144 S is stationary in . [Hint: Use (iii) of the last lemma: suppose the conclusion fails; then there is C with Ď 1145 C S (for) every cardinal . Let A , C S  . Let j, M, B be as in< (iii) above. P ˆ 1146 Let∩ C , B . By elementarity of the embedding j the following holds in M “C is c.u.b.in , whilst P 1147 C ∩ S ∩ = ∅”. B u t S M S - so< this is a contradiction.]= {⟨ ⟩∣ } ∪ { } = { ∣⟨ ⟩ } ∶ ∩ ∩ = ∅ ( ∩ ) = 1148 Definition 2.58 (Jensen) A cardinal is subtle if 1149 For any sequence A with all A and any c.u.b. C , there is a pair of , Cwith Ď Ď P 1150 A A ). ⟨ ∣ < ⟩ 1151 Lemma< ∧ 2.59∩(Jensen)= For a cardinal the following are equivalent: 1152 (i) is inefable ; (ii) for any sequence A with all A there is a set E ,sothat Ď Ď ⟨ ∣ < ⟩ A E is stationary.

{ < ∣ = ∩ } 1153 Definition 2.60 Acardinal satisfes the partition relation  df for any f  1154 there is an H , ot H ,whichishomogeneous for f < ,namelyforalln< — n Ď 1155 f“ H . %→ (< ) ⇐⇒ ∶ [ ] %→ ( ) ≥ ∶ [ ] %→ < [ ] ∣ = 38 Initial segments of the Universe

1156 Te extra strength here is that f must assign the same colour to each n-tuple from H (although for 1157 a diferent m n a diferent colour may be chosen for all m-tuples from H). Such cardinals become 1158 rapidly stronger than those considered above, and quickly enter the realm of ‘medium large cardinals’. 1159 Tis happens≠ as soon as crosses the threshold from countable to uncountable. Te cardinals here 1160 defned are in increasing strength, when measured in terms of where they are frst exemplifed in On: if 1161 is the least satisfying  then is the ’th inefable cardinal. Similar if is the frst inefable, 1162 it is the ’th subtle cardinal, and also< the ’th weakly compact cardinal. If is the frst weakly compact 1163 cardinal, then it is the ’th%→ Mahlo( ) cardinal. All the above are consistent with ‘V L’; not however the 1164 existence of a cardinal satisfying   : if such a cardinal exists we may prove that V L. < = %→ ( ) ≠ 1165 Chapter 3

1166 Formalising semantics within ZF

1167 Te study of frst order structures and the languages appropriate to them is the branch of mathematics 1168 called model theory. Like other parts of mathematics it can be formalised within set theory, and developed 1169 from the ZF axioms. Whereas most mathematicians would not be seeing any great advantage in having 1170 their area of mathematics in doing this, as set theorists we shall see that formalising that part of model 1171 theory that handles structures of the form X, , (or of X, , A ,...,A where A X), will be of P P  n i Ď 1172 immense utility. Amongst other results it is at the heart of Godel’s¨ construction of the constructible 1173 hierarchy, L. ⟨ ⟩ ⟨ ⟩ 1174 We have defned the notion of absoluteness of formulae between structures or terms rather generally. 1175 However we have not been very specifc about what kinds of concepts are actually absolute. We alluded 1176 to this problem at the end of Section 1.2, and in particular we noted the possible non-absoluteness of the 1177 power set operation. In general objects that have very simple defnitions tend to be absolute for transitive 1178 sets and classes (thus , x, y , ,“f is a function”, “x an ordinal”) whilst more complex ones are not 1179 y x ,“x is a cardinal”). ∅ { } ( = P( ) 1180 3.1 Definite terms and formulae

1181 Te defnite terms and formulae are amongst those that we are interested in being absolute between 1182 transitive ZF models. We address the question of which terms and formulae defning concepts can be 1183 so absolute. We− shall defne “defnite term (and formula)” frst and later show that such have this degree 1184 of being “absolutely defnite”.

1185 Definition 3.1 (Defnite terms and formulae) 1186 (A) We defne the defnite terms and formulae by a simultaneous induction on the complexity of formulae 1187 and of the terms’ defnition. 1188 (i) Any atomic formula x y, x yisdefnite ; P 1189 if , are defnite, then so are: ; ; y x = D P 1190 (ii) Any variable x is a defnite term. If s, taredefnite terms, so are: ¬ ( ∨ ) 1191 s, s, t , s t.

⋃ { } / 39 40 Formalising semantics within ZF

1192 (iii) Suppose t x,...,xn and t,...,tn are defnite terms. Ten t t x,...,tn xn is a defnite 1193 term. If x,...,xn is a defnite formula then so is t x,...,tn xn . ( ) ( / / ) (iv) If x, x,...,xn and t,...,tn are defnite, then so are the terms: ( ) ( / / ) ( y) x x, t x ,...,t x and t y, x y z .   n n  P

1194 (v) . ∩ { ∣ ( / / )} { ( ) ∣ } 1195 (vi) If t is defnite, and Fun t , then the canonical function term f given by the recursion 1196 f y, x t y, x, f z, x z y is defnite. ( ) P 1197 Note (1):( By⃗ (i)) = any( ∆⃗formula{ ( ⃗) of∣ is de})fnite. (iv) gives a form of “defnite separation” axiom, in the 1198 frst part, and a kind of “defnite replacement” in the second part. Note also that if s is a defnite term 1199 then in particular “x s”,“ y s L” are defnite formulae. P D P

1200 Lemma 3.2 ZF If t is a defnite term then: x t x V . @ P − 1201 Proof: Formally( ) this would be a proof by induction⃗( (⃗) on the) complexity of t; informally notice that the 1202 way we have defned defnite terms uses methods, such as at (ii) where the ZF axioms yield these classes 1203 directly as sets, or in the case of (iii) and (iv) an appeal to Ax.Subsets would− yield them as sets. In vi 1204 we appeal to the principle of recursion (which does not use Ax.Power) to ensure that f as defned there n 1205 is a function of V to V (for some n). Q.E.D.( ) 1206 1207 We shall be interested in terms and formulae that are absolute between any two transitive ZF models 1208 M, N. Such we shall call absolutely defnite, a.d. for short. We shall be particularly interested− in when 1209 they are so absolute between such an M and V. We shall readily be able to identify a whole host of terms 1210 and defning formulae as defnite. We shall also be showing that any defnite term or formula is a.d., and 1211 thus in one fell swoop be able to conclude they are absolute for such classes. As might be expected the 1212 proof proceeds by induction on the complexity of the term or formula.

1213 Theorem 3.3 Let t x be a defnite term, and x adefnite formula. Ten (a) t and (b) are a.d., that 1214 is they are absolute between any two transitive ZF models M, N. (⃗) (⃗−) Proof: We shall frst prove (a) and (b) by a simultaneous induction on the complexity of defnite terms and formulae. We do this by referring to the construction clauses (i)-(vi) Def. 3.1 in turn. It sufces to prove this absoluteness between V and any transitive class term model of ZF W (note V is also a transitive ZF term). So let W be a transitive class term with ZF W . Te atomic− formulae of (i) are trivially so absolute,− and the inductive steps in the more complex− formulae are trivial except for the bounded existential quantifer; assume y W and is absolute( for W) : P x y W x x y W x W x y W x x y W x y D P D P D P P D P D P

1215 where(( we use) the) transitivity↔ ( ( of W∧ and)) hence↔ that y( W, in∧ the) ↔direction( of∧ the third) ↔ equivalence.( ) Ď 1216 We remark that we have shown: ←

1217 Corollary 3.4 Let be a ∆ formula. Ten is a.d. Defnite terms and formulae 41

1218 For (ii) suppose s, t are defnite: W W W W 1219 s z y s z y z z W y s z y z y s z y s W D P P P D P P D P P 1220 since s W. s, t and s t are similar. ( )Ď = { ∣ ( )} = { ∣ ∧ ( )} = { ∣ ( )} = 1221 For⋃ (iii) suppose t,...,tn are defnite. Let z Fvbl t,...,tn . Let x,...,xn Fvbl ⋃t . Ten { } / Ě W Ě 1222 we make the inductive assumptions that for any z W ti z ti z ,andforanyx W that W ⃗ P { } { } (P ) 1223 t x t x . By Lemma 3.2, if ti z is defned for z W then we know that ti z W. ⃗ P ∶ (⃗) = (⃗) P ⃗ (⃗) = (⃗) (⃗) ⃗ (⃗) W W W W t t z x,...,tn z xn t t z x,...,tn z xn tW t z x ,...,t z x ( ( (⃗)/ (⃗)/ )) =  (  (⃗)/ n (⃗)/n ) t t z x ,...,t z x =  ( (⃗)/ n (⃗)/ n ) 1224 Te frst equality is just the defnition of relativisation= ( to(⃗W)/ and the next(⃗)/ two) are the inductive hy- 1225 potheses outlined. Entirely similarly,

W W W W W t x,...,tn xn t x,...,tn xn t x,...,tn xn t x,...,tn xn

W 1226 where( ( the/ new inductive/ )) hypothesis↔ ( is/ now that / x) ↔ x( / for any x/ W) ↔, and( is/ used in the/ fnal) P 1227 equivalence. Te frst equivalence is Lemma 1.22. (⃗) ↔ (⃗) ⃗ 1228 For (iv): suppose x, x,...,xn and t,...,tn are defnite, then: W 1229 y x x x, t x,...,tn xn ( ) W 1230 y W x x, t x,...,tn xn ( ∩ { ∣ ( / / / )}) W 1231 y W x W x, t x,...,tn xn = ∩ ∩ ({ P∣ ( / / )}) 1232 y W x W x, t x,...,tn xn (by (iii)) = ∩ ∩ { P ∣ ( ( / / )) } 1233 y x x, t x,...,tn xn since y W as Trans(W). = ∩ ∩ { ∣ ( / / )}Ď 1234 Assume z W and t is defnite. We make the inductive assumption that we have shown that = W∩ { ∣ P( / / )} 1235 t u, v t u, v W for any u, v W. Ten WP W P W 1236 t y, x y z t y, x y z t y, x y z  P  P  P 1237 using( ) that=z ( W )in the frst equality. { ( )∣Ď } = { ( ) ∣( ) } = { ( )∣ } 1238 For (v) we consider . We note that the following are expressible in a ∆ way and hence are absolute 1239 for W 1240 (a) x z x z z @ P 1241 (b)∶Trans x y x z y z x ; @ P @ P P 1242 (c) x = On∅ ↔ Trans( x≠ ) y, z x y z z y z y ; P @ P P P 1243 (d) Lim x( ) ↔x On x ( y ) x z x y z ; P @ P D P P 1244 (e) x ↔x ( On ( )Lim∧ x y( x Lim∨ y .∨ = )) P P @ P 1245 (f) x ( ) ↔x On Lim∧ ≠x∅∧ y x Lim (y ) ↔ P ∧¬ ( ) ∧@ P ¬ ( ) 1246 By (e) we have seen that x is given by a ∆ formula and hence is absolute for W.Nownotethat = ↔ ∧ (P ) ∧ ¬ ( ) 1247 W: suppose n is least for which n W. Ten  W so n m  df m m . However ĎW P R W P 1248 if m m then by Ax.Pair and Union (m m W where W W P 1249 (m m x W x m x m x W= ∅x m x =m + =x x ∪ m{ }x m P P P P P 1250 m m= . ∪ { }) ∪ { }) = { ∣( ∨ = ) } = { ∣( ∨ = )} = { ∣( ∨ = )} = ∪ { } 42 Formalising semantics within ZF

W W 1251 Hence W. But then x W x x W x (by (e) Ď P P P P 1252 = x x (since W . P Ď 1253 = .= { ∣( ) } = { ∣( )} 1254 Finally for (vi): we assume t{is∣( defnite,)} and Fun t , and) f is the canonical function term given by: 1255 f y, x t y, x, f z, x z y . P W 1256 We thus have the inductive hypothesis that t y,(x,)u t y, x, u for any y, x, u W. Let y, x ⃗ ⃗ ⃗ P P 1257 W. We( prove) = the( result{ by( -induction,) ∣ }) hence we also assume we have proven for any z y that W P ⃗ W ⃗ ⃗ P ⃗ 1258 f z, x f z, x W. Ten by (iv) we have: (f z, x ) z = y( )f z, x z y W. Ten: W P W P P P 1259 f y, x t y, x, f z, x z y ⃗ ⃗ P W ⃗ ⃗ 1260 ( ) = (= t )y, x, f z, x z y { ( ) ∣ } = { ( ) ∣ } ⃗ ⃗ ⃗ P 1261 ( ) = (t (y, x, {f (z, x ) ∣z y }) (by the above comment) ⃗ ⃗ P 1262 f (y, x { as( required.) ∣ } ) Q.E.D.(Tm.3.3) 1263 ( ⃗ { ( ⃗) ∣ }) ⃗ 1264 We now have= ( a very) powerful method for showing that all sorts of concepts and defnitions are ab- 1265 solute for transitive structures in which ZF holds. For example all the ordinal arithmetic operations are 1266 defned by recursive clauses from defnite terms.− We can formally justify this as follows.

Lemma 3.5 Suppose we defne:

f y, x t y, x if  y, x

( ⃗) = ( ⃗) ( ⃗) tn y, x if n y, x = ⋮ otherwise. = ( ⃗) ( ⃗) 1267 for some defnite t,...,tn, and mutually exclusive= (meaning∅ at most one of  y, x ,..., n y, x holds) 1268 but defnite ,..., n, then f y, x is defnite. ( ⃗) ( ⃗) 1269 Proof: Note that u un( ⃗)u,...,un so this is defnite. 1270 Ten: f y, x t y, x  y, x tn y, x n y, x . Q.E.D. ∪⋯∪ = ⋃{ } 1271 Corollary 3.(6 All⃗) the= { arithmetical( ⃗)∣ ( functions⃗)} ∪⋯∪ A { ( ⃗)∣ (;M⃗)} . ;E are defnite 1272 and hence a.d. ( ) = + ( ) = ( ) = Proof: For example:

A x if x ; A x A y  if x On Succ x ; P A (x) = sup A y y x if x = On∅ Lim x . ( ) = ( ) + P P ∧ ( ) 1273 Te frst and third conditions( ) on= the right{ we( have)∣ already} seen are∧ defnite( ) at (a), (c), (d) above. But 1274 Succ x y x y y y x x y y . We note that y y is defnite, and so by the D D P 1275 Teorem 3.3 Succ x is defnite. Te three conditions are mutually exclusive we can appeal to the last 1276 lemma( ) once↔ we( note= that∪ { the}) three↔ terms( ,=y ∪ {y ,and}) z where z is∪ a de{ f}nite set by 3.1 (iv) in place ( ) 1277 of t, t,and t are defnite. Te other functions are exactly the same. Q.E.D. 1278 ∅ ∪ { } ⋃ Defnite terms and formulae 43

1279 Note (1): x is not defnite: if it were we could conclude from Teorem 3.3 that for any transitive W 1280 set satisfying ZF that x W which is not true in general. P 1281 “x is countable”P( −) cannot be expressed by a defnite formula x : again if it were, we should have that W 1282 the concept is( absolute) for transitiveP( ) W satisfying ZF . We list some defnite concepts. − ( ) n ( ) 1283 Lemma 3.7 For any n: (i) x, (ii) x,...xn , (iii) x, y ; u , u  where u u , u  ; (iv)  1284 x ,...,x , (v) x y,(vi) ran(z), (vii) dom(z), (viii) z“x, (ix) z x, (x) z are all defnite terms.  n ˆ 1285 Te following relations are de⋃fnable by{ defnite formulae:} ⟨ ⟩ ( ) ( )− = ⟨( ) ( ) ⟩ 1286 ⟨ (xi) x ⟩ y; (xii) Trans y ; (xiii) Rel z ; Fun z ; (xiv) z x ↾ y; (xv) “z is a (1-1) function”; z is an onto Ď 1287 function; (xvi)“x is unbounded in ”; “z is a cofnal function”; “x is a closed and unbounded Ď 1288 set”; ( ) ( ) ( ) ( ) = 1289 (xvii) the terms TC x , (xviii) x ∶are%→ defnite terms. 1290 Tus all the above are a.d. ( ) ( ) 1291 Proof: Te frst two are simply repeated applications of operations defned to be defnite. Similarly 1292 (iii) x, y x , x, y ; (iv) x,...,xn was defned by repeated application of , and hence is 1293 defnite; (v) x y x z z y w, z w x z y . ˆ ˆ P  P P 1294 (vi):⟨ ran⟩ = {{z } {u }}z w ⟨z v z⟩ w v, u ; ⟨− −⟩ = P{ {D }∣P D }P= {{ ⟨ ⟩∣ }∣ } 1295 (vii): dom r ⋃u z w z v ⋃ z w u, v ; ( ) = { P ∣ D P D P ( = ⟨ ⟩} 1296 (viii): z“x v ⋃z u x w z⋃w u, v ; ( ) =P{ D ∣ P D P ( = ⟨ ⟩} 1297 (ix), (x) Exercise. ⋃ ⋃ = { ∣ ( = ⟨ ⟩)} 1298 (xi): x y z ⋃ x z y ,itisthus∆ and so defnite; Ď @ P P  1299 (xii) Trans y z y z y ↔ @ P( Ď) 1300 (xiii) Rel z z dom z ran z ; ( ) ↔ Ď ( ˆ ) 1301 Fun z Rel z x dom z u, v ran z v u x, u z x, v z ; ( ) ↔ @ P( ) @( ) P P R 1302 (xv) z x y Fun z x, y z; ( ) ↔ ( ) ∧ ( )P ( )( ≠ %→ (⟨ ⟩ ↔ ⟨ ⟩ )) 1303 (xvi) Exercise. ( ) = ↔ ( ) ∧ ⟨ ⟩ 1304 (xvii) TC x t x, TC y y x for a defnite t using the defnite recursion scheme. P 1305 (xviii): s z z z is defnite; then s v v u is defnite as then is t u s v v u (by P P 1306 (iv) and (ii) resp.( ) = in( Def.{3.1.( Using)∣ the}) defnite recursion scheme (vi) we get x t y y x . P 1307 (Here we are( just) = expressing∪ { } that x sup{ ( ) y∣  }y x .) ( ) = ⋃{ ( ) ∣ Q.E.D.} P ( ) = ({ ( ) ∣ })

