Forcing Axioms and Inner Models of Set Theory

Forcing Axioms and Inner Models of Set Theory Boban Veličković Equipe de Logique Université de Paris 7 http://www.logique.jussieu.fr/ boban 15th Boise Extravaganza in Set Theory Boise State University, Boise, ID April 1 2006 1

1 History and Motivation This is a report on an ongoing collaboration with Andres Caicedo (Caltech) and a student of mine Matteo Viale. Forcing axioms are natural combinatorial statements which decide many of the questions left open by the usual axioms ZFC of set theory. They assert a kind of saturation of the universe of sets, i.e. if a set satisfying certain properties can be found in a suitable generic extension of the universe then such a set already exists. Let K be a class of forcing notions and κ an uncountable cardinal. FA κ (K) is the statement. For every P K and a family D of κ dense subsets of P there is a filter G in P such that G D, for all D D. 2

Examples of forcing axioms (Martin and Solovay) Martin s Axiom (MA + CH) - provides a rich structure theory for the reals and subsets of ℵ 1. (Baumgartner and Shelah) Proper Forcing Axiom (PFA) Very successful in resolving questions left open by MA. The proof of the consistency of PFA uses a supercompact cardinal. (Foreman, Magidor, Shelah) Martin s Maximum (MM) - the provably maximal forcing axiom. Resolves questions left open by PFA such as the saturation of the nonstationary ideal NS ω1, Chang s Conjecture, etc. 3

(Goldstern, Shelah) Bounded versions of traditional forcing axioms. Have many of the same consequences, yet require much smaller large cardinal assumptions. Bounded Proper Forcing Axiom (BPFA), Bounded Martin s Maximum (BMM), etc. (Woodin) Axiom ( ). This is different from the other forcing axioms. Assuming large cardinals and working in L(R) Woodin defines a forcing notion P max. If G is a P max -generic the model L(R)[G] satisfies the following. Any Π 2 statement over the structure H ℵ2 which can be forced is already true. Woodin s model satisfies V = L(P(ω 1 )). It is not known if one can obtain such an axiom without going to the inner model L(R). 4

Question 1 Is there an inner model theory for forcing axioms? To what extent is a model of a forcing axiom determined by its cardinal structure? By a model we mean a transitive model containing all the ordinals. The motivation for this investigation comes from the following observation I made in 1986. Theorem 1 If V satisfies MM and M is an inner model of V such that ℵ M 2 = ℵ V 2 then P(ω 1 ) V M. In fact, this is a consequence of stationary set reflection and a result of Gitik. 5

Definition 2 Stationary Set Reflection: If κ ℵ 1 and S [κ] ℵ 0 is stationary then there is X κ with X = ℵ 1 and ω 1 X such that S [X] ℵ 0 is stationary. Shelah showed the following. Theorem 3 MM implies stationary set reflection. In fact, MM implies a much stronger form of reflection of stationary sets called SRP. Most of the consequences of MM which do not follow already from PFA are actually obtained using SRP. On the other hand, PFA does not even imply the weaker form of stationary set reflection. Theorem 4 (Gitik) Suppose M V are models of set theory and there is a real x V \ M. Then the set ([ω 2 ] ℵ 0 ) V \ M is stationary in V. 6

Proof of Theorem 1: For each α [ω 1, ω 2 ) choose in M a closed unbounded set C α [α] ℵ 0. Let S = α C α. By Gitik s result if R V M then E = [ω 2 ] ℵ 0 \ S is stationary. By MM there is α < ω 2 such that E [α] ℵ 0 is stationary, which is a contradiction. Thus, R M = R V. But then by almost disjoint coding P(ω 1 ) V M. 7

Assuming V W are models of Woodin s axiom ( ) and have the same cardinals. Then V = W. To what extent can we extend this to models of other forcing axioms? Question 2 If V W are models of some strong forcing axiom and V and W have the same cardinals. Is ORD ω 1 V = ORD ω 1 W? Why require V and W to have the same cardinals? We want to avoid situations like the following. 8

