GÖDEL S CONSTRUCTIBLE UNIVERSE

Transcription

1 GÖDEL S CONSTRUCTIBLE UNIVERSE MICHAEL WOLMAN Abstract. This paper is about Gödel s Constructible Universe and the relative consistency of Zermelo-Fraenkel set theory, the Continuum Hypothesis and the Axiom of Choice. It was written as part of an undergraduate research project course, and follows Jech s approach to the subject in [1]. Some of the proofs are my own, but many of the proofs are taken from Jech. I have, however, added my own explanations and commentary regarding the ideas and motivations behind many of the definitions and proofs, and have added some details as necessary. 1

2 2 Contents 1. Preliminaries Overview of Cardinal Arithmetic Cofinality and Konig s Theorem The Continuum Hypothesis and its Implications The Axiom of Choice 6 2. A Brief Intro to Model Theory Basic Model-Theoretic Notions and Results Models of Set Theory The Lévy Hierarchy Gödel s Constructible Universe The Constructible Sets The Absoluteness of Constructibility The Continuum Hypothesis and the Axiom of Choice Notes on Relative Consistency Proofs 27 References Preliminaries In this section we will briefly go over cardinal arithmetic, the statements of the Continuum Hypothesis and the Axiom of Choice, and some of their consequences Overview of Cardinal Arithmetic. Definition 1.1. We say that two sets have the same cardinality, X = Y if there exists a bijective mapping from X to Y. We call an ordinal α a cardinal number if α β for all β < α. Definition 1.2. Now, we define the alephs, which are the cardinal numbers, as follows: ℵ 0 = ω 0 = ω, ℵ α+1 = ω α+1 = ℵ + α, ℵ α = ω α = sup {ω β β < α} for limit ordinals α where α + is the least cardinal greater than α. Definition 1.3. Arithmetic operations on cardinal numbers are defined as: κ + λ = A B where A = κ, B = λ, and A, B are disjoint, κ λ = A B where A = κ, B = λ, κ λ = A B where A = κ, B = λ.

3 GÖDEL S CONSTRUCTIBLE UNIVERSE 3 The following facts are true for cardinal arithmetic: + and are associative, commutative, and distributive (κ λ) µ = κ µ λ µ κ λ+µ = κ λ κ µ (κ λ ) µ = κ λ µ If κ λ, then κ µ λ µ If 0 < λ µ, then κ λ κ µ κ 0 = 1; 1 κ = 1; 0 κ = 0 for κ > 0 Lemma 1.4. If A = κ, then P (A) = 2 κ. Proof. For every X A, let χ X be the the characteristic function on X. Then the function f taking X P (A) to χ X is a bijection from P (A) to {0, 1} A. Theorem 1.5 (Cantor). For every set X, X < P (X). Proof. Let f : X P (X). The set Y = {x X x / f(x)} is not in the range of f: if z X such that f(z) = Y, then z Y iff z / Y, a contradiction. Therefore there is no surjection from X to P (X), and X P (X). Also, f(x) = {x} is one-to-one, and so X P (X). Therefore, we can see that for every cardinal number ℵ α, and so ℵ α < 2 ℵα, ℵ α+1 2 ℵα Cofinality and Konig s Theorem. Let α > 0, β be limit ordinals. We say an increasing β-sequence {α ξ ξ < β} is cofinal in α if lim ξ β α ξ = α. Definition 1.6. If α is an infinite limit ordinal, the cofinality of α is defined as cf α = the least limit ordinal β such that there is a β-sequence that is cofinal in α. Definition 1.7. We say an infinite cardinal ℵ α is regular if cf ω α = ω α. It is singular if cf ω α < ω α. Definition 1.8. Let {κ i i I} be an indexed set of cardinal numbers. We define κ i = X i, i I i I κ i = Y i, i I where {X i i I} is a disjoint family of sets such that X i = κ i for all i I, and {Y i i I} is a (not necessarily disjoint) family of sets such that Y i = κ i for all i I. i I

4 4 MICHAEL WOLMAN Lemma 1.9. If λ is an infinite cardinal and κ i > 0 for all i < λ, then κ i = λ sup κ i. i<λ i<λ In particular, if λ sup i<λ κ i, we have κ i = sup κ i. i<λ i<λ Proof. Let κ = sup i<λ κ i and σ = i<λ κ i. On one hand, since κ i κ for all i, we have σ i<λ κ λ κ. On the other hand, since κ i 1 for all i, we have λ = i<λ 1 σ, and since σ κ i for all i, we have σ sup i<λ κ i = κ. Therefore σ λ κ. Lemma If λ is an infinite cardinal and {κ i i < λ} is a non-decreasing sequence of non-zero cardinals, then κ i = (sup κ i ) λ. i<λ Proof. Let κ = sup κ i. Since κ i κ for all i < λ, we get that κ = κ λ. i<λ κ i i<λ Now, consider a partition of λ into λ disjoint sets of size λ, λ = A j. For every j < λ, so we get that i<λ κ i = j<λ j<λ i A j κ i sup i A j κ i = κ, i A j κ i j<λ κ = κ λ. Theorem 1.11 (König). If κ i < λ i for every i I, then κ i < λ i. i I i I Proof. We will show that i κ i i λ i. Let T i, i I be such that T i = λ i for each i I. It is enough to show that if Z i, i I are subsets of T = i I T i and Z i κ i for all i I, then i I Z i T. For every i I let S i be the projection of Z i onto the ith coordinate, S i = {f(i) f Z i }. Since Z i < T i, we have S i T i. Now let f T be a function such that f(i) / S i for all i I. Obviously f / Z i for all i I, and so i I Z i T. Corollary κ < 2 κ for every κ. This is the same as theorem 1.5 above.

