Boolean-Valued Models and Forcing

Boolean-Valued Models and Forcing Abstract This introduction to forcing is based on Chapters 5 6 in T. J. Jech s book The Axiom of Choice and is written primarily for the Fraenkel-Mostowski Models reading group. Forcing is introduced via Boolean-valued models and generic extensions, and these techniques are used to prove the independence of the axiom of choice of ZF. This is then related back to the theory of FM models. Please inform me of any errors by sending an email to cn319@cam.ac.uk. 1 Boolean-valued models First we need to fix some notation. Let M be a fixed transitive model of ZFC sitting inside the universe V. Throughout this section we will be working inside M. Let B M be a complete Boolean algebra in M. This means that B is a set in M with the structure of a Boolean algebra, which is complete inside M, meaning that if A B and A M then A M and A M. Note that B need not contain the joins and meets of all its subsets just those that lie in M. We ll write for the Boolean operation defined by a b = a b. The B-valued model M B is defined by transfinite recursion: M B 0 =, M B α+1 = [M B α B], M B λ = α<λ M B α, M B = where [X Y ] denotes the set of functions f with dom(f) X and im(f) Y. We can embed M M B by x ˇx, where ˇx is the function defined recursively by dom(ˇx) = {ˇy : y x} and ˇx(ˇy) = 1 for all y x α Ord Very loosely speaking, it is intuitively helpful to think of elements of M B (which are functions) as being names for sets in another model. Then, for x M B and y dom(x), we can think of x(y) as being the probability that the set named y is a member of the set named x. We make this more precise by considering Boolean values of formulae with variables in M B ; that is, to each formula φ we assign an element of B. Suppose φ(x 1 x n ) is a formula with variables in M B. Its Boolean value φ(x 1 x n ) B B is defined by recursion on the complexity of φ: (i 1 ) x y B = z dom(y) (i 2 ) x = y B = z dom(x) y(z) z = x B (x(z) z y B ) z dom(y) (ii) B respects the Boolean operations,,,. 1 (iii 1 ) xφ B = { φ(x) B : x M B } (iii 2 ) xφ B = { φ(x) B : x M B } (y(z) z x B ) From now on, we drop the subscript. Before moving on, notice the similarity between (i 1 ) and the logical formula x y ( z y)(z = x) and, likewise, the similarity between (i 2 ) and the formula x = y ( z x)(z y) (z y)(z x) This is no coincidence. In fact, it is intuitively helpful to think of the order in B as being implication. This helps motivate and make sense of the following properties of the Boolean value: 1 Here I ve been lazy. What I mean is that, for example, φ ψ B = φ B ψ B, where is logical implication and is the Boolean operation on B. M B α 1

(1) x = x = 1 (2) x(y) y x (3) x = y = y = x (4) x = y y = z x = z (5) x = z x y z y (6) x = z y x y z (7) x = y φ(x) φ(y) (8) ( y x)φ(y) = y dom(x) {x(y) φ(y) } (9) ( y x)φ(y) = y dom(x) {x(y) φ(y) } Why are these useful? We say a formula φ(x 1 x n ) with variables in M B is valid in M B if φ(x 1 x n ) = 1. Theorem 1.1 ˆ Every axiom of the predicate calculus is valid in M B, and if φ is obtained via a rule of inference applied to valid formulae then φ is valid in M B. In particular, every theorem is valid in M B. ˆ Every axiom of ZFC is valid in M B. In particular, by the previous remark, we also have that every sentence provable from the ZFC axioms is valid in M B. 2 Generic extensions Now we step outside the model M and into the universe V. The goal is to extend M to a larger model M[G] obtained by adding to M a special subset G of our complete Boolean algebra B, and throwing in everything else that we need in order for this to be a model of ZFC (and nothing more). A subset G B is an M-generic ultrafilter if ˆ G is an ultrafilter, i.e. It is upwards-closed: for all b G, if b c then c G; It is closed under meets: for all a, b G, a b G; If b B then either b G or b G, but not both (so, in particular, 0 G). ˆ G contains the meets of all its subsets which lie in M, i.e. if A G and A M then A G. A generic ultrafilter is precisely the special subset we wish to adjoin to M. To construct M[G], we go via M B. The interpretation ι G of M B by G is defined recursively by ι G (x) = {ι G (y) : x(y) G, y dom(x)} We say that x M G is a G-name for x V if ι G (x) = x; so every function in M G is a name for some set in the universe. Then M[G] is defined to be precisely collection of sets in the universe which have G-names; that is, M[G] = {ι G (x) : x M B } So far this definition doesn t serve much use. But it turns out to be useful thanks to the following two theorems. Theorem 2.1 (a) M M[G] and G M[G] (b) If x, y are names for x, y, then x y if and only if x y G, and x = y if and only if x = y G Proof: 2