1308 Exercise 3.1 (i) Show that “z is a total( order) = of y{” can( ) be+ expressed∣ } in a ∆ fashion. 1309 (ii) Complete (ix),(x), (xvi), (xvii) of Lemma 3.7.

n n 1310 Lemma 3.8 Tefollowingaredefnite: x (for any n); x x n ;“xis fnite”. Hence df P 1311 z df x z xisfnite is defnite. < Ď = ⋃{ ∣ } < n 1312 Proof( ): By= induction{ ∣ on n we} defne F n, x x:  1313 F , x x . n  ∶ ( ) = 1314 F n , x x f n, y f F n, x y x ; ( ) = = ∅ P P 1315 F , x x + F n, x n . ( + ) = = { ∪ {⟨ P⟩}∣ ( ) ∧ } 1316 Tis is given by< defnite recursion clauses, and so F n, x is defnite for n . ( ) = = ⋃{ ( )∣ } ( ) ≤ 44 Formalising semantics within ZF

1317 “x is fnite” f x f is onto). And then: D P 1318 x z x is fnite x< f z x ran f Q.E.D. Ď D P 1319 ↔ ( < W 1320 { Note:∣ the absoluteness} = { ∣ of fniteness( = implies( ))} that if Trans W ZF then any fnite subset of W 1321 is in W. Tis need not be true of course for infnite subsets of W. − ( ) ∧ ( ) W W 1322 Exercise 3.2 Suppose Trans W ZF .Show V V W. [Hint: use that the rank function is defnite.] − 1323 Note: “cf ” along with( “xisacardinal)∧( ) ”or“( ”) are= not∩ defnite, and so not absolute for such W in 1324 general (but see the next exercise). Neither then is “x is a regular/singular cardinal.” However being a 1325 wellorder( is so) absolute as the next lemma shows.

1326 Exercise 3.3 Let be a limit ordinal; show that the following are absolute for V : (i) x (ii) “ is a cardinal” V 1327 (and hence Card Card ); (iii) cf (iv) “ is (strongly) inaccessible” (v) y V (vi) (vii) . P( ) ( ) = ∩ ( ) = ℵ ℶ 1328 Lemma 3.9 (i) “z is a wellorder of y” ; (ii) “z is a wellfounded relation on y” are absolutely defnite.

W 1329 Proof: Suppose Trans W ZF , z, y W. For (i) “z is a total order of y” can be expressed in a P W 1330 ∆ way (Exercise). Suppose (“z is a− wellorder of y” . Since we have Ax . Replacement holding in W we W W 1331 have that “ y, z is isomorphic( ) ∧ to( an ordinal) ” holds in W.If On and f y, z , then P 1332 dom f y, ran f ,“f is a bijection”, etc., are) all absolute for W. Hence f y, z , holds in 1333 V. Consequently⟨ ⟩ y, z is truly a wellorder. ( ) ( ∶ ⟨ ⟩ ≅ ⟨ <⟩) 1334 Conversely( ) = if( “z) is= a wellorder of y”withz, y W, then as for any w V with∶ ⟨ w ⟩ ≅y⟨we< have⟩ w has ⟨ ⟩ P P Ď 1335 a z-minimal element w say, then w W (as Trans W and no u W satisfes uzw. So if also w W P W P P 1336 then “w is an z-minimal element of w” . ( )) 1337 (ii) is only an amplifcation of (i), efected by defning an absolute rank function z of the wellfounded 1338 relation( z. We leave this to the reader. ) Q.E.D. 1339

1340 Te example of wellorder shows that being expressible by a ∆ formula is not a necessary condition W 1341 for absoluteness: wellorder in general is a Π-concept when literally written out. However if ZF 1342 holds then we have Ax.Replacement available to turn this Π concept into an existential statement− and ZF 1343 hence have that it is U-absolute for W.Wemaysaythatitisthus“∆ ”. I f W is not a model of su( fcient)  − 1344 Replacement then this argument can fail.

1345 3.1.1 The non-finite axiomatisability of ZF

1346 We use the Refection Teorem together with our absoluteness results to prove the non-fnite axioma- 1347 tisability of ZF. (We say a set of axioms T axiomatises S if T for every from S. A set S is fnitely 1348 axiomatisable if there is a fnite set T that axiomatises S.) ⊢ 1349 Theorem 3.10 (Te non-fnite axiomatisability of ZF) Let T be any set of axioms in , extending ZF, 1350 and T be any fnitesubsetof T ;iffrom T we can prove every axiom of T then T is inconsistent. 1351 In particular, with T as ZF, no fnite subset of ZF axioms will axiomatise all of ZF, unlessL ZF is incon- 1352 sistent. Formalising syntax 45

1353 Proof: Suppose T T were such sets of axioms, with all of T provable from T, for a contradiction. Ě V 1354 We have the assertion: ZF T T . Ten as T proves every axiom of ZF,it @ D    1355 proves the following instance of the Refection Teorem: ⊢ > ((V⋀⋀ ) ↔ ⋀⋀ ) 1356 T T T . @ D V 1357 However trivially T T ⊢ @ D > ((⋀⋀ ) ↔ ⋀⋀ ) 1358 since T T. Ten, by the principle of ordinal induction: ⊢ > ((V⋀⋀ ) ) V 1359 T  T   T ]. ⊢ ⋀⋀ D @ ˚ 1360 We are assuming that T proves all of ZF, so by the Soundness of frst order predicate logic, Teorem ⊢ [( ⋀⋀ ) ∧ V < (¬(⋀⋀ V ) ( ) V 1361 1.20, in the form that if T and T , then , we may deduce, T ZF  . 1362 Ten all our absoluteness results about transitive models hold for V for such a  as in .Also ⊢ ( ) ( )  ⊢ ( ) ˚ 1363 in particular : ⋀⋀ V 1364  ( ) T  V V V  V (Exercise 3.2) V 1365 Again using Soundness, since T T ,wehave ⊢ < → ( V ) VD = ∩ = 1366 T T  . D ⊢ (⋀⋀ ) 1367 However, then we have ⊢ (V (⋀⋀ ) ) 1368 T T which contradicts .SoT and hence T is inconsistent. Q.E.D.  D   ˚  ⊢ < (⋀⋀ ) ( ) 1369 3.2 Formalising syntax

1370 We shall consider the language ˙,˙ that we have been using to date, that can be interpreted in P 1371 -structures, that is any structure X, E with a domain a class of sets X and an interpretation E for the P = 1372 ˙ symbol. In what follows, we shallL = almostL always be considering the standard interpretation of the ˙ P P 1373 symbol, where it is interpreted as⟨ the true⟩ set membership relation. Te equality symbol ˙ will without 1374 exception be interpreted as true equality . Up to now the object language of our ZF theory has been 1375 foating free from our universe of sets, but we shall see how this language (indeed any reasonably= given 1376 language) can be represented by using sets,= just as we can represent the natural numbers , , ,...by the 1377 sets , , , ,... We make therefore a choice of coding of the language by sets in V. Te 1378 method of coding itself is not terribly important, there are many ways of doing this, but the essential 1379 feature∅ { is∅ that} {∅ we{ want∅}} a mapping of the language into a class of sets, where the latterL is ZF (in fact ZF 1380 or even much more simply) defnable). As we are mainly interested in the frst order language we give− 1381 the defnitions in detail just for that. In principle we could do this for any language, for any structures. L

1382 Definition 3.11 (Godel¨ code sets) We defne by (a meta-theoretic) recursion on the structure of formulae 1383 of the code set x y V . i  jP  i  j  1384 (i) xvi ˙v jy is   ; xvi ˙v jy is   ; L P 1385 (ii) x y is +x y, x+ y, x y ; + + = ⋅ ⋅ 1386 (iii) x y is , x y ; ∨ { i  { }} 1387 (iv) x v y is  , x y . D i ¬ { + } 1388 Note (a) atomic{ formulae} are the only ones coded by integers, (b) in each case, that if is non-atomic 1389 then the code set contains immediate subformula(e) codes as direct members; (c) Note the device in (iii) 46 Formalising semantics within ZF

1390 for ensuring which is the frst disjunct: we have w x y w  u, v, s w u x y v D P 1391 x y s u . Tis is thus ∆. 1392 Clearly given a code set u we may decode from it= in a unique∨ ↔ fashion,∣ ∣ = making∧ use of( the= primes∧ and= 1393 the prime∧ = { power}) coding. We give the formal counterpart of the above defnition using fnite functions 1394 from V , a defnite formula defning the characteristic function of the class of code sets of formulae of 1395 :

1396 DLefinition 3.12 Fml u, f , n  f V dom f n  f n u i P j  i  j  1397 k dom f i, j f k  < f k   @ P D P i  1398 m, l k f k ( f m ,)f= l ↔, f +m + ∧f k ( ),+=f m++ ∧ (i ) = f∧k  , f m ; D D P 1399 Fml∧ u, f , n( )[ otherwise( (. ) = ⋅ ∨ ( ) = ⋅ ∨ + < [ ( ) = { ( ) ( ) { ( )}} ∨ ( ) = { ( )} ∨ ( ( ) = { ( )}]] ) 1400 We thus( may) = think of the formula as represented by, or coded by, f n , where f is the function 1401 that describes its construction according to the last defnition, with dom f n . ( ) 1402 Definition 3.13 Fmla u  n f V Fml u, f , n ; Fmla( )u= + otherwise. D P D P < 1403 It should be noted( that) = both↔ the last two defnitions( are built) = up using( ) de=fnite terms, and so are 1404 defned by defnite formulae and thus a.d.

1405 3.3 Formalising the satisfaction relation

1406 We now formalise the (frst order) satisfaction relation due to Tarski, familiar from model theory.

1407 Definition 3.14 (i) Q h Fun h dom h ran h x n y x m nh m y . x df Ď D P D P @ 1408 (ii) If h Q ,andy x, then h y i is the function defned by: P x P 1409 j j i =h {y i∣ j ( )h∧j ( ) =h y∧ i (i)=y.( ) ∧ ( ≥ ( ) = )} @ P ( / ) 1410 Again Qx(is≠ def%→nite: we( may/ )( write) = ( )) ∧ ( / ) 1411 h Q h x y x h h n, y n dom h n . P x D  P D P  P  1412 Tus although Q does< not contain fnite functions, any h Q is essentially a fnite function with x P x 1413 a constant tail -↔ and this makes it de( fnite.= ∪ (Again:{⟨ ⟩∣ x, like∧ x (,is)not≤ de})fnite.) (ii) also specifes a 1414 defnite relation between i, x, y, and h. 1415 We next specify what it means for a fnite function h toP be( an) assignment of variables potentially 1416 occurring in a formula u to objects in x that makes u come out true in the structure x, . P

1417 Definition 3.15 (i) We defne by recursion the term Sat u, x ; ⟨ ⟩

Sat xv ˙v y, x h Q h i ( h )j ; i j P x Sat xv ˙v y, x h Q h i h j ; iP j P x P Sat (x = y, x) = Sat{ x y,∣x ( )Sat= (x )}y, x ; ( ) = { ∣ ( ) ( )} Sat x y, x Qx Sat x y, x ; Sat( x ∨v y, x) = h ( Q )y∪ x (h y i )}Sat x y, x ; D i P x D P P (Sat¬ u, x) = if/ Fmla( u )}. ( ) = { ∣ ( ( / ) ( ))]} ( ) = ∅ ( ) = Formalising defnability: the function Def. 47

1418 (ii) We write x, u h if h Sat u, x . P P 1419 Note: By design then we have x, x y h if it is not the case that x, x y h etc. (We write ⟨ ⟩ ⊧ [ ] P( ) P 1420 the latter as x, x y h .) If, uninterestingly, x then also Sat u, x . P ⟨ ⟩ ⊧ ¬ [ ] ⟨ ⟩ ⊧ [ ] 1421 Lemma 3.16⟨Sat ⟩/u⊧, x is[ de]fned by a defnite recursion.= ∅ Hence “ x, ( x y) =h ∅”isdefnite. P 1422 Proof: Tis should( be) pretty clear, but we give an explicit recursive⟨ term⟩ ⊧ t for[ ]Sat: 1423 Sat u, x h Q P x 1424 Fmla u  i  j  i  j  1425 i,(j ) =u{  ∣ h i h j u   h i h j D P P 1426 u( )= ∧v, w+ u +w v h Sat v, x +v u+ D P P P 1427  u[( h = Sat⋅ v,∧x v( ) u= ( )) ∨ ( = ⋅ ∧ ( ) ( ))]]∨ P iR P 1428 ∨[∣ i∣ = ∧ u [ y = x{ h} ∧y i ⋃{ Sat( v,)x∣ v }]]u ∨) . D P P D P P P 1429 T∨[e specif∧cation+ ⋃ here{ ( yields)∣ a def}]nite term Sat u, x t x, u, Sat v, x v u noting that we have ∨ ( ∧ ( ( / ) { ( ) ∣ } ]} P 1430 already established that all the concepts appearing⋃ here, such as “Qx”,“Fmla u ”, “ ”, etc. are defnite 1431 ( ) = ( { ( )∣ }) Q.E.D. 1432 By our work so far then then we may say that “the assignment h makes( the) formula true in the 1433 structure x, ” if x, x y h . Otherwise we say it is similarly “false”. P P 1434 If is a formula of with free variables amongst v j ,...,v jn and y,...,yn x then we abbreviate: ⟨ ⟩ ⟨ ⟩ ⊧ [ ] P 1435 x, x y y ,...,y x, x y h for any h Q with h j y all i n. P  n P P x i i 1436 Tis makes perfect sense,L since the intepretation of the formula in the structure only depends on the 1437 assignment⟨ ⟩ ⊧ to the[ free variables] ←→ of⟨ .If⟩ ⊧ has[ no] free variables at all, then( ) = it is deemed≤ a sentence and 1438 either Sat x y, x Q , in which we case we say the sentence is true in x, or else Sat x y, x x P 1439 in which case it is false. In each case we simply write x, x y or x, x y accordingly, as then P P 1440 assignment( functions) = h are superfuous. ⟨ ⟩ ( ) = ∅ ⟨ ⟩ ⊧ ⟨ ⟩/⊧

1441 3.4 Formalising definability: the function Def.

1442 Te following is the crucial function used to build up defnable sets.

1443 Definition 3.17 Def x w x x, u h w  ;Fmla u  h Q . df P P P x

1444 Lemma 3.18 “Def x (”isade) = f{{nite term.∣⟨ ⟩ ⊧ [ ( / )]} ( ) = ∧ } Proof: First note that we have shown that “ x, u h w  ” is defnite. Hence so is ( ) P x, u, h ⟨w ⟩x⊧ x[, ( /u)]h w  . df P P

1445 Hence x, u, h Fmla u ( h Q) = is{ defnite.∣⟨ ⟩ ⊧ [ ( / )]} Q.E.D. P x 1446 Teclass{ ( Def)∣x we( think) = of∧ as the “de} fnable power set of x”: it consists of those subsets y x so Ď that membership in y is given by a formula v, v,...,vm all of whose free variables are amongst those ( ) shown, together with a fxed assignment of some y,...,ym, and the members y y are determined ( ) P 48 Formalising semantics within ZF

by allowing v to range over all of x. Tose y that when added to the fxed assigment y,...,ym,cause y , y ,...,y to come out true in x, are then added to y. We may write slightly more informally:   m P [ Def] x z z w ⟨x, ⟩ w, y ,...,y , Fmla , y x P  m P < 1447 where it is implicitly( understood) = { ∣ = that{ ∣⟨ we should⟩ ⊧ [ have written]}x y for ( and) = it is⃗ lef unsaid} that the free 1448 variables of have all been assigned some value in x by the assignment displayed.