5 GÖDEL S CONSTRUCTIBLE UNIVERSE 5 Proof < }{{}} 2 2 {{.. }.. κ times κ times Corollary cf(2 ℵα ) > ℵ α. Proof. It is enough to show that if κ i < 2 ℵα for i < ω α, then i<ω α κ i < 2 ℵα, because then max{ℵ α, sup i<ωα κ i } < 2 ℵα and so sup i<ωα κ i < 2 ℵα. Let λ i = 2 ℵα. Then κ i < λ i = (2 ℵα ) ℵα = 2 ℵα. i<ω α i<ω α 1.3. The Continuum Hypothesis and its Implications. The Continuum Hypothesis (CH) is the statement that there is no cardinal between ℵ 0 and 2 ℵ0, or equivalently that every subset of R is either countable or equinumerous to R. More formally, CH is the statement that 2 ℵ0 = ℵ 1. Kurt Gödel proved in 1940 that CH is consistent with ZFC, the standard axioms of set theory. His proof consisted of creating a model of set theory by taking as a universe all of the constructible sets, and showing that CH is true in this universe. This proof will be the topic of section 2 of this paper. In 1963, Paul Cohen came up with a new method called forcing to construct models of ZFC where CH is false. This proved that the Continuum Hypothesis is a statement independent of the axioms of set theory. In fact, Easton showed that the only restriction on the cardinality of continuum (or of 2 κ for any κ) is that it must not contradict König s theorem (see ), i.e. cf c > ω. One of the consequences of CH is that we can order the real numbers such that any initial segment is countable. If CH is true, then by the well-ordering theorem, we can write R = {a α α < ω 1 }, and then for any β < ω 1, the set {a α α < β} is countable. Using this, we can construct a subset of R 2 that is measurable along every horizontal and every vertical slice, but isn t itself measurable. To do this, let I = [0, 1] be the unit interval, I = {a α α < ω 1 } a well-ordering of I. Then, consider the set E = {(x, y) I I x < y}, where we say x < y if x = a α, y = a β for some α, β ω 1, and α < β. Then for any a I, the set {(x, a) x < a} is countable and therefore has measure 0. In addition, the set {(a, y) a < y} has countable complement, and so since it differs from I by a null set, it has measure 1. Now let f be the indicator function on E. We then have that I I ( ( I I ) f(x, y)dy dx = ) f(x, y)dy dx = 1dx = 1 I 0dx = 0. I

6 6 MICHAEL WOLMAN Therefore, by Fubini s theorem, E is not measurable, although all its horizontal slices have measure 0, and its vertical slices have measure 1. Another interesting construction that requires CH is the Sierpiński set, an uncountable set of real numbers whose intersection with any null set is countable. To construct this set, take a collection of c-many sets of measure 0 such that any set of measure 0 is contained in one of them. Enumerate this set {X α α < ω 1 }. Then, for every β < ω 1, pick a β R \ α<β X α, such that a β a γ for γ < β (this is justified because α<β X α has measure 0, and so R \ α<β X α is uncountable, and in particular it is non-empty). Let A = {a α α < ω 1 }. Then A is uncountable, and for any null set X, X X α for some α < ω 1, and so A X A X α {a β β < α}. Since α < ω 1, A X is countable as desired The Axiom of Choice. If S is a family of sets, a choice function f on S is a function such that for every X S. f(x) X Definition 1.12 (Axiom of Choice). Every family of non-empty sets has a choice function. The Axiom of Choice (AC) is interesting because unlike the other axioms in ZFC, it tells us there exists a set without defining the set explicitly. Because of this, it is often interesting to see which statements can or can t be proven without it. Like the Continuum Hypothesis, the Axiom of Choice has been shown by Gödel and Cohen to be independent of ZF, and so many theorems are unprovable in ZF without AC. The Axiom of Choice has many equivalent forms in ZF, notably Zorn s Lemma and the Well-Ordering theorem. Theorem 1.13 (Well-Ordering Theorem). Every set has a well-ordering. Proof. Given a set X, consider a choice function f on P (X). Then, define inductively x α = f(x \ {a β β < α}) as long as X {a β β < α}. Then we can enumerate X = {a α α < ξ} where ξ is the smallest ordinal such that X = {a α α < ξ}. The well-ordering theorem has many relevant consequences. For example it allows us to define cardinality for all sets. It also allows us to create sets with many interesting properties, such as the Vitali set and the sets constructed in 1.3. In fact, any example of a non-measurable set will require AC to construct, because it is consistent with ZF minus AC that every subset of R is measurable. Another interesting set we can construct using AC is a subset A of R where neither A nor its complement contain a perfect set. To do this, first let P = {P α

7 GÖDEL S CONSTRUCTIBLE UNIVERSE 7 α < c} be an enumeration of the perfect sets. Then we define inductively, for α < c, a α P α \ ({a β β < α} {b β β < α}) b α P α \ ({a β β α} {b β β < α}). Then for any perfect set P α, and and so A = {a α α < c} is as desired. b α P α \ A, a α P α \ A c, Definition If P is a partially ordered set, then a chain in P is a totally ordered subset of P. Now we are ready to state another equivalent form of AC, Zorn s Lemma. Lemma 1.15 (Zorn s Lemma). Let P be a partially ordered set such that every chain has an upper bound in P. Then P contains at least one maximal element. One immediate consequence of this is the following: Theorem Every vector space has a basis. Proof. Let V be a vector space, P the set of all linearly independent subsets of V partially ordered by inclusion. Then any chain C in P has an upper bound in P (take C) and so has a maximal element, which is a basis for V. An interesting construction we can make from this is a set A R such that A is measurable but A + A = {a + b a, b A} isn t. To do this, we first find a Hamel basis of measure 0. A Hamel basis is a basis of R when considered as a vector space over Q. The existence of such a basis is guaranteed by the previous theorem, and we note that the dimension of this vector space is c. To find a Hamel basis of measure 0, consider the Cantor Set C. Since C + C = [0, 2], we see that C spans R. But then C contains a basis H for R and since C is a null set, so is H. Now define the following sets inductively: A 0 = {qh q Q, h H} A n+1 = A n + A n. Because H is a Hamel basis, we know that n ω A n = R. We also have that A 0 is a null set as it is the countable union of null sets. Since R has non-zero measure, we know that not every A n is null. Let N be the smallest number such that A N is not null. Then A N = A N 1 + A N 1 (since N 1), and A N 1 is measurable since it is null, so we only have left to show that A N is not measurable. By the definition of A N, we know that A N+1 = A N + A N = A N A N. If A N is measurable then by assumption it has positive measure. By the Steinhaus theorem, A N A N = A N+1 contains an interval. But then A N+1 = R, contradicting the fact that H is linearly independent. So A N is not measurable, as desired.

