Consistency of the Continuum Hypothesis

Consistency of the Continuum Hypothesis Bruce W. Rogers April 22, 2005 1 Introduction One of the basic results in set theory is that the cardinality of the power set of the natural numbers is the same as the cardinality of the real numbers, which is strictly greater than the cardinality of the naturals. In fact Cantor proved a more general theorem: for any set X, the cardinality of X is strictly less than the cardinality of the power set of X. Since there is an infinite set, say the naturals, each application of the power set gives a greater infinite number. For each infinite set X we assign a cardinal number ℵ X. It is equivalent to the Axiom of Choice that every set of alephs is linearly ordered. In fact, every set of cardinals is well ordered, so we can index the alephs with ordinals, α, β, in such a way that α < β iff ℵ α < ℵ β. The problem is that we don t know which cardinals go where in the ordering. We know that countably infinite sets have the smallest infinite cardinal, ℵ 0, and we know ℵ 0 is less than 2 ℵ 0, but we don t know if there are any cardinals between. Cantor could not find any sets whose cardinalities were greater than ℵ 0 but less than 2 ℵ 0, so Cantor hypothesized that 2 ℵ 0 is actually the next cardinal after ℵ 0, i.e. 2 ℵ 0 = ℵ 1. This statement is known as the Continuum Hypothesis [CH] since one can prove that 2 ℵ 0 is the cardinality of the continuum of real numbers, R. Equivalently, CH says that every uncountable set of real numbers has cardinality 2 ℵ 0. Proving CH would mean that mathematicians have a very good handle on the ordering of the cardinals, so much effort was spent trying to prove CH from the axioms of set theory. As it turns out, however, CH cannot be proven true or false from the Zermelo-Fraenkel [ZF] axioms of set theory. In other words, ZF cannot imply CH, and ZF cannot imply the negation of CH (if ZF is consistent to start with). Thus we say CH is independent of ZF in 1

the former case, and CH is consistent with ZF in the latter. Another way to think about consistency is that assuming CH along with the other axioms of ZF will not introduce a contradiction. The notions of consistency and independence also appear outside of set theory. In fact, for any sufficiently strong (and consistent) axiomatic system there must be a proposition, say P, such that neither P nor the negation of P is provable in the system. This follows from Gödel s incompleteness theorem. The most famous example of such a proposition is Euclid s Parallel Postulate. The Greeks did not think the Parallel Postulate was self-evident, so they attempted to prove it using the other axioms of geometry. This is impossible since the Parallel Postulate is independent from those axioms. In fact, the negation of the Parallel Postulate produces interesting noncontradictory geometries, which shows that the Parallel Postulate is independent. Consistency of the Parallel Postulate follows from the lack of contradictions in Euclidean plane geometry. In this paper we will show the consistency of CH with ZF. This proof was first given by Gödel, and our proof follows his loosely. Before we start, the term consistent needs to be defined a little more precisely. We do this by introducing the notion of model. 2 Models and the Axiom of Constructibility Definition. Let M be a set and ϕ(v 1,..., v n ) be a formula. Let x 1,..., x n M. Then we write M = ϕ(x 1,..., x n ) (read M models ϕ or M satisfies ϕ) according to the following inductive definition: (1) if ϕ is of the form v i v j, then M = ϕ if it is actually the case that x i x j ; (2) M = ϕ if it is not true that M = ϕ; (3) M = ϕ ψ if both M = ϕ and M = ψ; (4) M = v i (ϕ) if there exists ˆx M such that M = ϕ(x 1,..., ˆx,..., x n ). Basically, M = ϕ means we would think ϕ were true if M were the universe of all sets. We say M is a model for ZF if M satisfies every axiom of ZF. Now if there is a model for ZF in which CH is true, then CH is consistent with ZF. The rest of this section deals with defining such a model. Definition. The cumulative hierarchy of sets (V α ) is defined for all ordinals α by V 0 =, V α+1 = (V α ), and V β = α<βv α for limit ordinals β. 2