1449 Lemma 3.19 (i) x Def x ; (ii) Trans x x Def x ; P Ď 1450 (iii) z x z z Def x ; @ Ď P 1451 (iv) (AC) x ( Def) x x(. ) %→ ( ) (∣ ∣ < → ( )) 1452 Proof: (i) x w x, xv v y w and so x Def x . ∣ ∣ ≥ %→P ∣ ( )∣ = ∣ ∣ } P 1453 (iv) Assume x is infnite. Ten Qx has the same cardinality as x,namelyx .Also,F df u 1454 Fmla u = {is a∣⟨ countable⟩ ⊧ set.= Since[ ] Def x is the class( ) of subsets< of x given by a defnition in- ∣ ∣ = { ∣ 1455 volving a formula u Fmla together with a fnite parameter string y,...,yn we see that: Def x ( ) = } P ( ) 1456 F . Qx . x x . Tat x Def x follows from (iii). (ii) and (iii) are lef as an exercise. Q.E.D. ∣ ( )∣ ≤ 1457 E∣ xercise∣ ∣ ∣ =3.4 ∣Finish∣ = ∣ (ii)∣ and∣ (iii)∣ ≤ of∣ Lemma( )∣3.9.

1458 Exercise 3.5 Let x, be a transitive -model. Show that Trans Def x . If y, z x then is y, z Def x ?Is P P P P 1459 x ? [Hint (for the last question): If x , compute Def x and compare this with the given sets.] ⟨ ⟩ ( ( )) ⟨ ⟩ ( ) Exercise 3.6 Let us say that w is outright defnable in the set x, if for some formula with only free variable { } ( ) = ( ( ))P v then w is the unique element in x so that x, w . We may thus defne a variant on the Def function by: P ⟨ ⟩ Def x z z w x ⟨x, ⟩ ⊧ [w ] , Fmla , FVbl v , w x  P P  P 1460 of the sets outright de( f)nable= { ∣ in{ }x=, { , defnable∣⟨ ⟩ without⊧ [ ]} use of parameters.( ) = Show( ) that= { Def} x } for any x. P  ⟨ ⟩ ∣ ( )∣ ≤ 1461 Definition 3.20 We say that a set z is ordinal defnable˚ (“z OD˚”) if for some ,z Def V . P P  1462 (Tis defnition is just a placeholder for the ofcial - but equivalent - defnition of ordinal de(fnability) 1463 to come.)

1464 Exercise 3.7 (i) Show that: (a) On OD˚; (b) V OD˚; (c) x x OD˚ x OD˚ . (ii) Show Ď @ P @ P P ˚ 1465 that there is a (countable) set X so that for unboundedly many ordinals X Def V . [Hint: consider the P  1466 theory of each V : the set of all codes of sentences so that V , x(y. Tis is a→ subset{ } of V .] ) ( ) P ( ) ⟨ ⟩ ⊧ 1467 3.5 More on correctness and consistency

1468 Te next theorem illustrates that our defnitions are ‘correct’: we have formulated two ways of talking 1469 about a statement being ‘true in a structure’ W, frstly we considered relativised formulae and spoke W 1470 from an exterior perspective of ‘ holds or is true in W’ by asserting ‘ ’. Teformula from we 1471 consider to be in our language in which we wish to state our axioms about the structure consisting of our 1472 intuitive universe of sets. We have now a second interior method through the formalised version ofL the More on correctness and consistency 49

1473 language which consists of sets coding formulae as for x y above together with the satisfaction relation. 1474 Tis relation was between (codes of) formulae and structures or ‘models’. Te next theorem asserts that 1475 these two methods are in harmony.

1476 Theorem 3.21 (Correctness Teorem) Suppose is a formula of with free variables v v j ,...,v jm 1477 then: m Lx, ⃗ = 1478 ZF x y x x, x y y v y v P . @ @ P P − ⟨ ⟩ 1479 Tis would⊢ be a proof⃗ by[( induction⟨ ⟩ ⊧ on[⃗ the/⃗]) complexity←→ ( (⃗ of/⃗))(we] shall omit the details). It is again 1480 a theorem scheme, being one theorem for each . 1481 T● e ZF and ZFC axiom collections themselves have formal counterparts as sets: just as each formula 1482 is mapped to its code set x y as above, we can also fnd sets that collect together the code sets of those 1483 sentences that are axioms of ZF (or ZFC). Namely, there is an algorithm for listing the axioms of ZF 1484 as , ,..., n,...

1485 Definition 3.22 xZFy df u Fmla u  Ax  u Ax  u Ax  u . 1486 xZFCy is defned similarly by adding “ Ax  u ”. = { ∣ ( ) = ∧ ( ( ) ∨ ( ) ∨⋯∨ ( ))} 1487 In the above by ‘Axj u ’wemeanthat∨ u is( a) code set for an axiom of the type Axj. Tus Ax0 1488 is the Ax.Extensionality. If this latter axiom is written out using only , , etc. as then we have ( ) D 1489 Ax  u u x y. Te other axioms similarly must be written out in the formal language, and then 1490 coded according to our prescription. Some axioms are in fact axiom schemata:¬ ∨ infnite sets of axioms. 1491 So for( Ax) ←→ u =(for Ax.Replacement) we should demand that u conforms to the right shape of formula 1492 that is an instance of the axiom of replacement when written out in this correct manner. Ax  u will ( ) 1493 then be an infnite set, as will be xZFy . ( ) Lemma 3.23 (i) “u xZFy”, “u xZFCy”aredefnite. (ii) If is an axiom of ZF then P P ZF x y xZFy. P − 1494 Similarly if is an axiom of ZFC then ZF ⊢x y xZFCy . P − 1495 Tis would again be a proof by induction on⊢ the structure of . Te intuitive meaning that it captures 1496 is that “ZF xZFy”. Te point again is that the defnitions of xZFy and xZFCy are again defnite. Tese Ď 1497 details are uninteresting and somewhat tedious, but the idea that this can be done is very interesting. (ii) 1498 is again a theorem scheme, one for each axiom .

1499 Definition 3.24 “ x, xZFy” u xZFy x, u. (“ x, xZFCy” similarly.) P df @ P P P 1500 We have that, e.g. “ x, xZFy”and“ x, xZFCy” are defnite, and so a.d. Ten in this case we ⟨ ⟩ ⊧P ⇐⇒ P ⟨ ⟩ ⊧ ⟨ ⟩ ⊧ 1501 say that x, “is a model of ZF C ”. P ⟨ ⟩ ⊧ ⟨ ⟩ ⊧ Corollary⟨ 3⟩.25 (to the Correctness( ) Teorem) For any axiom of ZF (or ZFC then − x, ZF x, xZF y P ; ) P − − ⟨ ⟩ ⊢ (⟨ ⟩ ⊧ %→ ) 50 Formalising semantics within ZF

similarly x, ZF x, xZFCy P . P − ⟨ ⟩ 1502 Exercise 3.8 Suppose is strongly inaccessible.⊢ (⟨ Verify⟩ ⊧ that V%→, xZFC) y. P 1503 Exercise 3.9 E Let , be structures. We write ă if for every formula u, every h Q if u h ˚ ⟨ ⟩ ⊧ P 1504 then u h . Suppose that , are such that V , ă V , .Showthat is a strong limit cardinal and that P P A 1505 both V , , (V )(, are) modelsA B of ZFC. A B B ⊧ [ ] A ⊧ P[ ] P ⟨ ⟩ ⟨ ⟩ 1506 Exercise 3.10 E Suppose there is which is strongly inaccessible. Show that there is with V , a model ⟨ ⟩ ⟨ ˚ ⟩ P 1507 of ZFC,andwithcf . [Hint: Use the Refection Teorem proof on V , which we now have assumed to be 1508 a ZFC model, to( show)( ) that every formula of ZFC now “refects” down to a cub C set of⟨ ordinals.⟩ Now Ď 1509 intersect over all . T( )is= method shows that in fact there is a cub set of points with V , not only a model P 1510 of ZFC,butalso V , ă V , ] P P < ⟨ ⟩ 1511 Exercise 3.11 E . Let xSny denote the codes of Σn formulae of in the Levy hierarchy. Let n N.Showthat ˚⟨ ⟩ ⟨ ⟩ P 1512 there is a term cn On for a closed unbounded class of ordinals, so that ZF cn u xSny x V x Ď @ P @ P @ P 1513 ( )( ) L V , u x . Tis we should naturally, but informally, abbreviate as ‘ V , ăΣn V, ’. P ⊢P P ⃗ ( (⃗) ↔ 1514 Exercise 3.12 Suppose X, T for some set of sentences T including Ax.Ext. Show that there is a countable ⟨ ⟩ ⊧ [⃗]) P ⟨ ⟩ ⟨ ⟩ 1515 transitive x with x, T. [Hint: Te Downward-Lowenheim¨ Skolem Teorem says for any cardinal with P 1516 X there is a Y⟨ with⟩ ⊧Y, ă X, and Y . Ten use the Mostowski-Shepherdson Collapsing P P 1517 Lemma.] In particular⟨ ⟩ if⊧ there is an -structure which is a model of ZFC then there is a countable transitive one. ≤ ≤ ∣ ∣ ⟨ P⟩ ⟨ ⟩ ∣ ∣ =

1518 3.5.1 Incompleteness and Consistency Arguments

1519 In general when we say that a theory T is consistent we mean that for no sentence do we have T and 1520 T . We abbreviate this as “Con T ”. Of course if T is inconsistent then we may prove anything at all 1521 from T and we can then say (assuming that T is in a language in which we formulate arithmetic axioms)⊢ 1522 that⊢ “¬T  ” encapsulates the notion( ) that T is inconsistent. TeheartofGodel’s¨ argument is that it is 1523 possible to formulate the concept of a formal proof from an algorithmically or recursively given axiom ⊢ = T 1524 set T extending PA, in such a way that “v codes a proof from xTy of v”, a b b r e v i a t e d Pf v, v ,canbe T T 1525 represented in the theory T. Ten we may use “ v Pf v, x y ”, a b b r e v i a t e d a s “ Con ”, to capture D T 1526 the formal assertion that T is consistent. He then showed that T Con . In short we thus( formalise) the 1527 notions of “proof”, “contradiction”, “axiom” etc.¬ within the( theory= T), starting with the formalisation of 1528 syntax that we have already efected. We are not going here to go⊢/ down the route of investigating Godel’s¨ 1529 proof in its entirety, however we can rather easily obtain a weak version of Godel’sSecond¨ Incompleteness 1530 Teorem which sufces for our purposes. (Compare the proof of Teorem 3.10) More on correctness and consistency 51

1531 Theorem 3.26 (Godel)¨ Con ZF ZF x Trans x x, xZFy . D P 1532 Proof: Suppose abbreviates( the) ⇒ sentence⊢/ x( Trans( x) ∧ ⟨ x, ⟩ ⊧ xZFy)). Suppose that ZF . Ten: D P 1533 ZF z Trans z z, xZFy “ w Trans w w z w, xZFy ” D P @ ( ( ) ∧ ⟨ ⟩ ⊧ P ˚ z⊢, 1534 Let z satisfy the last formula. By the last Corollary for any axiom of ZF we have P . Tat is ⊢z, ( ( ) ∧ ⟨ ⟩ ⊧ ∧ ( z, ( ) ∧ ( ) < ( ) → ⟨ ⟩/⊧ ) ( ) z, 1535 ZF P .AsZF we shall have that P . In other words x Trans x x, ⟨ xZF⟩ y P . D P 1536 So let⟨ y ⟩ z satisfy this, namely ⟨ ⟩ ⟨ ⟩ ( ) P ⊢ z, ( ) ( ( ( ) ∧ ⟨ ⟩ ⊧ )) 1537 Trans y y, xZFy P . P z, 1538 But this is a defnite formula and⟨ so⟩ is absolute for the transitive structure z, as ZF P . Hence we ( ( ) ∧ ⟨ ⟩ ⊧ ) P 1539 really do have: − ⟨ ⟩ ⟨ ⟩ ( ) 1540 y z Trans y y, xZFy. P P 1541 But y z . Tis contradicts . Hence ZF is inconsistent. Q.E.D. ˚ 1542 ∧ ( ) ∧ ⟨ ⟩ ⊧ ( ) < ( ) ( ) T 1543 However, providing we have done our formalisation of xZFy and Pf v, v etc. sensibly, we shall 1544 have that in ZF we can prove the Godel¨ Completeness Teorem: that any consistent set of sentences in 1545 any frst order theory whatsoever has a model, and thus shall have: ( ) ZF 1546 ZF “Con X, E X X, E xZFy]” D ˚˚ 1547 But there is no indication that E should be the natural set membership relation on the countable set X, ⊢ %→ [∣ ∣ = ∧ ⟨ ⟩ ⊧ ( ) 1548 or that Trans X . X, E arise simply from the proof of the Completeness Teorem. In general E will not 1549 be wellfounded, and will be completely artifcial. ( ) 1550 Taking this line further: if there is a set which is a transitive model of ZF, let us assume x, V is P P 1551 such. We additionally assume such an x is chosen of least rank. Te assumed existence of x, implies ZF P 1552 Con , and as this latter assertion is expressed as a defnite sentence, and x, is a transitive⟨ZF ⟩model, ZF x, x, P ⟨ ⟩ 1553 we have Con P .By X, E X X, E xZFy P . Ten the model X, E − x can- ˚˚ D P 1554 not be a model with⟨ E⟩ wellfounded (it is an exercise to check this using⟨ ⟩ ⟨X ⟩ x - cf. Ex. 3.16 below). 1555 ( ) ( )( [∣ ∣ = ∧ ⟨ ⟩ ⊧ ) ⟨ ⟩ ( ) < ( ) 1556 What we shall attempt with Godel’s¨ construction of L is to show: 1557 (+) Con ZF Con ZF Φ 1558 where Φ will be various statements, such as AC or the GCH. ( ) ⇒ ( + ) 1559 A statement such as the above (+) should be considered as a statement about the two axioms sets 1560 displayed: if the former derives no contradiction neither will the latter. Te import is that if we regard 1561 ZF as “safe”, as a theory, then so will be ZF Φ. (One usually claims that these arguments about the 1562 relative consistency of recursively given axiom sets are theorems of a particular kind in Number Teory 1563 and themselves can be formalised in PA - but+ we ignore that aspect.)

1564 Exercise 3.13 E We say that a set x is outright defnable in a model M, E of ZFC if there is a formula v ˚ M 1565 with the only free variable shown, so that x is the unique set so that ( x holds. Suppose Con ZFC .Show 1566 that there is a model( )( )M, E of ZFC in which every set is outright defnable.⟨ ⟩ ( ) [ ]) ( ) 1567 Exercise 3.14 E⟨ Show⟩ that there is no formula v with just the free variable v so that y y is the ˚   1568 class of outright defnable (in V, ) sets. [Hint: use a form of Richard’s Paradox. Suppose there is such a . Te P 1569 least ordinal not( )( outright) defnable is a countable ordinal,( ) but now let v be “v is an ordinal”{ ∣v ( v)} v .   @    1570 Ten .] ( ) ( ) ∧ < ( ) = { ∣ ( )} 52 Formalising semantics within ZF

Alfred Tarski (1902 Warsaw -1983 Berkeley, USA)

1571 Exercise 3.15 E Suppose M, E is a model of ZF. Show that there is a N, E EMwith N, E a model ˚˚ 1572 of ZF. ′ ′ ( )( ) ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ 1573 Exercise 3.16 E Suppose that there are transitive models of ZF. Let x, be such, chosen with x ˚˚ P 1574 least. Ten (by Ex.3.15 ) if X, E x is such that X, E xZFy,showthat X, E cannot be an ‘ -model’, that X,E P 1575 is .((Tus)(X), E contains non-standard integers, and in particular codes⟨ ⟩ for non-standard formulae.( ) X,E 1576 More⟨ particularly⟩ still, xZF⟨ y ⟩ will contain non-standard⟨ ⟩ ⊧ axioms besides⟨ the standard⟩ ones.) ≠ ⟨ ⟩ ⟨ ⟩ 1577 Exercise 3.17 E Suppose the language ˙ is the standard language of set theory augmented by a single ˚˚ ( ) 1578 constant symbol ˙. Suppose we consider the following scheme of axioms Γ stated in ˙: for each axiom of V˙ 1579 ZFC we adopt the( axiom)( ) ˙: x Fr x L x x .(Tus is declared absolute for V˙.) Γ @ Ď 1580 consists of all the axioms ˙. Informally, taken together then, Γ says that V ă V whereL interprets ˙. However 1581 the existence of a satisfying the⃗( latter( ) relation⃗ %→ is( not[⃗ provable]) ←→ in ZFC[⃗])) (by the Godel¨ Incompleteness Teorem). 1582 Nevertheless show that Con ZFC CON ZFC Γ . Why does this not contradict Godel?¨

( ) ⇒ ( + ) 1583 Chapter 4

1584 The Constructible Hierarchy

1585 In this chapter we defne the constructible hierarchy due to Godel,¨ and prove its basic properties. Besides 1586 its original purpose used by Godel¨ to prove the relative consistency of AC and GCH to the other axioms 1587 of ZF, we can exploit properties of L to prove other theorems in algebra, analysis, and combinatorics. In 1588 set theory itself, properties of L can tell us a lot about V even if V L. 1589 ≠

Kurt Godel¨ (1906 Brno - 1983 Princeton, USA)

1590 4.1 The L -hierarchy

1591 We use the Def function to defne a cumulative hierarchy based on the notion of defnable power set 1592 operation: the Def function.