8 8 MICHAEL WOLMAN Proof of Zorn s Lemma. Using a choice function for non-empty subsets of P, construct a chain in P that leads to a maximal element of P by induction, letting a α = an element of P such that a α > a ξ for all ξ < α if one exists. If α > 0 is a limit ordinal, then C α = {a ξ ξ < α} is a chain and so a α exists by assumption. Then there is eventually a β such that there is no a β+1 P, a β+1 > a β. Then a β is a maximal element of P. 2. A Brief Intro to Model Theory In the following section, we will be working with models of set theory in order to show the relative consistency of ZF along with AC and CH. In order to do so, we must first establish some definitions and basic theorems regarding models Basic Model-Theoretic Notions and Results. There are two aspects of model theory, the syntax and the semantics. We will first define the syntax of model theory, and then define models, which will be the semantical part of model theory, and will give meaning to our syntactical notions. A language L is a set of symbols, usually distinguished as function, relation, or constant symbols, along with an arity function defined on those symbols, giving the arity of the functions and relations in the language. We usually write a language like this: L = {R,..., F,..., c,... }, where every R is a relation symbol, every F is a function symbol, and every c is a constant symbol (which can be viewed alternatively as a 0-ary function symbol). We define terms and formulas in a language recursively as sequences of symbols in our language, along with logical symbols ((, ), =,,,,,,,, ) and variables (usually denoted by lower-case letters like v, u, x, y, z). In particular, an L-term (a term in the language L) is defined as either a constant symbol in L, a variable symbol, or F (τ 1,..., τ n ), where F is an n-ary function symbol in L, and τ i are all L-terms. An L-formula is defined similarly as one of R(τ 1,..., τ n ) where R is an n-ary relation symbol in L and τ i are terms, τ 1 = τ 2 where τ 1 and τ 2 are terms, ϕ ψ, ϕ ψ, ϕ ψ, ϕ ψ, or ϕ, where ϕ, ψ are formulas, or xϕ or xϕ, where x is a variable symbol and ϕ is a formula. We call these first two types of formulas atomic formulas. Also, a formula is called a sentence if it has no free variables. Notation 2.1. We denote Form(L) to be the set of all L-formulas. If Γ is a set of L-formulas, we want to talk about what it means for Γ to prove something. For that, we need axioms and inference rules. We will consider the following axiom schemes: (1) Any instance of a tautology from propositional calculus, with variables replaced by formulas (2) ( xφ ψ) (φ xψ) where x is not free in φ

9 GÖDEL S CONSTRUCTIBLE UNIVERSE 9 (3) xφ φ(t/x), where φ(t/x) is the formula we get by replacing all free occurrences of x in ϕ by a term t, as long as no occurrence of x is within a quantifier bounding a variable occurring in t (4) Equality axioms: x = x as well as the following inference rules: (1) Modus Ponens φ (2) Generalization rule (x = y) (f(... x... ) = f(... y... )) (x = y) (φ(... x... ) φ(... y... )) φ ψ ψ φ xφ If Γ is a set of L-formulas and ϕ is an L-formula, we say Γ proves ϕ if there is a sequence of formulas ϕ 1,..., ϕ n such that ϕ n = ϕ and for all i n: either ϕ i Γ or ϕ i is an axiom or there is a j < i such that ϕ i = xϕ j or there are j, k < i such that ϕ j = (ϕ k ϕ i ). These last two points are applications of Modus Ponens and the Generalization rule. Notation 2.2. If Γ is a set of formulas, ϕ is a formula, we write Γ ϕ if Γ proves ϕ. Given a language L, a model M for L is a tuple M = (M, R M,..., F M,..., c M,... ), where M is a set, called the universe of M, R M is a relation in M with the same arity as R for each relation R in L, F M is a function in M with the same arity as F for each function F in L, and c M is a constant in M for each constant c in L. These are called the interpretations of L in M. The values of a term τ in M, denoted τ M, is defined by replacing all constants and functions in τ by their interpretations in M. The satisfaction of formulas is defined recursively as follows: M = τ 1 = τ 2 (we say M satisfies τ 1 = τ 2 ) if τ M 1 = τ M 2 ; M = R(τ 1,..., τ n ) if (τ M 1,..., τ M n ) R M ; M = ϕ ψ if M = ϕ or M = ψ (the other connectives are treated similarly); M = xϕ(x) if for all a M, M = ϕ(a/x) ( is treated similarly). We will want to talk about consistent models of ZF, and so we must first define what a theory is, and what it means to be a model of a theory, and what it means for a theory to be consistent. Definition 2.3. If Γ is a set of L-formulas, we say Γ has a model if there is a model M such that M = γ for all γ Γ.

10 10 MICHAEL WOLMAN Definition 2.4. A set of sentences is inconsistent if there exists a sentence σ such that Γ σ and Γ σ. A theory is consistent if it is not inconsistent. Notation 2.5. If Γ is a set of L-formulas, then Th(Γ) = {ϕ Γ ϕ}, i.e. the set of formulas that can be proven from Γ. Definition 2.6. A set Γ of L-formulas is a theory if Γ = Th(Γ). We can now properly define ZF as Th(Γ), where Γ is the set of all the axioms of ZF, and ZFC to be Th(Γ {AC}) Definition 2.7. A theory T is complete if for every sentence σ, we have either σ T or σ T. We will now state a few very interesting and important theorems without proof. Theorem 2.8 (Gödel s Completeness Theorem). A set of formulas Γ is consistent if and only if it has a model. This theorem is very important because it gives us an equivalence between the syntax and semantics of model theory, i.e. between proofs and satisfaction. Theorem 2.9 (Compactness Theorem). A set of formulas Γ is consistent if and only if every finite subset Γ 0 Γ is consistent. This is a consequence of the fact that proofs must be finite in length. Theorem 2.10 (Tarski). A truth definition does not exist. This theorem shows that in general, there is no definition that will tell us which sentences are true or false for all sentences. In particular, we can encode all formulas as objects in set theory. If we denote for a given formula ϕ its corresponding object ϕ, there is no definable relation T such that σ is true if and only if T ( σ ) for all sentences σ. Theorem 2.11 (Gödel s Incompleteness Theorems). ZFC is incomplete, and in fact one of the things that ZFC is unable to prove is its own consistency. In fact, this is true of any recursively enumerable theory containing a sufficient part of Peano Arithmetic (a theory encoding natural numbers). This means that we can t hope to come up with a stronger set of axioms that are consistent, because any set of axioms we can effectively describe is recursively enumerable. We can, of course, consider a complete theory containing ZFC, but this would not be interesting because we wouldn t know its axioms. The Incompleteness theorem explains why we will later show the relative consistency of ZF, CH and AC. ZF is not able to prove its own consistency, and so we cannot prove that ZF+AC+CH is consistent because we cannot prove that ZF alone is consistent. What we can do, however, is prove that if ZF is indeed consistent, then so is ZF+AC+CH. In fact, to prove this we will construct a model of ZF+AC+CH which, by Gödel s Completeness theorem, will prove that ZF+AC+CH is consistent. The reason why model is in quotes is that again, by the Incompleteness theorem, we are unable to actually construct a model of ZF (as this would be a proof that ZF is consistent). However we will construct a class model, a model whose universe is not really a set, but a proper class (a collection of sets describable by a formula,