Suppose V satisfies say MM and has a supercompact cardinal. We can first collapse cardinals and then force MM all over again to obtain a generic extension W of V such that V and W both satisfy MM but otherwise have little in common. If V has a proper class of completely Jónsson cardinals and G is generic for the class stationary tower forcing P and W = V [G] then there is an elementary embedding j : V W and we can arrange cp(j) to be arbitrary high and of cofinality ω in W. 9

These questions are closely related to the study of the Härtig quantifier. The logic L(I) is obtained by augmenting first-order logic with the binary quantifier I. If M = (M,...) is a structure, iff M = (M,...) = Ixy (φ(x), ψ(y)) { b M : M = φ(b) } = { c M : M = ψ(c) }. One is then interested in the expressive power of the logic L(I) under the assumption of strong forcing axioms. 10

In 1984 Foreman, Magidor and Shelah showed that MM implies that κ ℵ 1 = κ, for every regular κ ℵ 2. In 1987 Todorcevic and I showed that PFA implies that 2 ℵ 0 = ℵ 2. In 2003 Moore showed that BPFA implies 2 ℵ 0 = ℵ 2. His proof gives a well ordering of the reals which is 2 -definable over H(ℵ 2 ) with parameter a subset of ω 1. Can one improve this to a 1 well ordering of the reals? Does PFA imply that κ ℵ 1 = κ, for all κ ℵ 2, in particular, does PFA imply the Singular Cardinal Hypothesis (SCH)? Can one get well orderings definable without parameters? In 2003 Moore introduced the Mapping Reflection Principle (MRP) and deduced it from PFA. This principle plays a crucial role in our analysis. 11

2 The mapping reflection principle Definition 5 Let θ be a regular cardinal, let X be uncountable, and let M H θ be countable such that [X] ω M. A subset Σ of [X] ω is M-stationary iff for all E M such that E [X] ω is club, Σ E M. Recall the Ellentuck topology on [X] ω. Basic open sets are of the form [x, N] = {Y [X] ω : x Y N} where N [X] ω and x N is finite. Definition 6 A set mapping Σ is open stationary iff there is an uncountable set X = X Σ and a regular cardinal θ = θ Σ such that [X] ω H θ, dom(σ) is a club in [H θ ] ω and Σ(M) [X] ω is open and M-stationary, for every M dom(σ). 12

Definition 7 The Mapping Reflection Principle is the following statement. If Σ is an open stationary set mapping whose domain is a club, there is a continuous -chain N = (N ξ : ξ < ω 1 ) of elements dom(σ) such that for all limit ordinals ξ < ω 1 there is ν < ξ such that N η X Σ Σ(N ξ ) for all η such that ν < η < ξ. If (N ξ : ξ < ω 1 ) satisfies the conclusion of MRP for Σ then it is said to be a reflecting sequence for Σ. 13

Example. Suppose α = K 0 α K 1 α is a partition of α into clopen sets, for each α < ω 1. Let M H ℵ2 be countable. Then one of K 0 α and K 1 α is M- stationary. Let Σ(M) = [ω 1 ] ℵ 0 \ K i α where i is such that Kα 1 i is M-stationary. Let N be a reflecting sequence for Σ. Set C = {N ξ ω 1 : ξ < ω 1 }. Then C is a club subset of ω 1 and for every limit point ξ of C there is i such that C \ Kξ i is bounded in ξ. 14

Theorem 8 (Moore) PFA implies MRP. In fact, what Moore showed is that given a stationary set mapping Σ there is a proper poset which adds a reflecting sequence for Σ. Using a bounded version of MRP he showed the following. Theorem 9 (Moore) BPFA implies 2 ℵ 0 = ℵ 2. On the other hand Moore also showed that MRP implies the failure of κ, for all κ > ω 1. This was later improved by Sharon. Theorem 10 (Sharon) Assume MRP. Then κ,ω fails, for all κ ω 1. Thus, MRP has considerable large cardinal strength. 15