(a) By -recursion, ι G (ˇx) = {ι G (ˇy) : ˇy dom(ˇx)} = {y : y x} = x so each x M has a name, so M M[G]. We define the canonical generic ultrafilter G by dom(g) = {ǔ : u B}, G(ǔ) = u for all u B then so G has a name, so G M[G]. ι G (G) = {ι G (x) : G(x) G} = {ι G (ǔ) : u G} = G (b) Direct calculation. Theorem 2.2 If φ(x 1 x n ) is a formula and x i are names for x i for each 1 i n, then M[G] φ(x 1 x n ) if and only if φ(x 1 x n ) G Corollary 2.3 (a) M[G] is a model of ZFC; (b) M[G] is the least model N of ZF with M N and G N. We call M[G] the generic extension of M by G. At this point you might wonder what the point was in doing all this work: our goal is to prove the independence of the axiom of choice from ZF, and yet all we ve done so far is construct another model where choice holds! However, provided G M (as is usually the case), we have a strict inclusion M M[G]. The model N of ZF which we will construct in which the axiom of choice fails will satisfy M N M[G]; in particular, it cannot contain G as a set. Before we can do this properly, we need to introduce forcing. 3 Forcing Before we go too deep into forcing, let s review some of what we ve done. The intuition behind the Boolean values of formulae is to see φ B as being the probability that φ is true. So, in some vague sense, these formulae form a poset ordered by implication. Then if some formula φ is true, so are all its successors in this poset. What we do with forcing is take some poset (P, ) and embed it in a complete Boolean algebra B; then an element p P forces a formula φ if its image under the embedding lies somewhere below φ. We make this precise in what follows. Let (P, ) be a poset. We will call the elements of P forcing conditions, and if p q we say that p is stronger than q and that q is weaker than p. [Compare this to Section 1 where we said that can be thought of as meaning implies : indeed, p implies q if and only if p is stronger than q, so this terminology makes sense.] A collection D P of forcing conditions is dense if every forcing condition is weaker than one of its elements; that is, for all p P there exists d D with d p. We say p, q P are compatible if they share a stronger condition, i.e. if there exists r P with r p and r q; otherwise they are incompatible. [In the latter case, we write p q.] A subset Γ P is M-generic if ˆ Γ is upwards-closed: if x Γ and x y then y Γ; 3

ˆ Γ is compatible: if x, y Γ then x and y are compatible; ˆ If D P is dense and D M then D Γ. If B is a complete Boolean algebra and P B {0} is dense, then there is a correspondence between generic ultrafilters in B and generic subsets of P. More precisely: ˆ If G is a generic ultrafilter on B then Γ = G P is a generic subset of P ; ˆ If Γ P is generic then G = {u B : ( p Γ)(p u)} is a generic ultrafilter in B. Unfortunately, when we re dealing with most posets, we don t have for free that it is a dense subset of a complete Boolean algebra. However, with the help of some topological techniques, we can make it so. Let X be a topological space. A subset A X is regular open in X if it is equal to the interior of its closure. 2 We denote the collection of all such sets by RO(X). Then RO(X) is a complete Boolean algebra whose operations are defined by: ˆ A = A = int(a c ); ˆ A B = (A B) ; ˆ A B = A B. In particular, RO(X) is partially ordered by. We can endow an arbitrary poset (P, ) with the topology whose basic open sets are [p] = {q P : q p} Then P embeds into RO(P ) by e : P RO(P ), where e(p) = int(cl( [p])) This embedding has some useful properties: ˆ e preserves order: if p q then e(p) e(q); ˆ e preserves compatibility: if p and q are compatible then e(p) e(q) ; ˆ The image of P under e is dense in RO(P ). These conditions imply that the correspondence between generic subsets and generic ultrafilters can be carried over to this scenario. Indeed, if G RO(P ) is a generic ultrafilter then Γ = e 1 (G) P is generic; and if Γ P is generic then G = {u RO(P ) : ( p Γ)(e(p) u)} is a generic ultrafilter in RO(P ). We interchangeably write M[Γ] = M[G]. Returning to more familiar notation, let B = RO(P ), let M, M B, all be as before, and suppose φ(x 1 x n ) is a formula with free variables in M B. We say p P forces φ(x 1 x n ), and write p φ(x 1 x n ), if e(p) φ(x 1 x n ). Some useful properties of forcing are as follows: p φ if and only if for all q p, q φ p φ ψ if and only if p φ and p ψ p φ ψ if and only if for all q p there exists r q with r φ or r ψ p xφ if and only if for all x M B, p φ(x) p xφ if and only if for all q p there exist r q and x M B with r φ(x) 2 For example, in R, the set R {0} is open but it is not regular open because the interior of its closure is R. 4