11 GÖDEL S CONSTRUCTIBLE UNIVERSE 11 but which isn t itself a set). We will later see exactly what constructing such a model is proving, but for now the above theorems give an outline of how we will prove the relative consistency of ZF, CH and AC, and provides motivation for the following sections. One last note is that in fact, by proving that there is a model of ZF+AC+CH, we are simply proving that ZF does not prove the negation of AC or CH. However, since ZF is not a complete theory, this does not mean it proves AC and CH. In fact, as it turns out, these statements are independent of ZF (and of each other), meaning that if ZF is consistent, we can have models of ZF where AC and/or CH are true, and models of ZF where either of them are false. Now for a few more definitions: Definition A submodel N of M is a subset N M in the same language, with relations R M N, functions F M N and constants c M, such that N is closed under these functions and contains these constants. For example, in the language L = (, 1) of groups, any group M is a model in this language, where M is the group law and 1 M is the identity element in M. In this example, any subgroup is a submodel of M. It is not very interesting to simply consider submodels, because these may have a completely different structure and satisfy completely different formulas. Therefore, we consider the following types of models, which preserve some of the structure of our initial models. Definition Two models N and M are called elementarily equivalent if they satisfy the same sentences, and we write N M. Definition A submodel N M is called an elementary submodel, denoted N M, if for every formula ϕ and every a 1,..., a n N, N = ϕ(a 1,..., a n ) if and only if M = ϕ(a 1,..., a n ). We also say M is an elementary extension of N. Note that elementary extension is strictly stronger than elementary equivalence; if N M and N M, it is not necessarily true that N M. On the other hand, N M implies that N M. The following is an important criterion used for constructing elementary submodels: Theorem 2.15 (Tarski-Vaught test). Suppose N is a substructure of M. Then N is an elementary substructure if and only if for every L-formula ϕ(x, y) and every a N, if M = yϕ(a, y), then there exists some b N such that M = ϕ(a, b). Notation If A M for a model M, we will denote L(A) to be the language L along with a constant added for every element of A. The interpretation of these added constants in M will be the corresponding elements of A. Proof. The forward implication is obvious. We will prove the reverse implication by showing that N = σ M = σ for all L(N )-sentences σ by induction on the length of σ. Atomic formulas follow from N being a substructure of M. Connectives are easily seen by induction, and splitting the formulas into two (or one in the case of ) subformulas.

12 12 MICHAEL WOLMAN In the case of quantifiers, we only have to show this for, as follows by induction and the fact that xϕ is equivalent to x ϕ. If σ = xϕ(a, y) (where a N and ϕ is an L-formula), then if M = σ, M = ϕ(a, b) for some b N, and so N = ϕ(a, b) by the induction hypothesis, and so N = σ. The other implication is clear since N M and so any b N such that N = ϕ(a, b) in is necessarily in M, so M = ϕ(a, b) as well. Now we will prove a very useful theorem that allows us to construct elementary submodels of different sizes. Theorem 2.17 (Löwenheim-Skolem (lower)). Let L be a countable language, M an infinite L-structure. If A M, then there is a model N M such that A N and N = A + ℵ 0. Proof. Let κ = A + ℵ 0. We will construct N = {a α α < κ} by induction. For each α < κ, consider some formula ϕ α (x) Form(L(A {a i i < α})). If M = xϕ(x), choose some witness a α such that M = ϕ(a α ). Otherwise choose a α arbitrarily. If during this construction we consider all formulas in Form(L(A {a α α < κ})), then N M by the Tarski-Vaught test. In particular, for all a A, ϕ = (x = a) is such a formula, and so A N. To ensure we consider every such formula, we use a bookkeeping argument. We can write κ = β<κ I β for I β = κ. At steps α I β, consider formulas ϕ Form(L(A {a i i I γ })). Since every formula we want to consider is found here for some β < κ, we are guaranteed to look at every formula Models of Set Theory. The language of set theory is a simple language, consisting of only one binary relation symbol E. This is usually interpreted as meaning that xey if x is an element of y, and in this case we usually denote this relation by. However, it is important to note that any model in this language satisfying the axioms of ZF would be a valid model, even if this intuitive meaning doesn t hold in this model. In general, models of set theory have the form (M, E), where M is the universe and E is a binary relation on M. We will also consider models where M is a proper class, because as mentioned above, Gödel s Incompleteness theorem doesn t allow us to have set models of ZF. In general, when we don t care whether or not we are referring to a proper class or a set, we will call it a class. However, due to the differences between them, we may sometimes have to distinguish the two. When working with models of set theory, there are two important concepts we work with: relativization and absoluteness of formulas. Definition Let M, E be a class model of set theory, and ϕ(x 1,..., x n ) a formula in the language of set theory. The relativization of ϕ to M, E is the formula denoted by ϕ M,E (x 1,..., x n ) γ<β

13 GÖDEL S CONSTRUCTIBLE UNIVERSE 13 and defined inductively as follows: (x y) M,E xey (x = y) M,E x = y ( ϕ) M,E ϕ M,E (ϕ ψ) M,E ϕ M,E ψ M,E ( xϕ) M,E ( x M)ϕ M,E and similarly for the other connectives and. Notation For convenience, when E is, we will write ϕ M instead of ϕ M,E. When writing ϕ M,E (x 1,..., x n ) without quantifying over the x i s, we implicitly assume that the variables range over M. We will also sometimes write (M, E) = ϕ(x 1,..., x n ) instead of ϕ M,E (x 1,..., x n ), although we note that by Tarki s theorem on the undefinability of truth, the = relation is not definable in ZF. Definition If M is a model of set theory and ϕ is a formula, then we say that ϕ is absolute for M if for all x 1,..., x n M, ϕ M (x 1,..., x n ) ϕ(x 1,..., x n ). We will now talk about transitive models of set theory. Definition A set T is transitive if every element of T is a subset of T. The same definition holds for proper classes. If M is a transitive class, then we call (M, ) a transitive model of set theory. Transitive models are especially useful to study because of how they behave with certain types of formula, namely 0 and 1 -formulas, which we will define later. We will now prove a few theorems that will be very useful later on in the analysis of Gödel s Constructible Universe. Definition A sequence W α α Ord is called a cumulative hierarchy if W 0 = and (1) W α W α+1 P (W α ) and (2) if α is a limit, then W α = β<α W β. The universe of set theory is a cumulative hierarchy defined as follows: V 0 = V α+1 = P (V α ) V α = β<α V β for α limit. Note that if W α is a cumulative hierarchy, then each W α is transitive and W α V α. Theorem 2.23 (Reflection Principle). Let ϕ(x 1,..., x n ) be a formula, and W α be a cumulative hierarchy. Let W = α Ord W α. Then there are arbitrarily large (limit) ordinals α such that for all x 1,..., x n W α, We say that W α reflects ϕ. ϕ W (x 1,..., x n ) ϕ Wα (x 1,..., x n ). To prove this we first need the following lemma:

14 14 MICHAEL WOLMAN Lemma Let ϕ(u 1,..., u n, x) be a formula. For every set M 0 there exists an α such that if ( x W )ϕ(u 1,..., u n, x) then ( x W α )ϕ(u 1,..., u n, x) for every u 1,..., u n W α. Proof. For every u z,..., u n W, let C = {x W ϕ(u 1,..., u n, x)}. Then, if C is non-empty, let α be the least ordinal such that C W α. Otherwise let α = 0. Then let H(u 1,..., u n ) = C W α. Note that while C may be a proper class, H(u 1,..., u n ) is a set with the property that if ( x W )ϕ(u 1,..., u n, x) then ( x H(u 1,..., u n ))ϕ(u 1,..., u n, x). Now we will construct W α by induction. For each i ω, let where γ is the least ordinal such that M i+1 = W γ W γ M i {H(u 1,..., u n ) u 1,..., u n M i }. We let W α = i ω M i. Now, if u 1,..., u n W α, then there is an i ω such that u 1,..., u n M i and if ϕ(u 1,..., u n, x) holds for some x W, then it holds for some x M i+1 W α. Note that the proof of this lemma can be modified so that this lemma holds for finitely many formulas ϕ 1,..., ϕ n by considering functions H 1,..., H n and alternating between them. Using the same bookkeeping argument as in the proof of the Löwenheim-Skolem theorem, we can make sure this works as intended. Proof of theorem. Let ϕ(x 1,..., x n ) be a formula. We may assume that the universal quantifier does not occur in ϕ by replacing it with an existential quantifier (so x... becomes x...). Let ϕ 1,..., ϕ n be the subformulas of ϕ. Given a set M 0 we can find, by the previous lemma, an ordinal α such that W α M 0 and (1) ( x W )ϕ j (u,..., x) ( x W α )ϕ j (u,..., x), j = 1,..., n for all u, W α. We will show that W α reflects each ϕ j, and so in particular reflects ϕ, by induction on the complexity of ϕ j. Every W α reflects atomic formulas, and if W α reflects ψ and χ, then W α reflects ψ, ψ χ, ψ χ, ψ χ and ψ χ. So we can assume W α reflects ϕ j (u 1,..., u m, x) and we now have to show it reflects xϕ j (u 1,..., u m, x) as well. If u 1,..., u m W α, then W α = xϕ j (u 1,..., u m, x) ( x W α )ϕ Wα j (u 1,..., u m, x) ( x W α )ϕ j (u 1,..., u m, x) where the last equivalence holds by (1). xϕ j (u 1,..., u m, x)

15 GÖDEL S CONSTRUCTIBLE UNIVERSE 15 One other very useful theorem is the following, which allows us to construct and work with transitive models, which as we will see are very nice to work with. Definition Let E be a binary relation on a class P. For each x P, let ext E (x) = {z P zex} be the extension of x. We say E is well-founded if: (1) every non-empty set x P has an E-minimal element and (2) ext E (x) is a set for every x P. Definition A well-founded relation E on a class P is extensional if ext E (x) ext E (y) whenever x and y are distinct elements of P. Theorem 2.27 (Mostowski s Collapsing Theorem). (1) If E is a well-founded and extensional relation on a class P, then there is a transitive class M and an isomorphism π between (P, E) and (M, ), where M and π are unique. (2) In particular, every extensional class P is isomorphic to a transitive class M, and again this is unique. (3) In case (2), if T P is transitive, then π is the identity on T. Proof. We shall prove (1) as a general case of (2) (where E = in (2)). Since E is a well-founded relation, we can define π by induction, so that π(x) is defined in terms of π(z) s where zex. For each x P, let π(x) = {π(z) zex}. Let M = π(p ). Then π maps P onto M, and M is transitive by the definition of π. To show that π is one-to-one, we need the extensionality of E. Let z M be of the least rank such that z = π(x) = π(y) for some x y. Then ext E (x) ext E (y), and without loss of generality there is some u ext E (x) such that u / ext E (y). Let t = π(u). Since t z = π(y), there is a v ext E (y) such that t = π(v). But then t = π(u) = π(v) for some u v and t is of smaller rank then z (because t z), which is a contradiction. Therefore π is one-to-one. To show π is an isomorphism, we have that xey π(x) π(y) by definition. If π(x) π(y), then π(x) = π(z) for some zey. Since π is one-to-one, x = z and so xey. Uniqueness follows from the fact that if M 1 and M 2 are transitive classes and π is an -isomorphism between them, then π is the identity and M 1 = M 2 (which follows easily by induction). Then if π 1 : P M 1 and π 2 : P M 2 are isomorphisms, π 2 π1 1 : M 1 M 2 is an isomorphism, so M 1 = M 2 and π 1 = π 2. To prove (3), let T P be transitive. Then x P for every x T, so x P = x and we have that π(x) = {π(z) z x} for all x T. It follows now by induction that π(x) = x for all x T.

16 16 MICHAEL WOLMAN 2.3. The Lévy Hierarchy. In this section we will look at a hierarchy for organizing formulas of different forms. In particular, we will distinguish two types of formulas, 0 and 1 -formulas. Definition A formula of set theory is 0 if it has no quantifiers, or it is of the form ϕ ψ, ϕ ψ, ϕ, ϕ ψ or ϕ ψ for 0 -formulas ϕ and ψ, or it is ( x y)ϕ or ( x y)ϕ where ϕ is a 0 -formula. Essentially, 0 -formulas are exactly those whose quantifiers are bounded. This is an important concept because of the following lemma: Lemma If M is a transitive class and ϕ is a 0 -formula, then for all x 1,..., x n M, ϕ M (x 1,..., x n ) ϕ(x 1,..., x n ). This lemma is equivalent to the statement that 0 -formulas are absolute for all transitive models. This property is why we will later choose to work with transitive models. Proof. We will prove this by induction on the length of ϕ. If ϕ is atomic, this is obviously true. Similarly for connectives, this follows from our induction hypothesis. Let ϕ = ( u x)ψ(u, x, y 1,..., y n ), and assume ψ is absolute for M. We will now show that ϕ is absolute for M as well. If ϕ M is true, then we have that ( u(u x ψ)) M, so we have that ( u M)(u x ψ M ). Since ψ M ψ, using the same u M we get that ( u x)ψ. Conversely, if ( u x)ψ holds true, then since x M and M is transitive, u M and since ψ M (u, x, y 1,..., y n ) ψ(u, x, y 1,..., y n ), we get that u(u M u x ψ M ), and so (( u x)ψ) M holds as well. The same argument works for universal quantifiers. Note that this proof relied on the fact that M was a transitive model, and this is why we are choosing to work with transitive models. Lemma The following can be written for 0 -formulas, and so are absolute for transitive models: Proof. (1) x = {u, v}, x = (u, v), x is empty, x y, x is transitive, x is an ordinal, x is a limit ordinal, x is a natural number, x = ω. (2) Z = X Y, Z = X Y, Z = X Y, Z = X, Z = dom X, Z = ran X. (3) X is a relation, f is a function, y = f(x), g = f X. (1) x = {u, v} u x v x ( w x)(w = u w = v) x = (u, v) ( w x)( z x)(w = {u} z = {u, v}) ( w x)(w = {u} w = {u, v}) x is empty ( u x)u u x y ( u x)u y x is transitive ( u x)u x x is an ordinal x is transitive ( u x)( v x)(u v v u u = v) ( u x)( v x)( w x)(u v w u w) x is a limit ordinal x is an ordinal ( u x)( v x)u v