1593 Definition 4.1 (Godel)¨ (i) L ;L Def L , ;   P 1594 Lim L L . + 1595 (ii) L L On . = ∅ = (⟨ ⟩ ( ) %→ = ⋃{ ∣ < } = ⋃{ ∣ < } 53 54 Te Constructible Hierarchy

W 1596 Lemma 4.2 Te term L is defnite, and hence absolute for transitive W satisfying ZF . − 1597 Proof: Te Def function is defnite and the L function is defned by defnite( recursion) from it. 1598 Q.E.D. 1599 ↣ 1600 We thus have defned a class term function F L by a transfnite recursion on On,andsoalso 1601 the term L itself. It is natural to defne the notion of “constructible rank” or L-rank, by analogy with 1602 ordinary V-rank. ( ) =

1603 Definition 4.3 For x Lwedefne the L-rank of x, x the least so that x L . P L df P  + 1604 We give some of the basic properties of the L -hierarchy.( ) = Many are familiar properties common with 1605 the V -hierarchy: all of the following are true with L replaced by V .

1606 Lemma 4.4 (i) L L ; Ď 1607 (ii) L L ; < P%→ 1608 (iii) Trans L ; < %→ 1609 (iv) L ; ( ) 1610 (v) On L . = ( ) 1611 Hence Trans L and On L. = ∩ Ď 1612 Proof: We prove( ) this by a simultaneous induction for (i)-(v). Tese are trivial for . Suppose 1613 proven for and we show they hold for . (i): It sufces to prove that L L since by the inductive hypothesis, for we= already know Ď  L L . (Actually this is just an instance of+ Lemma 3.19(ii), noting that Trans L by (iii), but we prove Ď + it again.) Let x L . By (iii) for , Trans L and hence x L . < P Ď ( ) x y L L , ( x)v ˙v y y, x Def L , L P P P  P P . + 1614 (ii) Again it sufces= to{ show that∣⟨ L ⟩ ⊧L . However[ ]} L Def(⟨ L⟩), = by Lemma 3.19 (i). P  P P 1615 (iii) L  L hence x L  x L L  by (i). Ď P Ď + Ď (⟨ ⟩) 1616 (iv) By the inductive hypothesis L . By (ii) L L , hence L L  . Hence + P( ) + %→ + P 1617  L . For the reverse inequality note that: x L x L ,andso x L .  P + Ď + 1618 Tis means that ( ) = = ( ) < ( ) + ≤ ( + ) + %→ ( ) ≤ ( ) = 1619 L sup x  x L .  df P  (v) By the inductive hypothesis and (i) L L ,soitsufces to show that L in order to + + Ď Ď  P  show( that ) =  L{ (. T) +us: ∣ } ≤ + Ď  + + + + L On L L , xv ˙ Ony Def L , L . P P P P P P P  + 1620 Tat On L = { :∣ On L } = { On∣⟨ ⟩ ⊧  by[ (iv).]} But the(⟨ latter⟩) is= just .  Ď  Ď P 1621 We now assume Lim and (i)-(v) hold for . Ten (i)-(iii) and (v) are immediate. For (iv) : ∩ + + ∩ + { ∣ ( ) < + } + 1622 L sup x  x L sup . Conversely L L . Q.E.D. (P ) P < Ď ( ) = { ( ) + ∣ } ≤ { ∣ } = %→ ( ) ≥ TeL -hierarchy 55

1623 Lemma 4.5 (i) For all On, . P L 1624 (ii) For n Ln Vn. 1625 (iii) For all L . ( ) = ( ) = ≤ = 1626 Proof: (i) and (ii):≥ Exercise.∣ ∣ = ∣ For∣ (iii) we prove this by induction on .For this follows from (ii) 1627 and V . Suppose proven for . L Def L , L  by lemma 3.19 (iv).  P 1628 = For Lim : L L + as by the inductive hypothesis L for . 1629 ∣ ∣ = ∣ ∣ = ∣ (⟨ ⟩)∣ = ∣ ∣ = ∣ ∣ = ∣ + ∣ Q.E.D. ( ) ∣ ∣ = ∣ ⋃ < ∣ ≤ ∣ ∣ ⋅ ∣ ∣ = ∣ ∣ ∣ ∣ = ∣ ∣ ≤ ∣ ∣ < 1630 Exercise 4.1 (i) Verify that for all On, (ii) Prove that for n L V . P L n n 1631 Remark: (i) shows that as far as ordinals( ) go,= they( ) = appear at the same stage≤ in the= L-hierarchy as in 1632 the V-hierarchy. However it is important to note that this is not the case for all constructible sets: there 1633 are constructible subsets of that are not in L .

+ 1634 Definition 4.6 (i) Let T be a set of axioms in . Let W be a class term. Ten W is an inner model of T, W W 1635 if (a) Trans W ; (b) On W ; (c) T , that is, for each in T, . Ď 1636 (ii) If (i) holds we write IM W, T and if TL is ZF then simply IM W . ( ) ( ) ( ) ( ) ( ) L 1637 Theorem 4.7 (Godel)¨ L is an inner model of ZF, IM L . In particular ZF .

L 1638 Remark: again this is to be read as saying: for each( axiom) of ZF, (ZF ) . L 1639 Proof: We already have (a) and (b) by Lemma 4.4, so it remains to show ZF . We justify this by 1640 considering each axiom (or axiom schema) in turn. We use all the time, without⊢ ( ) comment the fact that 1641 each L is transitive. ( ) L 1642 Ax  Empty is trivial as L. P 1643 Ax1: Extensionality: Tis is Lemma 1.21, since we have Trans(L). 1644 Ax2: Pairing Axiom Let∅x=, y∅ L . Ten P 1645 x, y z L L , xv ˙v v ˙v y z , x , y  Def L L L. P P     P  Ď 1646 By Lemma 1.24 then Ax2 holds in L. + 1647 Ax{3 Union} = { Axion Let∣⟨x L⟩ ⊧. Tis= follows∨ = from[ / Lemma/ 1/.25]}once we( show:) = P 1648 x z L L , x v v ˙v v ˙v y z , x  Def L . P P D  P  P  P 1649 Ax4 Foundation Scheme Let a be a term. Ten: L L L L 1650 ⋃a = { x∣⟨ a x⟩ ⊧a ( ∧ a [ / / x]} a x( a) . But the right hand side D P D P 1651 of the equivalence here is simply an instance of the Foundation scheme in V and thus is true. Ax( 5≠Separation∅ %→ ( Scheme( ∩Again= ∅))) let a ↔be a( class≠ ∅ term.%→ Suppose( ∩ = ∅))

a z z , y ,...,yn n .

1652 Suppose x, y L . We apply Lemma 2=.40{ to∣ ( the/ hierarchy/ Z / L)}, Z L to obtain a so that P L L 1653 ZF z L z, y ,...,y z, y ,...,y ). @ P  n  n By the Correctness⃗ Teorem 3.21 = = > ⊢ (( ( )) ↔ ( ( )) ) z, y ,...,y L L , x y z, y ,...,y .  n P  n ( ( )) ↔ ⟨ ⟩ ⊧ [ ] 56 Te Constructible Hierarchy

1654 Hence, putting it all together: L L 1655 z x z, y ,...,y z x z, y ,...,y = P  n P  n 1656 z L L , x v ˙v y z, y ,...,y , x Def L . P P P n   n P 1657 { ∣ ( )} = { ∣ ( ) } L Ax6 Replacement Scheme Suppose+ f is a term, x L, and Fun f . Let L be the constructible = { ∣⟨ ⟩ ⊧ ∧ [ ]}P (L ) 1658 rank function. Ten by the Replacement Scheme (in V L f “x V. Let be its supremum. Ten L P 1659 f “x L . Let be sufciently large so that by the Refection T(eorem) Ď L L )( ○ ) 1660 ZF y, z L f z y f z y ). @ P ≥ 1661 Ten again using the Correctness theorem we have that L ⊢ (( ( ) = ) ↔ ( ( ) = ) ) 1662 f “x y L L , x v ˙v f v v y y , x  Def L . P P D P    P 1663 Ax7 Infnity Axiom Just note that L . P  1664 = { ∣⟨ ⟩ ⊧ ( ( ) = ) [ / / ]} ( ) Since we have shown the requisite sets are all+ in L we apply the appropriate cases of Lemma 1.25 and 1665 conclude Ax3,5,6,7 hold in L. We are thus lef with: L L 1666 Ax8 PowerSet Axiom x y y x x y z z x z y @ D @ D @ Ď P 1667 x L y L z L z x z y @ P D P @ P ( Ď ( = PP( ))) ↔ ( ( ↔ )) ↔ 1668 x L y L y x L . ↔ @ P D P ( ↔ ) 1669 So we verify the latter: let x L be arbitrary. x L V by Axiom of Power and Separation in V.By ↔ ( = P(P ) ∩ ) P 1670 Ax.Replacement L“ x L V. Let be its supremum. Ten, as required: P P( ) ∩ 1671 x L z L L , xv ˙ v y z, x Def L . Q.E.D. P P Ď  P 1672 P( ) ∩ P( ) ∩ = { ∣⟨ ⟩ ⊧ [ ]} ( ) 1673 Suppose we defne IM W to be the variant on IM W that, keeping (a) and (b), replaces (c) by 1674 the statement that “ x W y V x y Trans y Def y, W ” then a close reading of the @ Ď D P Ď P Ď 1675 last proof reveals that we in( fact) may show: ( ) ( ∧ ( ) ∧ (⟨ ⟩ )

1676 Theorem 4.8 Suppose W is a class term and IM W . then IM W .

1677 Exercise 4.2 (E) Prove this last theorem. ( ) ( ) ˚ 1678 Exercise 4.3 Show( ) that “x is a cardinal”and“x is regular” are downward absolute from V to L. Deduce that if L 1679 is a (regular) limit cardinal then is a (regular) limit cardinal .

( ) 1680 4.2 The Axiom of Choice in L

1681 Te very regular construction of the L -hierarchy ensures that the Axiom of Choice will hold in the con- 1682 structible universe L. Indeed, it holds in a very strong form: whereas the Axiom of Choice is equivalent 1683 to the statement that any set can be wellordered, for L there is a class term that wellorders the whole 1684 universe of L in one stroke. Essentially what is at the heart of the matter is that we may wellorder the 1685 countably many formulae of the language , and then inductively defne a wellorder  for L  using 1686 a wellorder for L . Tis latter wellorder gives us a way of ordering all fnite k-tuples+ of elements+ 1687 of L , and thus, putting these together, weL get a wellorder of all possible defnitions that< go into making < < 1688 up new objects in L . We shall additionally have that the ordering  end-extends that of . Tis 1689 means that if y L  L then for no x L do we have that y  x. Taking L On gives us P + P + P 1690 the term for a global wellordering of all of L. We now proceed to fll out< this sketch. < + / < + < = ⋃ < Te Axiom of Constructibility 57

1691 Let x V and suppose we are given a wellorder of x. We defne from this a wellorder .For P x Qx 1692 f Q we let lh f the least n so that m n f m f n . We then defne for f , g Q : P x df @ P x 1693 < < ( ) = ≥ ( ( ) = ( )) 1694 f g lh f lh g lh f lh g k lh f n kf n g n f k g k . Qx df D @ x

1695 Exercise< 4.4←→Check that( ) if< (WO) ∨then( ( ) = WO( .) Moreover∧ ≤ ( )(is def

1696 We now suppose we also< have fxed< an ordering , <,..., n,...of the countably many elements 1697 of Fml which have at least v amongst their free variables (we may defne such a listing from any map 1698 g Fml). We assume that the function f given by f n x ny is a defnite term.

∶ ←→ ( ) = 1699 Definition 4.9 We defne by recursion the ordering of L . ;letx, y L :  P  1700 x  y df < < = ∅ + 1701 x L y L x, y L x y x, y L n f QL x L , n, f < + P ↔ R P R D P D P 1702 m g QL y L , m, g n m m n f Q g . ( ∧ ) ∨ ( @ P∧ @< P ) ∨ ( ∧ ( = ( )L∧ 1703 Lim ; L df On . P ( = ( ) %→ < ∨ ( = ∧ < )))) ( ) %→ < = ⋃ < < < = ⋃ < 1704 Lemma 4.10 (i) is defnite; (ii) the ordering is a wellordering and end-extends if ; (iii) if L 1705 is an infnite cardinal then has order type ; L has order type On. Tus AC . < < < ≤ 1706 Proof: (i) f df is< defned by a defnite< recursion. (ii) By an obvious( ) induction on . (iii) 1707 Exercise. Q.E.D. ( ) = <

1708 Exercise 4.5 Show that ot L , for an infnite cardinal; deduce that ot L, L On.

( < ) = ( < ) = 1709 4.3 The Axiom of Constructibility

1710 Definition 4.11 Te Axiom of Constructibility is the assertion “V L” which abbreviates “ x x L .” @ D P

1711 Te Axiom of Constructibility thus says that every set appears somewhere= in this hierarchy. Since the 1712 model L is defned by a restricted use of the power set operation, many set theorists feel that the Def func- 1713 tion is too restricted a method of building all sets. Nevertheless, the inner model L of the constructible 1714 sets, possesses a very rich structure.

Lemma 4.12 (i) Let W be a transitive class term, and suppose ZF W . Ten − L W LifOn W( On) L if On W . ( ) = ∩ = = ∩ = W 1715 (ii) Tere is a fnite conjunction  of ZF axioms, so that in (i) the requirement that ZF can be replaced W 1716 by  and the conclusion is unaltered.− − ( ) ( ) 58 Te Constructible Hierarchy

Proof: (i) Te function term L is defnite. Hence is absolute for such a W. Note that in the case that On W On then indeed Lim . But in either case for any W L W L . Hence P P ∩ = L W ( ) L On W L On (W ) = P P

1717 which yields the above result.( ) = (⋃{ ∣ }) = ⋃{ ∣ ∩ } 1718 (ii)  is simply the conjunction of sufciently many axioms needed for the proof that the function 1719 term L is defnite, plus “there is no largest ordinal”. Q.E.D.

L 1720 Corollary 4.13 ZF V L .

L L L L L L 1721 Proof: Trans L (and)(ZF = .But) V L V L .AsV L and by Lemma 4.12 L L we 1722 are done. − Q.E.D. ( ) ( ) ( = ) ↔ = = ( ) = 1723 Theorem 4.14 Con ZF Con ZF V L .

1724 Proof: Suppose ZF( V ) ⇒L is inconsistent.( + = Suppose) ZF V L . L 1725 ZF ZF V L by the last Corollary and Teorem 4.7, then: L 1726 ZF +,= and hence: + = ⊢ ( ∧ ¬ ) L L 1727 ZF ⊢ ( + = .) Hence: L L 1728 ZF ⊢ ( ∧ ¬ ) . Hence ZF is inconsistent. Q.E.D. ⊢ ∧ (¬ ) 1729 Remark⊢ 4.15∧P.¬( Cohen) (1962)showedCon ZF Con ZF V L by an entirely diferent method, 1730 that of “forcing”. Tis method can be construed as either constructing models in a Boolean valued (rather 1731 than a 2-valued) logic; or else akin to some( kind) of⇒ syntactic( + method≠ ) of construction. (An entirely dif- 1732 ferent method was needed - see Exercise 4.7.) He further showed that Con ZF Con ZF AC 1733 and Con ZF Con ZF CH . His methods are now much elaborated to prove a wealth of “relative 1734 consistency” statements such as these. ( ) ⇒ ( +¬ ) ( ) ⇒ ( +¬ ) 1735 Theorem 4.16 (Godel¨ 1939) Con ZF Con ZF AC

L L L 1736 Proof: We have shown ZF AC( ),butalso⇒ ( ZF+ )ZF ,andthusZF ZF AC , Hence if L 1737 ZF AC for some then we should have ZF as in the last proof, and hence 1738 not Con ZF . ⊢ ( ) ⊢ ( ) ⊢ ( + ) Q.E.D. + ⊢ ∧ ¬ ⊢ ( ∧ ¬ ) 1739 Exercise(4.6) Suppose there is a transitive set model of ZFC. Show that there is a minimal (transitive) model of 1740 ZFC, that is for some countable ordinal , L  xZFCy and that L  is a subclass of any other such transitive set 1741 model of ZF . ⊧

1742 4.4 The Generalised Continuum Hypothesis in L.