Since the V α s are built up by taking power sets, it is fairly clear that in every V α, x y V α implies x V α, i.e. V α is transitive. Also, if α < β, V α is a proper subset of V β. It is also true that every set lives inside of some V α. This result is not intuitively obvious, so we will take the time to prove it. Propostion. Every set is a member of V α for some ordinal α. Proof: Suppose X is a set not in the cumulative hierarchy, and let Y be the transitive closure of X, i.e., Y = X n where X 0 = X and X n+1 = X n. Then Y is the smallest transitive set containing X. We claim there is a set Z that is not in the cumulative hierarchy, but all the elements of Z are. Since X Y, Y V α for any α by the transitivity of V α. Let S = {y Y y V α for all α}. If S =, then Y has the claimed property. If S, then it has an ɛ minimal element, Y 0, by the axiom of regularity. Since Y is transitive, we can let Z = Y 0. Now, for each x Z let α x be the smallest ordinal so that x V αx. Then let β = x Zα x. It follows that Z V β, hence Z V β+1, a contradiction. So we can (roughly) think of the union of all the V α s as the universe of sets. However, there are a couple of problems with this notion. First, we don t know how high the ordinals go, so we cannot take the union of every V α. Second, to make new sets we take the power sets of previous sets. This seems simple enough, but we don t have a specific mechanism to form all the subsets of a arbitrary set. One solution to the latter problem is to define the Gödel constructible hierarchy, in which we will give give definite methods for constructing sets. One way to think about the constructible hierarchy (roughly) is that is consists of those sets which we can build from logical and set theoretic notation with certain restrictions. To be a little more precise, there are ten Gödel operations on sets X and Y which we will use to define the constructible hierarchy. These are F 1 (X, Y ) = {X, Y } F 2 (X, Y ) = X Y F 3 (X, Y ) = ɛ(x, Y ) = {[x, y] X Y x y} F 4 (X, Y ) = X Y F 5 (X, Y ) = X Y F 6 (X, Y ) = X = {x y X where x y} F 7 (X, Y ) = π(x) = {x y([x, y] X} F 8 (X, Y ) = {[x, y] [y, x] X} F 9 (X, Y ) = {[x, y, z] [x, z, y] X} F 1 0(X, Y ) = {[x, y, z] [y, z, x] X} 3

Now for any set X the Gödel closure of X, denoted cl(x), is the smallest set which contains X and is closed under the Gödel operations. So to get cl(x) we take all the elements of X and apply all the Gödel operations, and then repeat. That is, if we define G(X) = {F i (x, y) x, y X and 1 i 10} X then cl(x) = G(X) G 2 (X) G 3 (X)... Next we use cl(x) to define another construction on a set X. Definition. For any set X, the set def(x) = cl(x {X}) (X) is called the set of logically definable subsets of X. Thus, def(x) consists of precisely those subsets of X which can be constructed from elements of X, along with X itself, in a finite number of steps using the Gödel operations. From the following lemma and theorem we see that def(x) is called the collection of logically definable sets for good reason. Note that a restricted formula is a logical formula whose quantifiers are all restricted, e.g. ( x y)(φ) is restricted but x(φ) is not. Also, we take φ(v 1,..., v n ) to mean v 1,..., v n are the variables of φ, none of which occur both free and bound. Lemma. Let ϕ be a restricted formula of n variables and let n 1. Then there is a function F of n variables such that F is a composition of Gödel operations and for any sets X 1,..., X n we have F (X 1,..., X n ) = {[x 1,..., x n ] X 1... X n ϕ(x 1,..., x n )}. Theorem. Let X be a transitive set. Then def(x) consists of precisely those subsets Y X such that for some formula ψ and some elements x 1,..., x n X we have Y = {x X ψ(x, x 1,..., x n )}. The proof of the lemma is an induction on the complexity of ϕ, where one formula is more complex than another if it has more quantifiers. For formulas with the same number of quantifiers, complexity is decided by length. Then, the theorem is a fairly immediate consequence. The main idea is that each element in def(x) is constructed in a finite number of steps using Gödel operations. We are finally ready to define the constructible hierarchy, L α. Definition. The constructible hierarchy, L α, is defined inductively for all ordinals α. Let L 0 =, L α+1 = def(l α ), and L β = α<βl α for limit ordinals β. There are a few interesting things we can say about L α. First, like V α, every L α is transitive and if α < β, then L α is a proper subset of L β. Also if X is a 4