17 GÖDEL S CONSTRUCTIBLE UNIVERSE 17 x is a natural number x is an ordinal (x is not a limit x = 0) ( u x)(u = 0 u is not a limit) x = ω x is a limit ordinal x 0 ( u x)x is a natural number Z = X Y ( z Z)( x X)( y Y )z = (x, y) (2) ( x X)( y Y )( z Z)z = (x, y) Z = X Y ( z Z)(z X z / Y ) ( z X)(z / Y z Z) Z = X Y ( z Z)(z X z Y ) ( z X)(z Y z Z) Z = X ( z Z)( x X)z x ( x X)( z x)z Z Now we have to show that: (a) z dom X is 0 (b) if ϕ is 0, then so is ( z dom X)ϕ (a) z dom X ( x X)( u X)( v u)x = (z, v) (b) ( z dom X)ϕ ( x X)( u x)( z, v u)(x = (z, v) ϕ) so now we get: Z = dom X ( z Z)z dom X ( z dom X)z Z and the same argument holds for Z = ran X (3) X is a relation ( x X)( u dom X)( v ran X)x = (u, v) f is a function f is a relation ( x dom f)( y, z ran f) where ((x, y) f (x, z) f y = z) (x, y) f ( u f)u = (x, y) y = f(x) (x, y) f g = f X g is a function g f ( x dom g)x X ( x X)(x dom f x dom g) Note that despite the fact that ordinals and limit ordinals and natural numbers are a 0 concept, in general cardinal numbers are not absolute. In fact, the following are not absolute: Y = P (X), Y = X, α is a cardinal, β = cf α, α is regular. Now we can define the Lévy Hierarchy of formulas: a formula is Σ 0 or Π 0 if it is 0. Inductively, a formula is Σ n+1 if it is of the form xϕ where ϕ is Π n, and it is Π n+1 if it is of the form xϕ where ϕ is Σ n. We say that a property (i.e. a class or relation) is Σ n (or Π n ) if it can be expressed by a Σ n (or Π n ) formula. A function F is Σ n (or Π n ) if the relation y = F (x) is Σ n (or Π n ). We also say a property is n if it is both Σ n and Π n. One interesting property of this hierarchy is that despite how it is defined, it is not purely syntactical. Some concepts can be written in multiple ways, but this requires a proof, which depends on the axioms we have. When we say a property P is Σ n, we mean that P can be expressed by a Σ n formula in ZF unless otherwise specified. Every proof is finite, and so every formula needs a finite set of axioms Σ of ZF to specify its place in the hierarchy. When M is a transitive model of Σ, then the relativization P M is unambiguously the formula ϕ M. We call such models adequate for P. Note that using the Reflection Principle and the Mostowski Collapse Theorem, we can always find an adequate model for a finite set of formulas.

18 18 MICHAEL WOLMAN Since 0 properties are absolute for all transitive models, Σ 1 properties are upward absolute: if P (x) is Σ 1 and M is a transitive model adequate for P, then for all x M, P M (x) implies P (x). Similarly, Π 1 properties are downward absolute, and consequently, 1 properties are absolute for transitive models. The absoluteness of 0 and 1 -formulas for transitive models is a very important property, which we will use in the next section. We will now prove a few lemmas that will enable us to easily classify properties in the hierarchy later on. Lemma Let n 1. (1) If P, Q are Σ n properties, then so are xp, P Q, P Q, ( u x)p, ( u x)p. (2) If P, Q are Π n properties, then so are xp, P Q, P Q, ( u x)p, ( u x)p. (3) If P is Σ n, then P is Π n. If P is Π n, then P is Σ n. (4) If P is Π n and Q is Σ n, then P Q is Σ n. If P is Σ n and Q is Π n, then P Q is Π n. (5) If P, Q are n, then so are P, P Q, P Q, ( u x)p, ( u x)p, P Q, P Q. (6) If F is a Σ n function then dom F is a Σ n class. (7) If F is a Σ n function and dom F is n, then F is n. (8) If F and G are Σ n functions, then so is F G. (9) If F is a Σ n function and P is a Σ n property, then P (F (x)) is Σ n. To prove this lemma we need the Collection Principle: X Y ( u X)[ vϕ(u, v, p) ( v Y )ϕ(u, v, p)]. Formulated differently, this says that if (C u ) u X is a collection of classes, then there is a set Y such that for every u X if C u then C u Y. To prove this, for every u X, letting α be the least ordinal such that C u V α (or 0 if C u = ), let Ĉu = C u V α, where C u = {v ϕ(u, v, p)}. Then Y = u X Ĉu is as desired. Proof. We will prove this for n = 1. The rest follows easily by induction. (1) Let where ϕ, ψ are 0 formulas. We have xp (x,... ) x zϕ(z, x,... ) P (x,... ) zϕ(z, x,... ), Q(x,... ) uψ(u, x,... ) v w v x w z w(v = (x, z) ϕ(z, x,... )). This is therefore a Σ 1 formula. We also have P (x,... ) Q(x,... ) z u(ϕ(z, x,... ) ψ(u, x,... )), P (x,... ) Q(x,... ) z u(ϕ(z, x,... ) ψ(u, x,... )), ( u x)p (u,... ) z u(u x ϕ(z, u,... )).

19 GÖDEL S CONSTRUCTIBLE UNIVERSE 19 To show that ( u x)p is Σ 1, we use the Collection Principle: ( u x)p (u,... ) ( u x) zϕ(z, u,... ), (2) follows from (1) and (3). (3) (4) (P Q) ( P Q). (5) follows from (1)-(4). (6) x dom F yy = F (x). (7) Since F is a function, we have y( u x)( z y)ϕ(z, u,... ). zϕ(z, x,... ) z ϕ(z, x,... ), zϕ(z, x,... ) z ϕ(z, x,... ). y = F (x) x dom F z(z = F (x) y = z). Since z = F (x) is Σ n and x dom F is Π n, the right-hand-side is Π n. (8) y = F (G(x)) z(z = G(x) y = F (z)). (9) P (F (X)) y(y = F (x) P (y)). Lemma E is a well-founded relation on P is a 1 property. Proof. E is a relation on P is a 0 -formula, and so is ϕ(e, P, X) = [ X P ( a X)a is E-minimal in X], and so E is a relation on P and Xϕ(E, P, X) is Π 1. On the other hand, E is well-founded if and only if there is a function f : P Ord such that f(x) < f(y) whenever xey. Therefore, we can write this as the Σ 1 formula E is a relation on P and f (f is a function ( u ran F )u is an ordinal ( x, y P )(xey f(x) < f(y)). Lemma Let n 1 and let G be a Σ n function, and let F be defined by induction as F (α) = G(F α). Then F is a Σ n function on Ord. Proof. Since Ord is a 0 class, it is enough to verify that the following is Σ n : y = F (α) if and only if f(f is a function dom f = α ( ξ < α)f(ξ) = G(f ξ) y = G(f)). But all of the properties and operations above are Σ 0 and G is Σ n, and so y = F (α) is Σ n. 3. Gödel s Constructible Universe The basic idea of Gödel s proof of the consistency of the Axiom of Choice and the Continuum Hypothesis with ZF is to take the class L of constructible sets as a model of ZF. The intuition behind this is that the constructible sets are well behaved, so it should be possible to order them (and therefore satisfy AC), and that the constructible sets are a small subset of all sets, and so we won t have enough sets to fit another cardinal between ℵ 0 and 2 ℵ0. In this section we will define L and prove it is a model of ZF. We will then show that in L every subset of N is in ℵ 1, and define a well-ordering on L, showing that L is a model of ZF+AC+CH, completing our proof.