1743 We frst prove a simple lemma, but one of great utility.

L L 1744 Lemma 4.17 (Te Condensation Lemma) Suppose x, ă L , where ZF (or just where P P  1745  is the fnite conjunction of axioms from Lemma 4.12). Ten there is with− x, L , . ⟨ ⟩ ⟨ ⟩ ( ) P ( P ) ≤ ⟨ ⟩ ≅ ⟨ ⟩ Te Generalised Continuum Hypothesis in L. 59

L 1746 Proof: By assumption on we have , and so by the Correctness Lemma, we have that L ,  P 1747 xV Ly. Hence x, xV Ly. Let x, y, be the Mostowski Shepherdson Collapse P ( ) P P ⟨ ⟩ ⊧ 1748 with Trans y . Ten y, x y xV Ly (as is an isomorphism). By the frst conjunct, Correctness = ⟨ ⟩ ⊧P y = ∶ ⟨ ⟩ %→ ⟨ ⟩ 1749 again, and Lemma 4.12, L L y LOn y. But by the second, this equals y itself. So we may take 1750 ( ) ⟨ ⟩ ⊧ ∧ = On y. ∩ Q.E.D. 1751 = ∩ = = ∩ L 1752 Note: It can be shown that the assumption that ZF can be very much reduced: all that is needed 1753 for the conclusion of the lemma is that Lim , and with− a lot more fddling around even this condition 1754 can be dropped, and we have Condensation holding( for) every L . ( ) L L 1755 Theorem 4.18 ZF Card H L . Hence ZF GCH and thus ZF V L GCH . P

1756 Proof: We have that⊢ (L ≤ V H%→already= and) hence the conclusion⊢ ( ) for . Assume+ = ⊢ L L L LP 1757 Card .If then by Lemma 4.5(iii) ( L . Hence (L H . Tus L H . L P Ď 1758 Now for the reverse inclusion= suppose= z H . Find an sufciently large= with z , TC(z < L ) < L P∣ ∣ = ∣ ∣ < ) ) ( P ) 1759 and by the Refection Teorem  .Asz H TC z H , we may apply the Downward ( P) P { } ( ) 1760 Lowenheim-Skolem¨ theorem in L and fnd x, ă L , with TC z TC z z x,and (L ) P %→ P ( ) Ď 1761 x max TC z ,  . ⟨ ⟩ ⟨ ⟩ ({ }) = ( ) ∪ { } 1762 As the transitive part of x contains all of TC z ,wehavethat z z where is the transitive (∣ ∣ = {∣ ({ })∣ ℵ } < ) 1763 collapse map mentioned in the Condensation Lemma, taking x, y, L , for some L P P P 1764 . However we know that ( x L ({ })by design. Hence z( )L= L . L L ∶ ⟨ ⟩P%→ P⟨ ⟩ = ⟨ ⟩ L 1765 As z H was arbitrary we conclude that L H .WethushaveshownH L .To ≤ P L ∣ ∣ = ∣ ∣ = ∣ ∣ < ) Ě L 1766 show GCH it sufces to show that for all infnite cardinals that  . However  and L L 1767 H( ) L . Hence L ( .ByCantor’s) Teorem+ we conclude( = ) L Ď 1768 .( )+ + + + ( = ) ≈ P( ) (P( ) = ) (∣P( )∣ ≤ ∣ ∣L= ) (∣P( )∣ = 1769 + Tis argument establishes that ZF GCH . If we additionally assume V L we have the conclu- ) 1770 sion of the Teorem. Q.E.D. 1771 ⊢ ( ) = 1772 Te proof of the next is identical to that of Cor. 4.16:

1773 Corollary 4.19 (Godel¨ 1939) Con ZF Con ZF GCH .

( ) ⇒ ( + ) W 1774 Exercise 4.7 (E) (Shepherdson) Show that there is no class term W so that ZFC IM W and ZFC ( CH . 1775 [Tis Exercise shows that Godel’s¨ argument was essentially a “one-of”: there is no way one can defne in ZFC alone 1776 an inner model and hope that it is a model of all of ZF plus, e.g. , CH.] ⊢ ( ) ⊢ ¬ )

L 1777 Exercise 4.8 Show that if there is a weakly inaccessible cardinal¬ then ZFC . Hence ZFC aweakly L D 1778 inaccessible cardinal.) [Hint: Use the fact that GCH .] ( ) ⊢/ ( 1779 Exercise 4.9 Show that if is weakly inaccessible( then) L xZFCy . [Hint: use the @ D 1780 Condensation Lemma and Downward Lowenheim-Skolem¨ Teorem.] < < ( > ∧ ⊧ ) 1781 Exercise 4.10 Assume V L. When does L V ?

= = 60 Te Constructible Hierarchy

1782 Exercise 4.11 (E)( Show that if is any limit ordinal, which is countable in L, then there is , countable in ˚  1783 L,sothat L L .[Tis shows that there are arbitrarily long countable ‘gaps’ in the constructible 1784 ) < hierarchy, where no new real numbers+ appear, although by the GCH proof all constructible reals will have appeared L 1785 by stage P( ). Hint:∩ = SupposeP( )∩V L, and look at the countable set X . Let A L , and, by the   P 1786 Downward Lowenheim-Skolem¨ Teorem, let Y X  be a countable elementary substructure+ of A: Y ă A. Ě 1787 Let Y( ) M be the transitive= collapse of Y and as in the GCH proof,= {M ∣ L< for} some =. Consider⟨ ⟩  .] ∪ + L 1788 Exercise 4.12 Show that (i) if is a weakly inaccessible cardinal, then is strongly inaccessible ; (ii) if is a ∶ %→ L = = ( ) 1789 weakly Mahlo cardinal, then is strongly Mahlo .[Hint: See Exercises 4.3 & 4.8. For (ii) show that the property 1790 of being cub in is preserved upwards from L to V.] ( ) ( ) 1791 L Exercise 4.13 (i) Let x, ă L  where   . Show that already Trans x and so x L for some . P L 1792 [Hint: For note that ( L  . Hence for x, in L , and thus in x, there is an onto map  P  1793 f L . Tus, as⟨ ⟩x f x we deduce= ( that) ran f L x. Deduce( that) Trans x=.] ≤ Ď P Ď 1794 < ∣ ∣ = ∣ ∣ = ) L (ii) Now let x, ă L  where   .ShowthatTrans x L  and so x L  L for some . ∶ %→˚ P ∧ ( ) = ( ) 1795 Exercise 4.14 Assume V L. N. Schweber defned a countable ordinal to be memorable if for all sufciently ( ) ˚ ⟨ ⟩ = ( ) ( ∩ ) ∩ = 1796 large , Def L , .Show:  P  P 1797 (i) Te memorable( ) ordinals form= a countable, so proper, initial segment of , .  P 1798 (ii) Let < be the least non-memorable(⟨ ⟩) ordinal. Show that is also the least ordinal so that for arbitrarily large 1799 , L ă L . ( )

< 1800 4.5 Ordinal Definable sets and HOD

1801 Godel’s¨ method of defning the inner model L of constructible sets was not the only way to obtain the 1802 consistency of the Axiom of Choice with the other axioms of ZF.Another model can be defned, the inner 1803 model of the hereditarily ordinal defnable sets or “HOD” in which the AC can be shown to hold. (Te 1804 GCH is not provably true there, and the absoluteness of the construction of L - which allowed us to show L HOD 1805 that L L is not available: it is consistent that HOD HOD.) We investigate the basics here. Tere 1806 is some evidence that Godel¨ was aware of this approach, as he suggested looking at the ordinal defnable 1807 sets for= a model of AC. However the construction requires≠ essential use of the Refection Teorem that 1808 was not proven until the end of the 1950’s by Levy and Montague. Some see these remarks of Godel¨ as 1809 indicating that he was aware of the Refection Principle, even if he did not publish a proof.

1810 Definition 4.20 We say that a set z is ordinal defnable (z OD) if and only if for some formula P 1811 v, v,...,vm with free variables shown, for some ordinals ,..., m then z is the unique set so that 1812 z, ,..., m . ( ) 1813 We next need to show that the expression z OD is defnable within ZF. (At the moment the last [ ] P 1814 defnition has loosely talked about “defnability (in V, )” - which is not defnable in V, .) We do P P 1815 this by showing it is equivalent to the alternative defnition given in Def.3.20, which involved only the ⟨ ⟩ ⟨ ⟩ 1816 defnable sets V and the defnable function Def x .

1817 Exercise 4.15 (Richard’s Paradox) Let T be the set of( those) z so that for some closed term x t (that is one 1818 without free variables) z x t . Show that there is no formula v (with just the one free variable shown), so 1819 that T z z ,andthusT is not defnable by such a formula. [Hint: as there are only countably{ ∣ } many closed 1820 terms, there will only be countably= { ∣ } many ordinals in T. Suppose for( a contradiction) that v does defne the set 1821 of elements= { ∣ of T[ ]}(meaning that it is true of just the elements of T). Consider the term .] (@ ) { ∣ ≤ ( [ ])} Ordinal Defnable sets and HOD 61

n n 1822 Exercise 4.16 Let ,..., n  On for some n. Ten there is so that Def  V . [Hint: Let be the n P n P 1823 wellorder of On as above at Ex. 1.10. Let express “ is the -least sequence so that Def V ”. −  @ R  1824 But if  were true,⃗ = it would refect to some . But then Def  V .] ⃗ ( ) < (⃗) ⃗ P < (⃗ ( )) Exercise(⃗4.)17 (Scott) For any formula v,...,vm  with⃗ free variables( ) v,...,vm ,

− − ZF ,..., ,..., (Def V )x ,...,x x ,...,x x ,...,x V . @  n D  n  P  @  m   m   m  − − − − − 1825 [Hint:⊢ Another use of the( Richard Paradox argument.( ) ∧ Expand Ex.4.16( using the formula) ↔ ( (  there: suppose)) ) the n 1826 displayed formula is false for some -least ,..., n . Let be any sufciently large ordinal that refects  . 1827 As in Ex.4.16, ,..., n  Def  V and V refects too.] P < − ∧ − ( ) 1828 Theorem 4.21 z OD is expressible by the single formula in ZF: z :“ z Def V ”. P OD D P 

1829 Proof: Let OD˚ denote the class of sets z satisfying the Defnition(3.)20,thatistheformula( ( )) OD z 1830 above. It sufces to show then OD˚ OD. is clear. Suppose x is the unique set satisfying x, ,..., . Ď  n  1831 By Ex. 4.17 there is with ,..., Def V and V refects with x V . Ten x, ,..., ( )  n  P  P  n  − 1832 = ( ) [ ] defnes x in V . But amalgamating the− defnitions of the sequence with that given by we have a def-− 1833 inition x in V without the use of ordinal( parameters.) Tus x Def V . [ Q.E.D.] P  ′ ⃗ [ ] ( ) 1834 Theorem 4.22 OD has a defnable wellordering.

HF 1835 Proof: We use a defnable wellorder of HF to impose a wellordering on the Godel¨ code sets of 1836 formulae with one free variable. As OD z z Def V for any z OD we can set z df < D P HF P 1837 the least so that z Def V . Let z be the least, in the ordering , formula with the single free P = { ∣ ( ( ))} ( ) = 1838 variable v, that defnes z in V z . Now defne ( ) < ( ) x z x, z OD x z x z HF . OD P x z

1839 One can check this< is a wellorder⇔ of OD∧. ( ( ) < ( ) ∨ ( ( ) = ( ) ∧ < )) Q.E.D.

1840 Lemma 4.23 Let A be any class that has a defnable set-like wellorder given by some v, v (“set-like” 1841 meaning for any z A, z A z, z is a set). Ten A OD. P P Ď ( ) 1842 Proof: By assumption we{ can de∣ f(ne by)} recursion a rank function r z sup r y  y A y, z . P 1843 Ten ran r On. But now for each z A for some we have r z and we may defne z as “that Ď P 1844 unique z with r z ”. ( ) = { ( )+ ∣ ∧ (Q.E.D.)} ( ) ( ) = ( ) = 1845 Corollary 4.24 L OD Ď

1846 Proof: By Ex. 4.5 the ordering L of L is both defnable and a wellorder in order type On.Itisthus 1847 “set-like” as described above. Hence L OD. Q.E.D. < Ď

1848 Corollary 4.25 Con ZF Con ZF V OD .

( ) ⇒ ( + = ) 62 Te Constructible Hierarchy

1849 Proof: V L implies V OD by the last corollary. So this follows from Teorem 4.14. Q.E.D. 1850 OD 1851 On the= other hand it is= not provable in ZF that V OD (or that OD is transitive; even AxExt 1852 may fail, see Ex.4.20 below). Indeed we cannot prove that OD is an inner model of ZFC. Ten we need 1853 to consider the closely related subclass of hereditarily ordinal= defnable sets. ( )

Definition 4.26 (Te hereditarily ordinal defnable sets - HOD)

z HOD z OD TC z OD. P P Ď

1854 We thus require not only that z be in OD⇔but this fact∧ propagates( ) down through the -relation below P 1855 z. By defnition HOD is a transitive class of sets containing all ordinals.

1856 Exercise 4.18 Show z HOD z OD y z y HOD .Showthat OD HOD. P P @ P P HOD ↔ ∧ ( ) P(HOD) ∩ = P( ) ∩ 1857 Theorem 4.27 ZFC , that is for each axiom of ZFC, we have .

1858 Proof: By transitivity( ) of HOD we have HOD and AxExtensionality holds in HOD. It is easy to P 1859 check that x, y HOD x, y , x HOD. Likewise as any ordinal is in HOD (e.g. , by induction P P 1860 using the last exercise), so is HOD.∅ For AxPower: suppose x HOD.Itsufces to show that HOD P P 1861 x x HOD→ { OD}. (Again⋃ see the last exercise. Te frst equality is obvious as “y x”is P Ď 1862 ∆ .) But notice that x HOD x OD. (If y x y OD then y HOD by the exercise.)  Ď P P 1863 SoP( it) sufces= P to( show) ∩ x OD OD. Let x ; then x OD V . Let be as above. P  Ď  OD 1864 As x OD there areP(, ),...,∩ with= P( x) ∩ x x, .BytheRe∧ fection Teorem on and P  n OD 1865 we can fnd , Pwith( )z∩ x OD V= ( )x x,P( )z∩ y y y x .    D ⃗ OD Ď For AxSeparation: let a be a class term,{ } and= { let∣x (HOD)}. We require that aHOD x HOD. Suppose ⃗ P ⃗ P a z z, y> for some ,some= P( y) ∩ HOD⇔.BytheRe⊧ (fection( )T∧eorem= { we∣ can(fnd) ∧ a sufciently}) large P which is refecting for and the defning formula for HOD,andwithaHOD x ∩ V . Ten we have: = { ∣ ( ⃗)} ⃗ P u aHOD x V u z x z, y HOD . ∩ P

1866 From the right hand side here,= we see∩ that⇔ u is⊧ def=nable{ over∣ (V ⃗but) using} the parameters x and y. 1867 However these are all in OD and so we may replace them by their (fnitely many) defnitions using just ⃗ 1868 ordinal parameters, thereby rendering the right hand side a term purely with ordinal parameters. Hence 1869 u is in OD and thence in HOD (as u x HOD). Ď Ď 1870 AxReplacement is similar: let F be a function given by a term, and let x HOD. We require HOD HOD P 1871 F “x HOD.InV, by AxReplacement, let F “x V , but then also is a subset of V HOD. P Ď 1872 It thus sufces to show V HOD HOD, as then the AxSeparation will separate out from HOD V HOD P 1873 exactly the set F “x. Tis is the next Exercise. ∩ 1874 Finally for AxChoice, we∩ show that the Wellordering Principle holds in HOD:byTeorem 4.22∩ we 1875 already have an OD-defnable wellordering of OD HOD, .Ifx HOD, then u, v u, v Ě OD P P 1876 x, u OD v is in HOD and wellorders x. Q.E.D. < {⟨ ⟩∣ 1877 Exercise< 4.}19 Show that for any , V HOD HOD. P ∩ Criteria for Inner Models 63

OD 1878 Exercise 4.20 Show that the following are equivalent: (i) V OD, (ii) V HOD, (iii) Trans OD , (iv) AxExt . 1879 [Hint: Use that for any V OD V OD OD.] P P = = ( ) ( ) 1880 Exercise 4.21 Show that HOD ∧ is∩ the largest subset of with a defnable wellorder. [Hint: Use Lemma 1881 4.23 and Ex. 4.18.] ∩P( ) P( ) 1882 Exercise 4.22 Suppose that W is a term defning an inner model of ZF and there is a defnable global wellorder 1883 of W (that, as in L, there is a formula defning a wellorder W of the whole of W in order type On). Show that 1884 W HOD. (Consequently HOD is the largest inner model W with a defnable bijection F On W.) Ď < 1885 Exercise 4.23 Defne “Π-OD” (and Π-HOD) just as we did for OD and HOD but now∶ restrict↔ the formulae 1886 allowed in defnitions to be Π only. Show that Π-OD OD and Π-HOD HOD. Now do the same for Σ-OD 1887 and Σ-HOD. = = 1888 Exercise 4.24 * Show that there is a single formula  v with just the free variable shown, so that OD is the V 1889 class of all those x so that x Def  V for some ,thatisforsome , x z  z . P ( ) 1890 Again it is consistent that V ( L) HOD, V L HOD and V{ } =L{ ∣HOD( )as} well as further com- HOD 1891 binations such as HOD may or may not equal HOD. CH may fail in HOD (see the next Exercise). = = ≠ = ≠ ≠ HOD n 1892 Exercise 4.25 E Tis shows that we may have CH . Let C n  n  . Suppose ˚ HODP + 1893 C On  (this can be shown consistent with ZFC), then CH . ℵ P ( )( )) (¬ ) = { ∣ = ℵ + + } 1894 ∣{ We∣ can de}f∣ ≥neℵOD and HOD as before but now we allow sets(¬ z )x as parameters in our defnitions x x P 1895 as well as ordinals. HODx will be an inner model of ZF as before, but it will only be a model of Choice 1896 if there is an HODx-defnable wellorder of x itself to start with.