finite transitive set, def(x) = (X) so that L α = V α for all α ω. However, L α is a proper subset of V α for every ordinal α > ω. In fact, we have that L α = α for every infinite ordinal α. Proposition. L α = α for every infinite ordinal α. At this point we may wonder if we get all the sets in the constructible hierarchy as we do in the cumulative hierarchy. The assumption that we do, that is if X is a set, then X L α for some α, is called the Axiom of Constructibility. Axiom of Constructibility. V = L V = L seems harmless enough since no one knows of a set that is not constuctible. However, since there is no proof that every set is in L, most set theorists are wary of V = L. Also, V = L implies both the Axiom of Choice and CH, so if L were a model for ZF, the Axiom of Choice and CH are consistent with ZF (and L also models ZFC). Fact. L = α OrdL α models ZF. The remainder of this paper will show how V = L implies CH. 3 Preliminary Results The basic idea now is to show that every constructible subset of ω belongs to L Ω, where Ω is the first uncountable ordinal. Assuming V = L we then have (ω) L Ω, and from above L Ω = Ω = ℵ 1 (Note: We define the cardinal number of a set as the least ordinal which can be mapped bijectively to the set. Thus Ω = ℵ 1, the first uncountable cardinal.) If X is a constructible subset of ω, we will show it is in L Ω by constructing a set M which models X L Ω while at the same time making sure what M models is really true. This will require some powerful tools: absoluteness, adequacy, and reflection. Absoluteness is what will ultimately insure the set M we construct will model true things, i.e. M = ϕ implies ϕ. Definition. Let M be a transitive set and ϕ(x 1,..., x n ) be a formula. We say ϕ is absolute for M if M = ϕ(x 1,..., x n ) if and only if ϕ(x 1,..., x n ) for all x 1,..., x n M. Lemma. Every restricted formula is absolute. 5

Proof: Let ϕ(v 1,..., v n ) be a restricted formula, M a transitive set, and x 1,..., x n M. If ϕ is atomic, a negation, or a conjunction, it is clear from the definition of = that ϕ is absolute. So suppose ϕ is of the form ( v i v j )(ψ). Then M = ϕ iff there exists x i M such that M = (x i x j ) and M = ψ(x 1,..., x n ). By transitivity, this is equivalent to saying there exists x i x j such that M = ψ(x 1,..., x n ). By induction, the last condition holds iff ψ(x 1,..., x n ) is actually true, so the whole thing is equivalent to the truth of ϕ. Hence, M = ϕ iff ϕ. The following expressions are absolute since they are defined by restricted formulas: x y, x =, x is transitive, x is an ordinal, x = ω, z = F i (x, y) for 1 i 10. Unfortunately, the expression x = L α is not absolute, so we define adequacy to get around that problem. Definition. Let φ be the expression (1) for all x and y and 1 i 10 there exists z such that z = F i (x, y); (2) for all x there exists a set y which equals cl(x); and (3) for every ordinal α there exists a family {L β β < α} such that L 0 =, L β+1 = def(l β ), and L β = γ<βl γ for limit ordinals β. We call a set M adequate if M = φ. Gödel devised adequacy to be a condition under which x = L α acts like it is absolute, so the following result is not surprising. Lemma. If M is transitive and adequate, then for every ordinal α M, M = (x = L α ) implies that x = L α. So now we have some conditions on a set M which guarantees the truth of any formula modeled by M. Next we address which sets can model various formulas. Specifically, we will have a version of the Löwenheim-Skolem theorem called the reflection principle. The reflection principle states that if ϕ if satisfied by a countable set, then there is a nice countable set which models ϕ. While the set is nice, it is not transitive. Definition. A set M is called extensional if for any distinct x, y M there is z M which belongs to one or the other of x and y, but not both. In other words, if x, y M and x y, then M = (x y). Lemma. (Reflection Principle) Let M 0 be a countable set and let ϕ(v 1,..., v 2 ) be any formula. Suppose ϕ(x 1,..., x n ) is true for some x 1,..., x n M 0. Then there is a countable set M such that M 0 M; M is extensional; and M = ϕ(x 1,..., x n ). 6