The power behind forcing comes from the following theorem, which is essentially a restatement of Theorem 2.2. Theorem 3.1 Let φ(x 1 x n ) be a formula, Γ P an M-generic set, and x i names for x i in M B. Then M[Γ] φ(x 1 x n ) if and only if there is some p Γ with p φ(x 1 x n ). 4 Symmetric submodels Symmetric submodels can be motivated by looking at how we made the axiom of choice fail in a permutation model: we took our model of ZFA+( AC) to consist of those sets whose symmetric groups lie in a given normal filter of subgroups of a group of -automorphisms of the universe (with atoms). The reason why we can t just do that here is because there are no nontrivial -automorphisms of V : indeed, every -automorphism of the universe with atoms fixes the kernel V. Nevertheless, although there are no nontrivial automorphisms of V or of M, there are nontrivial automorphisms of M B. So instead of permuting sets in M[G], we ll permute their names, and then pass to a submodel where choice fails. So, let B be a complete Boolean algebra, G be a generic ultrafilter on B, and M and M B be as usual. An automorphism π of B is a bijection π : B B which preserves the Boolean operations of B. We can then extend π to an automorphism of M B by recursion: dom(π(x)) = {π(y) : y dom(x)}, π(x)(π(y)) = π(x(y)) for all π(y) dom(π(x)) Notice that π(ˇx) = ˇx for all x M, since ˇx(ˇy) = 1 for all y x and we must have π(1) = 1. The induced automorphism of π has a useful homogeneity property: Lemma 4.1 If φ(x 1 x n ) has variables in M B then φ(π(x 1 ) π(x n )) = π( φ(x 1 x n ) ) This can be proved by direct calculation using induction on the complexity of φ. After the following definitions we will have all the machinery we need to prove the independence of the axiom of choice from ZF, provided we make smart chioces along the way. Let G be a group of automorphisms of B. We say a collection F of subsets of G is a normal filter if ˆ F is closed upwards: if H F and H K G then K F; ˆ F is closed under intersections: if H, K F then H K F; ˆ F is closed under G-conjugations: if H F and π G then π 1 Hπ F. We say a name x M B is (F-)symmetric if sym G (x) = {π G : π(x) = x} F Let HS F denote the class of hereditarily (F-)symmetric names; that is, HS F and if dom(x) HS F and x is F-symmetric then x HS F. Finally, define N F = {ι G (x) : x HS F } to be the class of sets with hereditarily F-symmetric names. Notice that, by our observation, ˇx HS F for each x M, since sym G (ˇx) = G F, and so M N F M[G] 5

and, usually, both of these inclusions will be strict. This is useful because... Theorem 4.2 N F is a model of ZF The proof is in Jech s book; working through it isn t as illuminating as seeing how it is used. Notice the lack of C in Theorem 4.2: it is in N F that we hope the axiom of choice will fail, and we ensure this by choosing G and F cleverly. 5 A model of ZF where choice fails The following construction is due to Cohen. Let M be a model of ZFC, and define a poset (P, ) by P = [ω ω 2] fin = {(f : ω ω 2) : dom(f) < ℵ 0 } with f g if and only if f extends g. Let B = RO(P ) and let G be a generic filter over B. We define M B and M[G] as in the previous sections. A permutation π of ω induces an automorphism of P given by dom(π(p)) = {(n, m) : (π(n), m) dom(p)}, π(p)(n, m) = p(π(n), m) and thus induces an automorphism of B by π(u) = {π(p) : p u} Let G be the group of automorphisms of B which are induced by permutations of ω in the above sense, and define a normal filter F = {H G : ( e ω)( e < ℵ 0 fix G (e) H)} where fix G (e) = {π G : π e = id e }. Let N F be as in Section 4. Theorem 5.1 N F AC Corollary 5.2 The axiom of choice is independent of ZF. Proof of Theorem 5.1: For each n ω define x n = {m ω : p(n, m) = 1 for some p G} P(ω) = R and let A = {x n : n ω}. We ll show that A and all its members lie in the model N = N F, but that there is no bijection ω A in N. This proves what we want, since if A were to have a choice function then there would certainly be a bijection ω A. We ll do this in a few steps. Step 1: x n N for all n ω, and A N. Each x n has a name x n M B defined by dom(x n ) = { ˇm : m ω}, x n ( ˇm) = u n,m = {p P : p(n, m) = 1} Then π(u n,m ) = u π 1 (n),m, and so we have sym G (x n ) = fix G ({n}) F, and hence x n N. And A has a name A M B defined by dom(a) = {x n : n ω}, A(x n ) = 1 for all n ω Then π(a) = A, and so we have sym G (A) = G F and hence A N. Step 2: The reals in A are pairwise distinct. 6