1897 4.6 Criteria for Inner Models

1898 It is possible to give a defnition for when a class term W defnes an inner model, IM W ,fortheZF 1899 axioms which is formalisable in ZF.Wefrst give an equivalent axiomatisation of ZF. ( ) 1900 Definition 4.28 We set Z F to be the theory that consists of the Axioms Ax0-4, Ax7-8 and: 1901 Ax5 (∆ -Separation Scheme)∗ For every ∆ -term a: x a V   P 1902 where∗ by a ∆-term a we mean a term a x x, y where is a ∆-formula. 1903 Ax6 (Collection Scheme) For every formula : y v ∩v, y z w y z v w v, y . @ D @ D @ P D P ∗ = { ∣ ( ⃗)} 1904 Te weakening of the Ax.5 is made up for by the strengthening⃗ ( ⃗) %→ of Ax6 which( ⃗ is less about( the⃗ range) of 1905 functions than ‘collecting’ together the ranges of relations on sets z. Note we could have expressed Ax6 , 1906 somewhat more awkwardly, as: “For any term r if yr“ y then z w y z r“ y w ”. ∗ @ @ D @ P

1907 Theorem 4.29 ZF ZF˚ and ZF˚ ZF, and thus the{ two} ≠ theories∅ are equivalent.( { } ∩ ≠ ∅)

1908 It is unknown whether⊢ weakening⊢ Ax5 alone to Ax5 but keeping Ax6 is a theory equivalent to ZF 1909 in this sense. ∗

Theorem 4.30 For any term W IM W is equivalent to the set of formulae:

ZF( W) Trans W , On W . Ď ∪ { ( ) } 64 Te Constructible Hierarchy

1910 Theorem 4.31 If W is any term, and (i) Trans W ,On W, (ii) x W Def x W, and (iii) W is WĎ @ P Ď 1911 supertransitive,thatis x W z x z W , then ZF ,andsoIM W . @ Ď D Ě P ( ) ( ) 1912 Te utility of the last theorem is that( ofen) it is a simple matter to( verify) (i)-(iii) for any given W. W 1913 For example, HOD is easily seen to have these three properties. Whilst the statement “ZF ” (or indeed 1914 “IM W ”) is metatheoretic in nature: it requires assertions of the infnitely many formulae contained W 1915 in “ZF ”, the three assertions (i)-(iii) are in our formal language. Tis shows that a term W being an 1916 inner( model) is truly a frst order expression about W.

1917 Exercise 4.26 Show that there is a fnite set of axioms of ZF so that if On W and W is a transitive class Ď 1918 model of just these axioms then it is a model of all the axioms of ZF. Why does this not contradict the non-fnite 1919 axiomatisability of ZF, Teorem 3.10?

M 1920 Exercise 4.27 Show that if M is a class term, and IM M ,and CH then ZF is inconsistent.

( ) (¬ ) 1921 4.6.1 Further examples of inner models

1922 Relative constructibility

1923 Tere are several ways to generalise Godel’s¨ construction of L. 1924 1925 (I) Te L A -hierarchy. 1926 ( ) 1927 Here we start out, not with the empty set as L but with the set A:

Definition 4.32 L A A A ; L  A Def L A , ; ( ) = ∪ { } P Lim L+ A L A . L(A) = L(⟨ A( ) ⟩On) . ( ) → ( ) = ⋃{ ( ) ∣ < } 1928 In this model the arguments for L can( be) straightforwardly= ⋃{ ( ) ∣ used< to} show that all axioms of ZF are 1929 valid in L A . However the Axiom of Choice need not hold, unless in L A there is a L A -defnable 1930 wellorder of A. Of course if V L then A L and the construction of L inside the ZF-model L A LP A 1931 reveals that( “)V L” holds, in which case AC trivially holds. Matters become( ) more interesting( ) when 1932 V L, and an important model= here is when (A) . Te model L contains all the reals (and so( the) 1933 structure of mathematical= analysis). Consequently anything defnable in the structure of analysis resides 1934 in the≠ model. Moreover anything obtained by ‘iterated= defnability( over) analysis’ is also here: it would be 1935 defnable using ordinals and the set of reals. Tus it is thought, the broadest methods of defnability over 1936 analysis would produce sets in this model. Consequently it is in some sense a laboratory for generalised 1937 defnability in analysis. However it is not thought in general that there must be wellorder of that is 1938 defnable over , or indeed in L .(Tis was one approach that Cantor took to look at CH:totryto 1939 fnd a defnable wellorder of ; but it is consistent with the axioms of ZF that there is no such wellorder.) 1940 Consequently when. set theorists( investigate) L they do not assume that AC holds there, although it

( ) Criteria for Inner Models 65

1941 is taken to hold hold in the wider universe V. 1942 1943 (II) Te L A -hierarchy. 1944 [ ] 1945 Te next hierarchy instead enlarges the language of set theory to incorporate a one place predicate 1946 symbol A˙. Tus A x either will or will not be true of sets x. Te Def operator is enlarged to an operator 1947 Def A˙ that now defnes new sets over some structure in this new language ( ) Definition 4.33 L A ; L  A Def A˙ L A , , A ; [ ] = ∅ P Lim L+ A L A L[A] = L (⟨A [ ] On⟩). ( ) → [ ] = ⋃{ [ ] ∣ < } 1948 Te predicate A is usually taken to be[ a set] in= V⋃,{ but[ the] ∣ defmeasurable cardinal’. [ ]

1952 Definition 4.34 (L )L is the above hierarchy where is a -complete ultraflter on in the 1953 sense of the discussion at the end of Section 2.1.2. [ ] [ ] P( ) L 1954 Te inner model L is much studied is a -complete ultraflter on and moreover, is the 1955 least inner model with this property. It has an absolute construction property similar[ ] to L within in any L 1956 other inner model[ with] such a ultraflter( or ‘measure’ on .ItcanbeshownthatP( )) GCH , although 1957 the Condensation Lemma strictly speaking, fails in L . [ ] ( ) L A [ ] L A 1958 Exercise 4.28 Show for any A that ZF and that ZFC . [Hint: Just modify the same arguments for L.] ( ) [ ] ( ) ( ) 1959 Exercise 4.29 (i) Show that in L A ,forA ,thatforany ,  .(Tus, in L A the GCH holds ‘above Ď 1960 ’.) [Hint: Again modify the argument for L; this can only work above since+ A could be completely general, and 1961 we have no knowledge how L A[ may] look.] ≥ = [ ] 1962 (ii) However improve the last exercise, by showing that in L A ,forA ,thatforany ,  . Ď [ ] + + [ ] = ≥ = 1963 Higher Order Constructibility

1964 We do not give the details, but for the reader familiar with notions of higher order logics, in particular n n 1965 n’th-order logics for n , we may construct L using n’th order logical defnablility Def (where our  1966 previous Def is now Def . Remarkably these notions do not form a hierarchy for n , but instead all 1967 collapse: < ≥ n 1968 Theorem 4.35 (Myhill-Scott) For n  L HOD.

≥ = 66 Te Constructible Hierarchy

1969 4.7 The Suslin Problem

1970 It is well known (in fact it is a theorem of Cantor) that if X, is a totally ordered continuum that satisfes 1971 (i) X, has no frst or last end points ; 1972 (ii) X, has a countable dense subset Y (that is ⟨x, z<⟩ X y Y x y z ); @ P D P 1973 then⟨ X<,⟩ is isomorphic to , . 1974 (By⟨continuum<⟩ one requires that for any bounded subset of an interval( < in

1984 Definition 4.36 A tree T, is a partial ordering such that x T y y x is wellordered. @ P 1985 (i) Te height of x in T , ht x ,isot y y x , (also called the rank of x in T). 1986 (ii) Te height of T is⟨sup<⟩ht x x T ; ({ ∣ < }) ( ) ({P ∣ < } <) 1987 (iii) T df x T ht x . P { ( ) ∣ } 1988 Tus T=consists{ of∣ the( bottommost) = } elements of the tree, and so are called root(s) (we shall assume 1989 there is only one root). A chain in any partial order T, T is any subset of T linearly ordered by T 1990 and an antichain is any subset of T no two elements of which are T comparable. For a tree T a subset 1991 b T is a branch if it is a maximal linearly ordered (and⟨ < so⟩ wellordered) set under . A branch need< Ď T 1992 not necessarily have a top-most element of course. < − < 1993 Definition 4.37 Let be a regular cardinal. A - Suslin tree is a tree T, such that 1994 (i) T ; 1995 (ii) Every chain and antichain in T has cardinality . ⟨ <⟩ ∣ ∣ = 1996 We shall be concerned with - Suslin trees (and we< shall drop the prefx“ ”). Konig’s¨ Lemma states 1997 that every countable tree with nodes that “split” fnitely, has an infnite branch. Tis paraphrased says, a 1998 fortiori, that there are no -Suslin trees. 1999 It turns out (see Devlin [1]) that the Suslin Hypothesis is equivalent to: 2000 (SH): “Tere are no -Suslin trees” 2001 (Although this requires proof which we omit.) So do such trees exist?

2002 Theorem 4.38 (Jensen) Assume V L; then there is an -Suslin tree.

2003 Hence: =

2004 Corollary 4.39 Con ZF Con ZFC CH SH

( ) ⇒ ( + +¬ ) Te Suslin Problem 67

2005 It turns out that there is a construction principle for Suslin trees that is in itself of immense interest: 2006 it can be considered a strong form of the Continuum Hypothesis. It has been widely used in set theory 2007 and topology and has been much studied.

2008 Definition 4.40 (Te Diamond Principle). is the assertion that there exists a sequence 2009 S  so that (i) S @ Ď ◇ 2010 (ii) X  X S is stationary. ⟨ @∣ <Ď ⟩ ( ) 2011 thus asserts{ that∣ ∩ there= is} a single sequence of S ’s that approximate any subset of  “very ofen”. 2012 In particular note that CH: if x is any real then x S for “stationarily” many . Tus ◇ Ď 2013 the sequence incorporates an enumeration of the real continuum with each real occurring  many 2014 times in that enumeration.◇ %→ However it does much more beside.= < ◇ L 2015 Theorem 4.41 (Jensen) In L, holds. Tat is ZF .

2016 Proof: Assume V L. We have◇ to defne a -sequence⊢ (◇) S  . We defne by recursion S , C 2017 for : S , C is the L-least pair of sets S, C so that 2018 (a) S = ◇ ⟨ ∣ < ⟩ ⟨ ⟩ Ď 2019 (b)< C is⟨ c.u.b. in⟩ ; < ⟨ ⟩ 2020 (c) C S S @ P 2021 if there is such a pair, and S , C = , otherwise. ( ∩ ≠ ) 2022 Tus, somewhat paradoxically, S , C is chosen to be the L- least “counterexample” to a - 2023 sequence of length . ⟨ ⟩ ⟨∅ ∅⟩ 2024 Let S . As we are assuming⟨ ⟩ V L, we have just constructed< H L . Looking◇  P   2025 a little more closely, since L , we have actually defned by a recursion which only involved  Ď  2026 S = ⟨ ∣ < ⟩ = S = inspecting objects in L  which had certain defnite properties. L  is a model of ZF so these properties 2027 P( ) S − are absolute between L  and V which is L by assumption. In short the recursion as defned in L  defnes 2028 the same as in V:  S , C L S , C and indeed L . @   2029 If is not a -sequence then: S < (⟨ ⟩) = ⟨ ⟩ (S) = S 2030 (1) Tere is an L-least pair S, C with 2031 (a)SS ; (b)◇ C and C cub in ; (c) C S S . Ď  Ď   @ P 2032 Given that we< have L ⟨the⟩ quantifers in (1) are referring only to sets in L .  thus holds P   2033 ( ∩ ≠ ) relativised to L  . Expressing that in semantical terms we have: 2034 (2) L , “ S, C Sis the -least pair with ( )  P L 2035 (a) S ; (b) C and C cub in ; (c) C S S .” Ď  Ď   @ P 2036 ⟨ ⟩ ⊧ ⟨ ⟩ < By appealing to the Lowenheim-Skolem¨ Teorem we can fnd X L  with: ( ∩ ≠ ) Ď 2037 (3) X, ă L , with , S, C , X, X, and X . P  P  P Ď 2038 By Exercise 4.13 (ii) we have that X L  is transitive and so in fact is some L for some .If 2039 we now⟨ apply⟩ the⟨ Mostoski-Shepherdson⟩ S ⟨ ⟩ Collapsing Lemma∣ we∣ = have there is a and a with: 2040 (4) L , X, with L ∩id. < P P 2041 (Recall that as L X and is transitive will be the identity on L .) ∶ ⟨ ⟩ ≅ ⟨ Ď ⟩ ↾ = 2042 (5) , and if S, Caresuchthat S S, C C, then S S , C C .   2043 Proof: X  P  (− ) = − ( ) = ( ) = = ∩ = ∩ ( ) = { ( ) ∣ ∩ } 68 Te Constructible Hierarchy

2044 X P  2045  X . = { ∣  ∩ } 2046 Similarly S S S X = ∩  = P 2047 − −S = ( ) = {P ( ) ∣ ∩ } 2048 − S (using (4)) = { (P) ∣ ∩ } 2049 S . = { ∣ ∩ } 2050 Tat C C is entirely the same. Q.E.D.(5) = ∩ 2051 (6) If then = . = ∩   2052 Proof: Note that− S  , S  S = (S) S S ↾  2053 − , S −  (⟨ ∣ < ⟩) =  ({⟨ ⟩∣ < }) 2054 − , S − = { (⟨ ⟩) ∣ < ( )} 2055 = −, S − since both , S L . P 2056 = {⟨ ( ) ( )⟩∣ < } {⟨ ⟩∣ < }  2057 Similar equalities hold for C  . Hence S , C  . Q.E.D.(6) 2058 Appealing to (2)and(4) we− have: − (⟨ ∣ < ⟩) (⟨⟨ ⟩∣ < ⟩) = S ↾ 2059 (7) L , “ S, C is the -least pair with P L 2060 (a) S ; (b) C and C cub in ; (c) C S S .” ⟨ Ď ⟩ ⊧ ⟨ Ď ⟩ < @ P 2061 As L L L and L is an end-extension of L and since (a)-(c) are absolute for transitive ZF ( ∩ ≠ ) 2062 models, we have that (a)-(c) are really true in V of S, C, i.e. : − < = (< ) < < 2063 (8) S, C is the L-least pair with 2064 (a) S ; (b) C and C cub in ; (c) C S S . ⟨ Ď ⟩ <Ď @ P 2065 Tat is, S, C really are the candidates to be chosen at the next, ’th, stage of the recursion: ( ∩ ≠ ) 2066 (9) S, C = S , C . 2067 Now note that C as C C is unbounded in the closed set C. Also, using (5), S S S . ⟨ ⟩ ⟨ P ⟩ 2068 Tis contradicts (1)! Q.E.D. = ∩ ∩ = = 2069 Exercise 4.30 (*) Formulate a principle which asserts similar properties for a sequence S where 2070 is any regular cardinal, and prove that it holds in L ◇ ⟨ ∣ < ⟩ 2071 Exercise 4.31 (**) Show that implies the existence of a family A  of stationary subsets of , such 2072 that the intersection of any two of them is countable. ◇ ⟨ ∣ < ⟩