We are almost ready to construct the desired set M. The last speed bump is that the reflection principle provides extensional sets, but for absoluteness we need a transitive set. This problem is taken care of in the next theorem, known as Mostowski s collapsing theoem. Theorem. Let M be an extensional set. Then there is a transitive set N which is isomorphic to M. That is, there is a bijection f : M N such that x y iff f(x) f(y). For any formula ϕ(v 1,..., v n ) and any x 1,..., x n M, we have M = ϕ(x 1,..., x n ) iff N = ϕ(f(x 1 ),..., f(x n )). Proof: We will define f by transfinite induction and let N be the range of f (so f is onto). By the Axiom of Regularity, there is a ɛ-minimal element, x, of M; since M is extensional, x is unique. Define f(x) =. For any y M, we may inductively assume that f is defined on every element of y which belongs to M. Then define f(y) = {f(z) z y M}. Suppose f is not one-to-one. Then the set S = {z M z = f(x) = f(y) for some x y} is nonempty and hence contains a minimal element z = f(x) = f(y) such that x y. Since M is extensional, without loss of generality, we may assume that there is w M so that w x y. Thus, f(w) f(x) = z, and by minimality of z it is not the case that f(w) = f(v) for any other v M. Then f(w) f(y), therefore, f(x) f(y), a contradiction. So f is bijective. Clearly, x y implies f(x) f(y). So suppose f(x) f(y). Then by the definition of f(y), f(x) = f(z) for some z y. Since f is one-to-one, x = z y. So f is an isomorphism onto N = f[m]. The fact that f preserves the satisfaction of formulas is a routine induction on the length of the formula. We can now put all the pieces together and show that V = L implies CH. 4 Consistency of CH Theorem Assume V = L. Then 2 ℵ 0 = ℵ 1. Proof: Let X ω. We claim that X L Ω. Once we show the claim we are done, since we will have (ω) L Ω. Then ℵ 0 < (ω) = 2 ℵ 0 L Ω = Ω = ℵ 1. So now we prove the claim. Since V = L, for some α we have X L α. By the reflection principle we can find a countable set M such that 7

(1) M is extensional; (2) ω M, X M, and α M; (3) M is adequate; and (4) M = (α is an ordinal and X L α ). By Mostowski collapse there exists an isomorphism f : M N where N is transitive. Since ω M, an easy induction shows that f(n) = n for all n ω. It also follows from the definition of f in the previous thereom that f(x) = X. Also, by (4), N = (α is an ordinal), hence by absoluteness β = f(α) actually is an ordinal. By (3), N is adequate, and by (4) N = (X = f(x) L f(α) = L β ). Thus, X L β. Now, β N and N is transitive and countable, so β is countable. Hence, X L β L Ω. By transitivity, X L Ω. References [Co] Cohen, P. Set Theory and the Continuum. The Benjamin/Cummings Publishing Company, London, (1966). [Gö] Gödel, K. The Consistency of the Continuum Hypothesis. Princeton University Press, Princeton, (1940). [Je] Jech, T. Set Theory. Academic Press, New York, (1978). [We] Weaver, N. Lecture notes in Set Theory. (1996?). 8