20 20 MICHAEL WOLMAN 3.1. The Constructible Sets. Given a set model (M, ), we say a set X is definable over (M, ) if there exists a formula ϕ F orm and some a 1,..., a n M such that X = {x M (M, ) = ϕ[x, a 1,..., a n ]}. Let def(m) = {X M X is definable over (M, )}. We can clearly see that M def(m), def(m) P (M), and if M is transitive, then M def(m). Definition 3.1. We will now define the class L of constructible sets by transfinite induction: (1) L 0 = (2) L α+1 = def(l α ) (3) L α = β<α L β if α is a limit ordinal (4) L = α Ord L α Definition 3.2 (Axiom of Constructibility). The statement V = L is the Axiom of Constructibility, where V is the class of all sets, and L is the class of all constructible sets. This axiom is the statement that every set is constructible. We will later prove that this axiom implies AC and CH, and that V = L is true in L, which, along with the proof that L is a model of ZF, will complete the proof that ZF is consistent with AC and CH. We also note that from the definition, we can see that L α forms a cumulative hierarchy, that each L α is transitive, and if α < β then L α L β. It follows that L is a transitive class. Lemma 3.3. For every α, α L α and L α Ord = α. Proof. We will prove this by induction on α. For α = and for α limit this is obvious. At step α + 1, we want to show that α L α+1, so that α is a definable subset of L α. Since x is an ordinal is a 0 formula, we have that α = {x L α x is an ordinal} = {x L α L α = x is an ordinal} L α+1 Lemma 3.4. For all α, L α V α. If α ω, then L α = V α. If α is infinite, L α = α. Proof. We will prove every part of this lemma by induction. First note that the first part of the lemma is obvious for 0 and for limit ordinals, and for any α, L α+1 = def(l α ) P (L α ) P (V α ) = V α+1. For the second part, again we note this is clear for n = 0. Suppose L n = V n. Let x V n+1. Then x V n, say x = {a 1,..., x n }. This is clearly a definable subset of V n = L n, and so x L n+1. For the last part, suppose α is infinite. L ω = V ω = ω, and if L α α, then L α+1 L <ω α ω α = α + 1, where L <ω α is the set of all finite sequences of L α, and this first inequality holds because each element of L α+1 is determined by a formula of set theory (of which there are countably many) and a finite tuple of elements of L α.

21 GÖDEL S CONSTRUCTIBLE UNIVERSE 21 For α limit, L α = ω β<α L β That α L α is clear since α L α. β α. ω β<α Now, using the fact that many formulas are 0 from the previous section, we will prove that L is a model of ZF. Theorem 3.5. L is a model of ZF Proof. We have to show that for every formula σ of ZF, σ L holds in L. Since L is a transitive class, every 0 formula is absolute for L. Extensionality: The formula (( u X)u Y ( u Y )u X) X = Y is a 0 formula, and so it holds in L. Pairing: Let a, b L, c = {a, b}. Pick α such that a, b L α. Since {a, b} is definable over L α, c L α+1, and so c L. Since c = {a, b} is 0, the Pairing Axiom holds in L. Separation: Let ϕ be a formula. Given X, p L, we have to show that the set Y = {u X ϕ L (u, p)} is in L. By the Reflection Principle (which we can apply to the cumulative hierarchy L α ), there exists an α such that X, p L α and Y = {u X ϕ Lα (u, p)}. Therefore Y = {u L α L α = u X ϕ(u, p)} and so Y L. Union: Given X L, let Y = X. Since L is transitive, we have that Y L. Pick some α such that X L α and Y L α. Y is definable over L α by the 0 formula x X, so Y L. Since Y = X is 0, the Axiom of Union holds in L. Power Set: Given X L, let Y = P (X) L. Let α be such that Y L α. Y is definable over L α by the 0 -formula x X and so Y L. We claim that Y = P L (X), meaning that Y is the power set of X holds in L. But x Y x X is a 0 -formula true for every x L. Infinity: We want to show that L = S( S ( x S)x {x} S). But since we just showed pairing and union are the same in L and in V, and so is (since it is a 0 property), it is clear that since ω L this holds for S = ω. Replacement: If F is a function in L, then for every X L there exists an α such that {F (x) x X} L α. Using separation, we can get Y = {F (x) L α x X}, and since L α L, Y L as well. Regularity: If S L is non-empty, let x S be such that x S =. Then x L and the 0 -formula x S = holds in L The Absoluteness of Constructibility. We will now introduce a new definition of definable sets through something called Gödel Operations. We will then use this to prove that the function α L α is a 1 function, and we will use this fact to prove both that L = V = L and an important lemma called Gödel s Condensation Lemma. These are the last steps required before proving that L is a model of AC and of CH.