2073 Theorem 4.42 (Jensen) implies the existence of a Suslin tree.

2074 Proof: We shall construct◇ by recursion a tree T of cardinality , using countable ordinals. In fact 2075 we shall have that T  itself, the construction thus delivers T . T will be the union of its levels T 2076 all of which will be countable, and T  T where (a) T T and (b) T is the tree = < 2077 ordering constructed so far on T . We shall ensure≤ that every T -branch is countable,< and likewise < = < < < = < < 2078 every maximal antichain. Ten T, T will⋃ be Suslin. Te recursion will⋃ ensure a normality condition: < < 2079 for every T, and if T , then for every  there is T with T ; every node then in P P ⟨ < ⟩ P 2080 the tree has tree-successors of arbitrary height below . < < < 2081 We let T  T  and T  . Assume Lim and T , T defned for all . Ten

2082 T T and T T . Normality as described above, is< then trivially conserved. ↾ = = { } < = ∅ ( ) < < < < ↾ = ⋃ < < = ⋃ < < Te Suslin Problem 69

2083 Assume now  and that Succ . We assume that T ,and T have been defned. We

2084 thus have defned T where  . For each T we allot in turn the next< sequence of ordinals = + ( ) P ↾ < 2085 available i i . (We thus go through T say by induction on the ordinals T and we defne + = P 2086 T by adding to the ordering T (which equals in the obvious sense T ) the pairs , i (and also { ∣ < } 2087 the< pairs , i for those T < to complete the ordering.) Tus at successor≤ stages of the tree it is < < < ⟨ ⟩ 2088 infnitely branching. Again normality≤ is obvious. Tis defnes T and T T . ⟨ ⟩ < 2089 Finally if  but Lim we need to defne T and make a careful≤ choice< of which maximal 2090 branches through T (thus those of order type that we may↾ extend with< impunity=< to have nodes at = + ( ) ↾ 2091 level , i.e. in T ,thusfxing T . Tis is where we use . < ) 2092 Case 1 S is a maximal< antichain in the tree so far defned: T , T . Ď < ◇ 2093 In this case for any T there must be some S with either T < or T . Either way ( ) P P ⟨ < < ⟩ 2094 by the normality of the tree T , T so far, we pick a branch b through

2103 enumerate T is a maximal antichain in T . < n  < n n + < 2113 C is thus unbounded in . Tat C is closed∩ is immediate. > Q.E.D. Claim  < + < < 2114 By our requisite property that ∩S is= a -sequence, now∩ that C is cub and A , there  Ď  2115 must be C with S A . Tus S is a maximal antichain in T . However at precisely this P ⟨ ∣ < ⟩ ◇ 2116 point in the construction we would have chosen T so that every element< of T , and so every element 2117 of A T , has a tree successor= ∩ at height in T . Note that all elements< of the tree at greater heights < 2118 are extensions of the tree above these elements on T . Tus A is a maximal antichain in ! < T 2119 But A∩ must be A and be countable! Contradiction! Q.E.D. > ∩ < 2120 Exercise∩ 4.32 (**) Show that implies the existence of two non-isomorphic Suslin trees.

2121 One could further ask whether◇ SH depends on CH. It is completely independent of CH as the fol- 2122 lowing states. 2123 Con ZF) implies the consistency of any of the following theories:

● ( 70 Te Constructible Hierarchy

2124 ZF+CH+SH; ZF+CH+ SH; ZF CH SH: ZF CH SH 2125 Te second of these is Cor. 4.39 above. Te other consistencies can be shown by using variations on 2126 Cohen’s forcing methods, for¬ which see+¬ [4]. Some+¬ of the+¬ arguments+ are very subtle.

Ronald Jensen 2127 2128 Appendix A

2129 Logical Matters

2130 A.1 The formal languages - syntax

2131 We outline formal frst order languages of predicate logic with axioms for equality. We do this for our 2132 language which we shall use for set theory, but it is completely general: P 2133 L = L 2134 (i) set variables; v, v,...,vn,... (for n N) P 2135 (ii) two binary predicates: ˙, ˙; an optional n-ary relation symbol Rv˙ v (other languages would P  n 2136 contain further function symbols F˙i and relations symbols R˙ j of diferent -arities). = ⋯ 2137 (iii) logical connectives: , 2138 (iv) brackets: (,) ∨ ¬ 2139 (v) an existential quantifer: . D 2140 2141 Aformulaisfnite string of our symbol set; the formulae of (‘Fml’) are defned inductively in a way 2142 similar for any frst order language. L 2143 1)x yandx yaretheatomic formulae where x, ystandforanyofthevariablesv , v . (If we opt for P i j 2144 variants where we have the relation or function symbols, then Rv vn and Fv vn vn  are also atomic.) = 2145 2) Any atomic formula is a formula; ⋯ ⋯ = + 2146 3)If and are formulae then so is and , x where x is any variable; D 2147 4) is only a formula if it is so by repeated applications of 1)-3). 2148 ¬ ( ∨ ) 2149 Inherent in the induction is the idea that a formula has subformulae and that a formula is built up 2150 from atomic formulae according to some fnite tree structure. Further, given the formula we may identify 2151 the unique tree structure. Indeed we think of this as an algorithm that given a symbol string tests whether 2152 it is a formula by winding the recursion backwards to try to discover the underlying tree structure. Using 2153 this fact we can then perform recursions over the class of formulae using the clauses 1)-3) as part of our 2154 recursive defnition. Clause 4) then ensures that our recursion will cover all formulae.

2155 Definition A.1 For aformulawedefne 2156 (A) the set of variables of , Vbl by:

( ) 71 72 Logical Matters

2157 Vbl v v Vbl v v v , v ; Vbl Rv v v , v ,...,v ; i j i P j i j  n   n 2158 Vbl Vbl ; Vbl Vbl Vbl ; Vbl x Vbl x . D 2159 (B)( the set= of) =free variables( )of= {, FVbl} would( ⋯ be) obtained= { exactly as} above but changing the clause 2160 for x (to:¬ FVbl)) = x ( ) FVbl(( ∨ x)) = ( ) ∪ ( ) ( ) = ( ) ∪ { } D D 2161 (C) is a sentence if FVbl . ( ) ( ) = ( ) − { } 2162 By the above remarks in (B)( ) we= have∅ defned the free variable set for all formulae. Note the crucial 2163 very fnal clause in B concerning the quantifer. Te set of ofcial logical connectives is minimal, it is D 2164 just and . But it is well known that the other connectives, , , can be defned in terms of them, as 2165 can ,from and (. We) shall use formulae freely involving these connectives, without comment. Here @ D 2166 is another¬ ∨ example. ∧ → ↔ ¬ 2167 Definition A.2 For a formula, we defne the set of subformulae of , Subfml ,by: 2168 Subfml v v Subfml v v Subfml Rv v ; i j i P j  n 2169 Subfml Subfml x Subfml ; ( ) D 2170 Subfml( = ) = Subfml( )Subfml= ( ⋯, ).= ∅ (¬ )) = ( ) = ( ) ∪ { } 2171 Deductive(( systems∨ )) = ( ) ∪ ( ) ∪ { } 2172 A deductive system of predicate calculus is (I) a set of axioms from which we can make pure logical 2173 deductions together with (II) those rules of deduction. Tere are many examples. Te following is the 2174 simplest to explain (but rather difcult to use naturally) but this allows us to prove things about the 2175 system as simply as possible. 2176 (I) Axioms of predicate calculus (for a language with relational symbols, and equality): 2177 For any variables x, y and any , , in Fml: 2178

2179

2180 → ( → ) 2181 ( x →x( → ))y x→ where(( →y is) →free( for→ x (this)) has a slightly technical meaning). @ 2182 (¬x → ¬ ) → ((¬ →x ) (where→ ) x Fr ) @ @ R 2183 x x( )x→ ( / ) @ 2184 x (y → ) →x, x( → x,)y ( ) 2185 (II)( Rules= ) of Deduction 2186 (1)=Modus→ ( Ponens( ).From→ ( )) and deduce: . 2187 (2) Universal Generalisation:From deduce x . @ 2188 ( → ) 2189 In general a theory is a set of sentences, T, in a language (such as ). A proof of a sentence is then P 2190 a fnite sequence of formulae: , , , n such that for any formula i on the list either: (i) i is L 2191 an instance of a pure axiom of predicate calculus; or (ii) i is in T; or (iii) i follows from one or more 2192 earlier members of the list by an application⋯ = of a deduction rule. 2193 In which case we shall say that the list is a proof from the set of axioms T,andwriteT .If 2194 T then we shall call this a proof in frst order logic alone. We shall want to be able to say that it is 2195 a mechanical, or algorithmic, process to check a proof . Given a fnite list which purports to be⊢ a proof, = ∅ 2196 it is indeed a mechanical process to check (i) or (iii) for any i on the list. In order to ensure the whole Semantics 73

2197 process is algorithmic it is usual (and overwhelmingly the case) that the set of axioms T is either fnite 2198 itself, or for any formula there is an algorithm or recursive process which can decide whether is in 2199 T or not. In which case we say that T is a recursive set of axioms.

2200 A.2 Semantics

2201 We have defned so far only syntactical concepts. We have not associated any meaning,orinterpreta- 2202 tion to our language. We want to know what it means for a sentence to be true (or false) in an interpre- 2203 tation. If I wish to express the commutative law in group theory say, then I may write something down 2204 such as x y x, y y, x (with a binary function symbol v , v ). For this to be true in the @ @ i j 2205 group G say, we need that for any interpretation of the variables x, y as group elements g, h in G that 2206 g.h h.g holds(○( for the) = group○( multiplication.)) ○( ) 2207 We can give a recursive defnition of what it means for a sentence to be ‘true in a structure’,but as can = 2208 be seen, the recursion involves at the same time defning satisfaction of a formula by an assigment of 2209 elements to the free variables of , again by recursion on the structure of formulae. We’ll keep with the  2210 example of a group G, e, , for a language containing a binary function symbl F , a unary function  2211 symbol I and a constant symbol− E which are interpreted as , ,ande respectively. Ten ‘I v j vk’and ⟨ ⋅ ⟩ ○ 2212 ‘F vi , v j vk’ now count as atomic formulae. − Vbl ⋅ ( ) = 2213 We let QG G be the set of maps from fnite sequences of variables of the language to G to G. ○( ) = < 2214 For a formula let Vbl , be the set of all variables occurring in .Forh Q , v dom h and P G i P 2215 g G we let h =g i be the function that is defned everywhere like h except that h g i v g. P i 2216 ( ) ( ) ( / ) ( / )( ) = 2217 Definition A.3 (i) We defne by recursion the term Sat , G ; 2218 Sat vi v j, G h QG h i h j ; P  ( ) 2219 Sat I v j vk, G h QG h j h k ( = ) = { P ∣ ( ) = ( )} 2220 Sat F vi , v j vk, G h QG h− i h j h k ; ( ( ) = ) = { P∣ ( ) = ( )} 2221 Sat , G Sat , G Sat , G h QG dom h Vbl Vbl ; ( ○( ) = ) = { ∣ ( ) ⋅ ( ) = P( )} Ě 2222 Sat , G QG Sat , G ; ( ∨ ) = ( ( ) ∪ ( )) ∩ { ∣ ( ) { ( ) ∪ ( )}} 2223 Sat vi , G h QG dom h Vbl vi & g G h g i Sat , G ; (¬D ) = / P ( )} Ě D P P 2224 Sat u, G if u is not a formula. ( ) = { ∣ ( ) ( ) ∪ { } ( ( / ) ( ))]} 2225 (ii) We write G, e, , h if h Sat , G . P ( ) = ∅ −   2226 Note: By design⟨ then we⋅ have⟩ ⊧ [G,]e, , ( h )if it is not the case that G, e, , h etc. (We  2227 write the latter as G, e, , h .) − − ⟨ ⋅ ⟩ ⊧ ¬ [ ] ⟨ ⋅ ⟩ ⊧ [ ] 2228 If is a sentence then− we write  ⟨ ⋅ ⟩/⊧ [ ]  2229 G, e, , if for some h QG with dom h Vbl G, e, , h P Ě  2230 − (equivalently for all h QG with dom h Vbl G, e−, , h ). ⟨ ⋅ ⟩ ⊧ P ( ) Ě ( ) ⟨ ⋅ ⟩ ⊧ [ ] 2231 If T is a set of sentences in a language, and A is a structure appropriate for− that language, we write ( ) ( ) ⟨ ⋅ ⟩ ⊧ [ ] 2232 A Tif for all in T A .

⊧ ⊧ Definition A.4 (Logical Validity) Let T be a theory in a language; then T if for every structure

∪{ } ⊧ 74 Logical Matters

A appropriate for the language, A T A .

Theorem A.5 (Godel¨ Completeness Teorem)⊧ Predicate⇒ ⊧ Calculus is sound, that is,

T T .

2233 and is moreover complete,thatis T T⊢ ⇒. ⊧

2234 Te substantial part here is the Completeness⊧ ⇒ ⊢ direction: it is an adequacy result in that it shows that 2235 Predicate Calculus is sufcient to deduce from a theory T all those sentences that are true in all structures 2236 that are models of a particular theory. Tat it deduces only those sentences true in all structures satisfying 2237 the theory, is the soundness direction. 2238 Tis theorem should not be confused with Godel’s¨ Incompleteness Teorems. Tese concerned whether 2239 sets of axioms T were consistent, that is whether from the axioms of T we cannot prove a contradiction 2240 such as ‘ ’. Here if we take T to be PA - Peano Arithmetic the accepted set of axioms for the natural 2241 number structure N N, , Succ , then Godel¨ showed that there was a suitable mapping x y tak- = 2242 ing formulae in the language appropriate for N, into code numbers of these formulae (called godel¨ codes). = ⟨ ⟩ ↣ 2243 If formulae could be coded as elements of N,socanafnite list of formulae - in other words potential 2244 proofs. Using the fact that the axioms of PA are capable of being recursively listed, he showed that there 2245 was a formula defning a function on pairs of numbers, F n, k ,to  with: 2246 ( ) / 2247 PA n kF n, k ,  @ @ P 2248 F n, k  n is a code number of a proof from PA of the formula with x y k. 2249 ⊢ ( ) { }∧ ( ) = ⇔ = 2250 He then showed that if PA is consistent, then in fact PA nF n, x y . Te right hand @ 2251 side here is a statement about F and numbers, but has the interpretation that “PA is a consistent system 2252 (in other words that ‘  is not deducible’). Tis is commonly⊢/ abbreviated( = ‘Con) =PA ’; so he showed 2253 that even if PA is consistent, PA Con PA (the Second Incompleteness Teorem). In fact the theorem 2254 has wider applicability as= he noted afer considering Turing’s work on computability:( for) any consistent, 2255 computably given set of axioms T⊢/ say, if( from) T we can deduce the Peano axioms, then T Con T . 2256 Te axioms of set theory ZF, if consistent, are of course such a T. ⊢/ ( )

ni 2257 Exercise A.1 Let x be any set, and fi V V for i be any collection of fnitary functions (meaning n 2258 that n ; show that there is a y x which is closed under each of the f (thus f “ iy y for each i)and i Ě i i Ď 2259 y max , x . [Hint: no need for a∶ formal%→ argument here:< build up a y in many stages y y at each k Ď k  2260 step applying< ) all the fi . ∣ ∣ ≤ { ∣ ∣} + ] Definition A.6 Let A A, , Ri , Fj be any structure for any (frst order) language A.WewriteB ă A (“B is an elementary substructure%→ of%→ A”), where B B, , Ri B, Fj B , to mean that every formula = ⟨ = ⟩ L v,...vn  of the language of A, and every n-tuple of elements%→ y%→,... ,yn  from B, then = ⟨ = ↾ ↾ ⟩ − − ( ) A y vL,... ,yn  vn  B y v ,... ,yn  vn  .

⊧ [ / − / − ] ⇔ ⊧ [ / − / − ] A Generalised Recursion Teorem 75

2261 Te Tarski-Vaught criterion yields when one substructure B is an elementary substructure of A.

Lemma A.7 (Tarski-Vaught criterion) B ă A if for all formulae v,...,vn ,

b ,...,b B a AA a, b b B B ( a, b .) @  n P D P D P ( ⊧ ( ⃗) → ⊧ ( ⃗))

Definition A.8 (Skolem Function) Let x x, y ,...,y be any formula in the language A appro- D  n priate for the structure A. Suppose there is a wellorder of the domain A. Te skolem function h for is the (partial) function: ( ) L ◁

h y,...,yn the -least x such that x, y,...,yn .