Suppose, for contradiction, that we have p P and i j with p (x i = x j ). By definition there is some m ω with (i, m) dom(p) and (j, m) dom(p). For this m, define q p to be the function extending p which satisfies q(i, m) = 0 and q(j, m) = 1. Then q (x i x j ). But q p, so this would mean p (x i x j ), contradicting our assumption. Hence x i = x j = 0 for all i j, and so the x i are distinct. Step 3: There is no bijection ω A in N. Suppose, for contradiction, that such a bijection exists, and let its name be f. Then there must be some p G with p (f is a bijection ˇω A). Let e ω be a finite subset with fix G (e) sym G (f). Then there exists i ω, q p and n e such that p (f(ǐ) = x n ). Choose n ω e with (n, m) dom(q) for each m ω, and let π be the permutation of ω which swaps n and n and fixes everything else. Then q and π(q) are compatible, so r = q π(q) q is a well-defined function extending q. Moreover, π fix(e), π(ǐ) = ǐ and π(p) (π(f)(ǐ) = π(x n )), so and in particular, r (f(ǐ) = x n f(x i ) = x π 1 (n) ) r (x n = x π 1 (n) ) So we must have p (x n = x π 1 (n)). But this contradicts our assumption since, by Step 2, x n x π 1 (n) since π 1 (n) n. 6 Relation to FM models In Cohen s model of ZF+( AC) in Section 5, it is quite remarkable how similarly the set A behaves like the set of atoms in a permutation model. It is natural to look for some kind of correspondence between symmetric extensions of a transitive model M of ZFC and permutation models of ZFA. 3 Unfortunately, whereas atoms are structureless and it is (in general) impossible to distinguish between them, subsets of ω carry too much structure and we can usually distinguish between them. However, if we look for a different kind of set to correspond to the set of atoms, we will have more luck. This is put precisely in the following theorem. Theorem 6.1 (First embedding theorem) Let U be a model of ZFCA, let A U be the set of atoms, let M be the kernel of U and let α be an ordinal in U. Then, for every permutation model V U of ZFA, there exists a symmetric extension N M (which is a model of ZF) and a set à N such that P α (A) V is -isomorphic to P α (Ã)N We will not prove this in full here; but we will construct a generic extension M[G] of M explicitly. Let G be a group of permutations of A and let F be a normal filter of subgroups of G such for which V is the class of hereditarily (G, F)-symmetric sets in U. Note that this is not the same construction as in Section 4; it is a model of ZFA, not ZF(C)! Let κ be a regular cardinal with κ > P α (A) in U. Define 3 For more on what this is, see Chapter 4 in Jech s book. P = {p [A κ κ 2] : dom(p) < κ} 7

where A M has the same cardinality of A. [From now on we ll abuse notation by identifying A with A ; but we must really use A so that P lies in M.] P is ordered by reverse-extension, i.e. p q if and only if p extends q. Let B = RO(P ) and let G be a generic ultrafilter of B. We ll construct Ã, and its name Ã, as follows: ˆ For a A and ξ < κ, define x a,ξ κ by x a,ξ = {η < κ : p(a, ξ, η) = 1 for some a A} Then x a,ξ has name x a,ξ M B, given by dom(x a,ξ ) = {ˇη : η < κ}, x a,ξ (ˇη) = u a,ξ,η = {p P : p(a, ξ, η) = 1} ˆ For a A, define Then ã has name ã M B, given by ã = {x a,ξ : ξ < κ} dom(ã) = {x a,ξ : ξ < η}, ã(x a,ξ ) = 1 for all ξ < κ ˆ At long last, à is defined by and this has name à given by à = {ã : a A} dom(ã) = {ã : a A}, Ã(ã) = 1 for all a A That is, we have a set (Ã) of sets (ã) of sets (x a,ξ) of ordinals (η), and these are all sets in M[G] since they all have names. In order to obtain the -isomorphism, we need some more identifications; fortunately, these are much more painless. For x U, define x M G by recursion: Then each x has a name x M B, given by x = {ỹ : y x} dom( x) = {ỹ : y x}, x(ỹ) = 1 for all y x The following results, whose proofs are in Chapter 6 of Jech s book, complete the proof: Step 1 x y if and only if x ỹ and x = y if and only if x = ỹ. Step 2 There is a group G of permutations of B and a normal filter F of subgroups of G for which, for all x U, x V if and only if x HS F. [Note to self, put the construction of G and F here some time.] Step 3 For all x U, x V if and only if x N = N F. Step 4 Pα (A) V = P α (Ã)N. Clive Newstead November 2012 8