22 22 MICHAEL WOLMAN The axiom schema of Separation tells us that for any set X and any formula ϕ, there exists the set Y = {x X ϕ(x)}. It turns out that for 0 formulas, the construction of Y from X can be described by finitely many elementary operations. Theorem 3.6 (Gödel s Normal Form Theorem). There exist operations G 1,..., G 10 such that if ϕ(u 1,..., u n ) is a 0 -formula, then there is a composition G of G 1,..., G 10 such that for all X 1,..., X n, G(X 1,..., X n ) = {(u 1,..., u n ) u i X i ϕ(u 1,..., u n )}. The operations G 1,..., G 10 and any composition of these operations are called Gödel Operations. Definition 3.7 (Gödel Operations). G 1 (X, Y ) = {X, Y }, G 2 (X, Y ) = X Y, G 3 (X, Y ) = ε(x, Y ) = {(u, v) u X v V u v}, G 4 (X, Y ) = X Y, G 5 (X, Y ) = X Y, G 6 (X) = X, G 7 (X) = dom X, G 8 (X) = {(u, v) (v, u) X}, G 9 (X) = {(u, v, w) (u, w, v) X}, G 10 (X) = {(u, v, w) (v, w, u) X}. Proof. The proof is done by induction on the complexity of 0 -formulas. To keep things simple, we can consider formulas only of this form: (1) the only logical symbols in ϕ are,, and restricted ; (2) = does not occur; (3) the only occurrence of is u i u j, where i j; (4) the only occurrence of is where i m. ( u m+1 u i )ψ(u 1,..., u m+1 ) We can rewrite any 0 formula in this form: we can restrict logical symbols to, and ; x = y can be replaced by ( u x)u y ( v y)v x; x x can be replaced by ( u x)u = x; the bound variables in ϕ can be renamed so that the highest index is quantified. Now let ϕ(u 1,..., u n ) be a formula of the above form and assume that the theorem holds for all subformulas of ϕ. Case I. ϕ(u 1,..., u n ) is atomic, of the form u i u j for i j. We prove this case by induction on n. Case Ia. n = 2. We have and {(u 1, u 2 ) u 1 X 1 u 2 X 2 u 1 u 2 } = ε(x 1, X 2 ) {(u 1, u 2 ) u 1 X 1 u 2 X 2 u 2 u 1 } = G 8 (ε(x 1, X 2 )).

23 Case Ib. Then Case Ic. GÖDEL S CONSTRUCTIBLE UNIVERSE 23 n > 2 and i, j n. Let G be such that {(u 1,..., u n 1 ) u k X k u i u j } = G(X 1,..., X n 1 ). {(u 1,..., u n ) u k X k u i u j } = G(X 1,..., X n 1 ) X n. n > 2 and i, j n 1. Let G be such that {(u 1,..., u n 2, u n, u n 1 ) u k X k u i u j } = G(X 1,..., X n ). Then we have that since Case Id. and so and {(u 1,..., u n ) u k X k u i u j } = G 9 (G(X 1,..., X n )) (u 1,..., u n, u n 1 ) = ((u 1,..., u n 2 ), u n 1, u n ). i = n 1, j = n. By Ia we have {(u n 1, u n ) u n 1 X n 1 u n X n u n 1 u n } = ε(x n 1, X n ) {((u n 1, u n ), (u 1,..., u n 2 )) u k X k u n 1 u n } Since we get that = ε(x n 1, X n ) (X 1 X n 2 ) = G(X 1,..., X n ). ((u n 1, u n ), (u 1,..., u n 2 )) = (u n 1, u n, (u 1,..., u n 2 )) (u 1,..., u n ) = ((u 1,..., u n 2 ), u n 1, u n ), {(u 1,..., u n ) u k X k u n 1 u n } = G 10 (G 10 (G(X 1,..., X n ))). Case Ie. i = n, j = n 1. This is similar to Id. Case II. ϕ(u 1,..., u n ) is a negation, ψ(u 1,..., u n ). Find a G such that Then {(u 1,..., u n ) u i X i ψ(u 1,..., u n )} = G(X 1,..., X n ). {(u 1,..., u n ) u i X i ϕ(u 1,..., u n )} = X 1 X n G(X 1,..., X n ). Case III. ϕ = ψ 1 ψ 2. Let G 1, G 2 be such that for k = 1, 2. Then {(u 1,..., u n ) u i X i ψ k (u 1,..., u n )} = G k (X 1,..., X n ) {(u 1,..., u n ) u i X i ϕ(u 1,... u n )} = G 1 (X 1,..., X n ) G 2 (X 1,..., X n ). Case IV. ϕ(u 1,..., u n ) is the formula ( u n+1 u i )ψ(u 1,..., u n+1 ). Let χ(u 1,..., u n+1 ) be the formula ψ(u 1,..., u n+1 ) u n+1 u i. We consider χ less complex than ϕ, and so by the induction hypothesis we can find a G such that {(u 1,....u n+1 ) u i X i χ(u 1,..., u n+1 )} = G(X 1,..., X n+1 ) for all X 1,..., X n+1. We claim that {(u 1,..., u n ) u i X i ϕ(u 1,..., u n )} = (X 1 X n ) dom(g(x 1,..., X n, X i )).

24 24 MICHAEL WOLMAN Denoting u = (u 1,..., u n ) and X = X 1 X n, for all u X we have and the rest follows. ϕ(u) ( v u i )ψ(u, v) v(v u i ψ(u, v) v X i ) u dom{(u, v) X X i χ(u, v)} The following lemma shows that Gödel operations are absolute for transitive models. Lemma 3.8. If G is a Gödel operation then the property Z = G(X 1,..., X n ) can be written as a 0 formula. Therefore, any 0 -formula can be written as a Gödel operation and vice-versa. Proof. We will prove this by the complexity of G in four cases: (1) u G(X,... ) is 0. (2) If ϕ is 0, then so are u G(X,... )ϕ and u G(X,... )ϕ. (3) Z = G(X,... ) is 0. (4) If ϕ is 0, then so is ϕ(g(x,... )). We proved (3) for most of G 1,..., G 10 in Lemma The rest we prove as follows: Z = ε(x, Y ) Z X Y ( z Z)( w z)( x w)( y w)(z = (x, y) x y) Z = G 8 (X) ( x X)( y Y )( z Z)(x y z = (x, y)) ( z Z)( x X)( u ran X)( v dom X)(x = (v, u) z = (u, v)) ( x X)( u ran X)( v dom X)( z Z)(x = (v, u) z = (u, v)) G 9 and G 10 are done similarly. To prove (1) and (2) we will show this for a few examples. The rest follow similarly. For (1), consider u F (X,... ) G(X,... ). This can be written as x F (X,... ) y G(X,... )u = (x, y). For (2), consider the formula u {F (X,... ), G(X,... )}ϕ(u), which we can write as ϕ(f (X,... )) ϕ(g(x,... )). We get (3) from (1) and (2) since Z = G(X,... ) ( u Z)u G(X,... ) ( u G(X,... ))u Z. To prove (4), let ϕ be a 0 -formula. Then G(X,... ) occurs in the formula ϕ(g(x,... )) in the form u G(X,... ), G(X,... ) u, Z = G(X,... ), u G(X,... ), or u G(X,... ). Since G(X,... ) u can be replaced by ( v u)v = G(X,... ) and the rest are 0 properties from (1)-(3), ϕ(g(x,... )) is a 0 property. If ϕ is a formula, then ϕ M is a 0 -formula, and so by Theorem 3.6 there is a Gödel operation G such that for every transitive set M and all a 1,..., a n, {x M M = ϕ(x, a 1,..., a n )} = {x M ϕ M (x, a 1,..., a n )} = G(M, a 1,..., a n ).