2262 Notice that there are as( many skolem) ≈ functions◁ as formulae in the( language -) which will be countable 2263 in the cases of interest to us. Te following theorem will be used in applications.

2264 Theorem A.9 Lowenheim-Skolem¨ Teorem Let A be any infnite structure for any language as above 2265 of cardinality .SupposeX A. Ten there is a elementary substructure B of A , B ă A,withX B Ď Ď Ď 2266 A B max X , .

Proof∧ ∣ ∣:=Te idea{∣ is∣ to} fnd the closure of X under the fnitary skolem functions h . Let H be the set of such functions. Ten H we are told is . Let X X, and let

∣ ∣ Xn  h “Xn =h H ; Y Xn. P n + = ⋃{ ∣ } = ⋃< 2267 Te idea is that by closing up in this way we have ensured that the Tarski-Vaught criterion can be applied. 2268 However Xn  Xn X . Hence B n Xn satisfes B X max , X . Now if we 2269 take any y,...,yn B we shall have that y,...,yn Xm for some m . But then if A z, y then ∣ + ∣ = P⊗ ∣ ∣ = ⊗ ∣ ∣ = P ∣ ∣ = ⊗ ∣ ∣ = { ∣ ∣} 2270 x Xm  B x, y . ⋃ Q.E.D. D P < ⊧ ( ⃗) + ( ⊧ ( ⃗)) 2271 Corollary A.10 Any infnite structure A has a countable substructure B ă A.

2272 A.3 AGeneralised Recursion Theorem

2273 Definition A.11 If A, R is a partial order, we let A y y A yRx .Wesometimeswrite x df P 2274 Ax pred A,R x if we wish to be clear about which order on A is concerned. ⟨ ⟩ = { ∣ ∧ } ⟨ ⟩ 2275 =Ax is thus( the) set of R-predecessors of x that are in A.

2276 Definition A.12 If A, R is a wellfounded relation, R is said to be is set-like on A, if for every x A, P 2277 Ax df pred A,R x df y y A yRx is a set. ⟨ ⟩ P = ⟨ ⟩( ) = { ∣ ∧ } 76 Logical Matters

One can prove a recursion theorem for wellfounded relations, but observe that such relations are not necessarily transitive orderings. We remedy this by defning R˚ -thetransitive closure or transitivisation of R in A, where for x, y A we want to put P

xR˚ y xRy n  z A,...,z A xRz Rz Rz Ry . df D D  P n P   n

2278 Tis is a somewhat informal↔ defnition,∨ but> the intention is clear:( xR˚ y if⋯ there is) a fnite R-path using 2279 elements from A from x to y.

2280 Definition A.13 A, R be a relation. For x Awedefne R x df z x Az.Welet P  n  n P 2281 x A ; x A z x . R x R z P R ⟨ + ⟩ n ⋃ = ⋃ 2282 For x, y AwesetyR˚xif y R x n .R˚ is called the ancestral or transitive closure of R. ⋃ = P ⋃ = ⋃ { ∣ P ⋃ } P n  2283 Te reader should check that a) ⋃{⋃x ∣ y }A z A z A yRz Rz Rz Rx , and b) R P D  P D n P   n 2284 with R as the -relation itself y ˚ x +y TC x . P P ⋃ P = { ∣ ⋯ ( ⋯ )} ↔ ( ) 2285 Lemma A.14 Let A, R be a relation. Ten: 2286 (i) R˚ is transitive on A. If x A then R˚ is transitive on Ax x . ⟨ ⟩ P 2287 (ii) If R is set-like, then so is R˚. ∪ { }

2288 Proof: (i) is obvious. (ii) We frst note that R z is a set; this is because z is a set and R is set-like on 2289 A which implies that Ax is a set for each x z, and the Axiom of Unions allows us to conclude that P⋃ n  2290 x z Ax V. Hence by induction, so is each R z, and then another application of Replacement and P P n 2291 Union ensures that x n V; but this+ latter set is then the set of R˚ predecessors of x. R P P 2292 ⋃ ⋃ Q.E.D. ⋃{⋃ { }∣ }

2293 Theorem A.15 (Transfnite Induction on Wellfounded Relations). Suppose A, R is a wellfounded relation, 2294 with R set-like on A. Let t A be non-empty class term. Ten there is u t which is R-minimal amongst Ď P 2295 all elements of t. ⟨ ⟩

2296 ˚ ˚ Proof: Let Ax df pred A,R˚ x be the set of R -predecessors of x. Note this is a set and is a subset of 2297 A. Let x be any element of t, and let u be an R-minimal member of the set t A˚ x . Q.E.D. ⟨ ⟩ x 2298 = ( ) ( ∩ ) ∩ { } 2299 One should note that we do need to prove the above theorem, since the defnition of A, R being 2300 wellfounded (Def. 1.14) entails only that every non-empty set z A has an R-minimal element. Te Ď 2301 theorem then says that this holds for classes t too. ⟨ ⟩

2302 Theorem A.16 (Generalized Transfnite Recursion Teorem) Suppose A, R is a wellfounded relation, with R set-like on A.IfG V V V then there is a ˆ unique function F A V satisfying: ⟨ ⟩ ∶ → ∶ → xF x G x, F A . @ x ( ) = ( ↾ ) A Generalised Recursion Teorem 77

2303 Proof: We shall defne G as a union of approximations where u V is an approximation if (a) Fun(u); P 2304 (b) dom u A is R-transitive - meaning y dom u A˚ dom u ; and (c) y dom u u y Ď P y Ď @ P 2305 G y, u A . We call an approximation u an x-approximation if x dom u .Sou satisfes the defn- y P 2306 ing clauses( ) for F throughout its domain. Notice that( ) if→u is an x-approximation,( ) then v is( also) ( an) =x ( ↾ ) ( ) 2307 approximation, where v u x A˚x. (It is the smallest part of u which is still an x-approximation.) 2308 (1) If u and v are approximations, and we set t dom u dom v then u t v tandisan 2309 approximation. = ↾ { } ∪ 2310 Proof: Note that for any y t, A˚ t so t is R-transitive.= Let( )Z∩ y ( t) u y ↾v y= . ↾ P y Ď P If Z let w be an R-minimal element of Z (by the wellfoundedness of R). Ten u Aw v Aw, hence: = { ∣ ( ) ≠ ( )} ≠ ∅ ↾ = ↾ u w G w, u Aw G w, v Aw v w .

2311 Tis contradicts the choice( of) w= .So( Z ↾ and) =u, v( agree↾ on )t,= the( common) part of their domains. 2312 Tis fnishes (1). Exactly the same argument establishes: = ∅ 2313 (2) (Uniqueness) If F, F are two functions satisfying the theorem then F F. 2314 (3) (Existence) Such an F exists. 2315 Proof: Let u B u u is an approximation . B is in general a proper= class of approximations, P 2316 but this does not matter as long as we are careful. As any two such approximations agree on the common 2317 part of their domain,⇔ we may{ ∣ defne F B and obtain:} 2318 (i) F is a function ; 2319 (ii) dom F A. = ⋃ Proof (ii): Let C be the class of sets z A for which there is no z-approximation. So if we suppose for a P contradiction( that) = C is non-empty, by Teorem A.15, then it will have an R-minimal element z such that y A u u is a y-approximation). But now we let f be the function: @ P zD y y y ( f y A f is a y-approximation dom f y A˚ . P z y By (1) for a given⋃y such{ ∣ an f y is∧ unique, and moreover the∧f y all( agree) = on{ the} ∪ parts} of their domains they have in common. Note that the domain of f is R-transitive, being the union of R-transitive sets y dom f for y z. Hence A˚ dom f and thus z dom f is also R-transitive. We can extend f P z Ď to ( ) ( z) { } ∪ ( ) f f z, G f Az z 2320 and f is then a z-approximation. However we assumed that z C, contradiction! Hence C and = ∪ {⟨ ( ↾ )⟩}P 2321 (ii) holds. Q.E.D. 2322 = ∅ 2323 For some applications it is useful to note that the AxPower was not used in the proof of this the- 2324 orem, and it can be proved in ZF .For A, R a wellfounded relation, we can defne a rank function 2325 A On by appealing to the− last theorem: x sup y  y A yRx . Clearly A,R A,R A,R P 2326 this satisfes xRy x y⟨,and⟩ x is onto On if A is a proper class, or an initial ⟨ ⟩ A,R A,R A⟨,R ⟩ ⟨ ⟩ 2327 segment∶ of→On, i.e. an ordinal, if A V. ( ) = { ( ) + ∣ ∧ } → ⟨ ⟩( ) < ⟨P ⟩( ) ⟨ ⟩( ) 2328 Exercise A.2 If A, R a wellfounded set-like relation, x A, and x ,showthat y y P A,R @ D P 2329 A yR˚x A,R y . ⟨ ⟩ ⟨ ⟩( ) = < ( ∧ ∧ ⟨ ⟩( ) = ) 78 Logical Matters

2330 Exercise A.3 If A, R a wellfounded set-like relation, show that A,R is (1-1) if and only if R˚ is a total order.

2331 Exercise A.4 (i) If A, R a wellfounded set-like relation, and B ⟨ A⟩,showthat x x for any ⟨ ⟩ Ď B,R A,R 2332 x B. Show that additionally equality holds if A˚ B where A˚ is as in the proof of Teorem A.15 above. P x Ď x ⟨ ⟩ ⟨ ⟩ 2333 (ii) If A, R , A,⟨S are⟩ wellfounded set-like relations, and S R,showthat (x ) ≤ (x) for any Ď A,S A,R 2334 x A. P ⟨ ⟩ ⟨ ⟩ ⟨ ⟩( ) ≤ ⟨ ⟩( ) 2335 Bibliography

2336 [1] K. Devlin. Constructibility. Perspectives in Mathematical Logic. Springer Verlag, Berlin, Heidelberg, 2337 1984. 36, 66

2338 [2] F.R. Drake. Set Teory: An introduction to large cardinals, volume 76 of Studies in Logic and the 2339 Foundations of Mathematics. North-Holland, Amsterdam, 1974. 36

2340 [3] T. Jech. Set Teory. Springer Monographs in Mathematics. Springer, Berlin, New York, 3 edition, 2341 2003. 36

2342 [4] K.Kunen. Set Teory: an introduction to independence proofs. Studies in Logic. North-Holland, 2343 Amsterdam, 1994. 3, 70

79 80 BIBLIOGRAPHY 2344 Index of Symbols

2345 , ˙ , , 4 2376 V , 10 P A 2346 ˙, ˙, 4 2377 ∆, 11 LP L L ⃗ 2347 FVbl, 4 2378 Σn, Πn, 11 = 2348 y x , 4 2379 T, 11 W W 2349 x , 5 2380 , t , 12 ( / ) n 2350 V, 5 2381 ⋀⋀ , 15 { ∣ } 2351 , 5 2382 , 15 < 2352 , 5 2383 cf∗ , 18 ∅Ď < 2353 , , 5 2384 Reg, Sing, Card, LimCard, 18 ( ) 2354 s, 5 2385 , 21 ∪ ∩ 2355 s t, 5 2386 , 24 ¬ ℶ 2356 , 5 2387 , 24 / < 2357 , 5 2388 ∑H , 27 ⋃ ∏ < 2358 t,...,tn , 6 2389 HF, H , 28 2359 ⋂ 2390 x, y , 6 HC, H  , 29 { } 2360 x, x,...,xn , 6 2391 T, T , 37 ⟨ ⟩ 2361 x z, 6 2392 x y, 45 ⟨ ˆ ⟩ ⟨ < ⟩ 2362 , 6 2393 Fml, 46 2363 dom r , ran r , feld r , 6 2394 Fmla, 46 P 2364 r“u, 6 2395 Qx, 46  ( ) ( ) ( ) 2365 r , 6 2396 Sat u, x , 47 2366 r− s, 6 2397 , 47 ( ) 2367 Fun, 7 2398 Def, 47 ○ ⊧ 2368 f a b, f a   b, 7 2399 Def, 48 2369 f a b, f a b, 7 2400 OD, 48 onto ( − ) a ∶ %→ ∶ %→ 2370 b, 7 2401 xZFy, xZFCy, 49 ∶ %→ ∶ ←→ 2371 f , 7 2402 Con T , 50  T 2372 ZF , 8 2403 Con , 50 ( ) 2373 Trans∏ , 9 2404 L , L, 54 2374 TC, 10 2405 L x , 54 2375 , 10 2406 IM, 55 ( ) 81 82 INDEX OF SYMBOLS

2407 2417 Qx , 57 SH, 66 2408 , L, 57 2418 , 67 2409 < V L, 57 2419 Vbl, 72 2410

2427 absolutely defnite, a.d., 40 2457 Collapsing Lemma 2428 absoluteness 2458 General, 27 2429 upward, downward, 31 2459 Mostowski-Shepherdson, 26 2430 Aronszajn trees, 37 2460 Condensation Lemma, 59 2431 assignment function, 46 2461 conservative theory, 8 2432 Axiom of Choice, 2 2462 consistency of AC, 58 2433 Axiom of Choice 2463 consistency of GCH, 59 2434 in L, 56 2464 consistency of SH, 66 2435 Axiom of Constructibility in L, 57 2465 constructible hierarchy, L, 54 ¬ 2466 constructible rank, L x , 54 2436 beth function, 21 2467 Continuum Hypothesis, CH, 1 2468 Correctness theorem, (49) 2437 Cantor, G, 1 2469 countable chain condition, c.c.c., 66 2438 cardinal 2470 cumulative hierarchy, V , 10 2439 inefable, 36, 37 2440 regular, 18 2471 deductive system, 72 2441 strong limit, 35 2472 defnability function Def, 47 2442 strongly Mahlo, 36 2473 defnite term and formula, 40 2443 subtle, 37 2474 diagonal intersection, 21 2444 sum, product, 24 2475 diamond principle, , 67 2445 weakly compact , 36 2446 weakly inaccessible, strongly inaccessible, 2476 Fodor’s Lemma, 23 ◇ 2447 35 2477 formula 2448 weakly Mahlo, 36 2478 ∆ , 11 2449 Cartesian product, 6 2479 of , 4, 71 W 2450 closed and unbounded, c.u.b. 2480 relativisation of, , 12 2451 flter, 22 2481 free variable,L 4 2452 set, 19 2482 function 2453 closed formula, 4 2483 (1-1), 7 2454 closed term, 5 2484 Fun, 7 2455 cofnality, 18 2485 bijection, 7 2456 Cohen, P, 2 2486 cofnal, 17

83 84 INDEX

2487 normal, 20 2520 rank function, , 10 2488 n-ary, 7 2521 Refection Teorem, Montague-Levy, 33 2489 onto, 7 2522 relation 2523 n-ary relation, 6 2490 Godel¨ code sets, 45 2524 extensional, 26 2491 Godel’s¨ Completeness Teorem, 74 2525 relativised constructible hierarchies, L A , 64 2492 Godel’s¨ Second Incompleteness Teorem, 50 2526 relativised constructible hierarchies, L A , 65 2493 generalised cartesian product, 7 2527 Richard’s Paradox, 60 ( ) 2494 Generalized Transfnite Recursion Teorem, 77 [ ] 2495 global wellorder of L, L, 57 2528 satisfaction relation, 47, 73 2529 scheme of -induction, 9 P 2496 hereditarily countable,

2502 inner model, IM, 55 2536 Tarski-Vaught criterion, 31, 75 2537 term 2503 Konig’s¨ Teorem, 24 2538 class term, 5 2539 term 2504 Levy hierarchy, 11 W 2540 relativisation of, t , 13 2505 logical connectives, 4 2541 Transfnite Induction on Wellfounded 2506 logical validity, 73 2542 Relations, 76 2507 Lowenheim-Skolem¨ Teorem, 75 2543 transfnite recursion along , 9 P 2544 transitive 2508 ordered n-tuple, 6 2545 -model, 11 2509 ordered pair, 6 P 2546 set, Trans, 9 2510 ordinal defnable, OD, 48, 60 2547 tree property, 37 2511 outright defnable, 48

2548 Urelemente, 9 2512 power set operation, 6 2549 ultraflter, 22 2513 quantifer 2550 von Neumann-Godel-Bernays,¨ NBG, 8 2514 bounded, 11 2515 quantifer 2551 wellfounded relation, 10 2516 existential, 4 2517 frst order, 4, 8 2552 Zermelo-Fraenkel Axioms, 7 2518 second order, 8 2553 Zermelo-Fraenkel Axioms 2519 unicity, , 7 2554 not fnitely-axiomatisability, 45