We will discuss the following two theorems. Theorem 11 (Caicedo, V.) Assume V W are two models of set theory, ℵ V 2 = ℵ W 2 and BPFA holds in both V and W. Then P(ω 1 ) W V. Theorem 12 (Caicedo, V.) Assume BPFA. Then there is a 1 -definable well-ordering of P(ω 1 ) with parameter a subset of ω 1. The length of the wellordering is ω 2. Sketch Let C = ( C ξ : ξ < ω 1 & lim(ξ)) be a fixed C-sequence, i.e., C ξ is cofinal in ξ of order type ω, for all limit ξ < ω 1. We define a robust coding of reals by ordinals, in fact by triples of ordinals < ω 2. We can then use almost disjoint coding relative to a fixed ω 1 -sequence of reals to code subsets of ω 1 by reals. 16

Let ω 1 < β < γ < δ be fixed limit ordinals and suppose N M δ are countable sets of ordinals. Assume that {ω 1, β, γ} N, that sup(ξ N) < sup(ξ M) and sup(ξ M) is a limit ordinal, for every ξ {ω 1, β, γ, δ}. We define a finite binary sequence s βγδ (N, M). This sequence is defined using the oscillation of N and M relative to β, γ and δ and the fixed parameter C. Finally, we say that the triple (β, γ, δ) codes a real r if there is a continuous increasing sequence of countable sets (N ξ : ξ < ω 1 ) whose union is δ such that for every countable limit ordinal ξ there is ν < ξ such that r = s βγδ (N η, N ξ ). ν<η<ξ 17

Let r be a given real. To show that r is coded by a triple of ordinals we first use MRP. We define an open stationary set mapping Σ r such that a reflecting sequence for Σ r gives us a triple of ordinals coding r. Σ r is defined on the set of all countable elementary submodels of H ℵ4 containing C. For M dom(σ r ) we let { N [M ω 4 ] ω : s ω2 ω 3 ω 4 (N, M ω 4 ) r }. Σ r (M) is open in the Ellentuck topology. The hard part is showing that it is M-stationary. This is done using Namba type games inside M. Let M be the transitive collapse of M and let π be the collapsing map. Then, from the point of view of M, obviously ω M 2, ω M 3 and ω M 4 are different regular cardinals, but from the outside they are actually countable ordinals and the C-sequence provides a witness to this. 18

Not every triple of ordinals < ω 2 codes a real, but given ordinals ω 1 < β < γ < δ < ω 2 of cofinality ω 1 we can define a family of clopen partitions α = K 0 α K 1 α, for α < ω 1 as in the example presented above. Now, using a bounded version of MRP we can show that (β, γ, δ) codes something, although not necessarily a real. Now, given V W such that ℵ V 2 = ℵ W 2 and both V and W satisfy MRP we show that R W V as follows. 19

Fix a C-sequence C in V as a parameter. Then any real r W is coded in W by some triple (β, γ, δ) and this is witnessed by a continuous increasing sequence N = (N ξ : ξ < ω 1 ) of countable subsets of δ whose union is δ. In V there is another continuous increasing sequence P = (P ξ : ξ < ω 1 ) witnessing that (β, γ, δ) codes something. But there is a club of ξ < ω 1 such that N ξ = P ξ and the coding is local enough to guarantee that on any limit point of this club, P actually codes r, so r V. 20

Let V I be the set of Gödel numbers of the validities for the logic L(I) with the Härtig quantifier. Theorem 13 (Caicedo, V.) Assume BPFA. Then V I is not projective. Using the result of Steel that PFA implies AD L(R) and a result of Solovay saying that AD L(R) and the existence of R imply that there is a real which is ordinal definable in L(R ) but not ordinal definable in L(R) we obtain the following. Theorem 14 (Caicedo, V.) Assume PFA. V I is not ordinal definable in L(R). Then 21

3 The P -ideal dichotomy In the Fall of 2005 my student Matteo Viale showed that PFA implies the Singular Cardinal Hypothesis. He used some ideas of Moore and deduced SCH from MRP. Recently he found another proof of SCH using another consequence of PFA, the P- ideal dichotomy. Since this proof seems optimal we present the main ideas. Definition 15 A P-ideal I on a set X is an ideal containing all finite subsets of X such that for every sequence {A n } n of elements of I there is A I such that A n A for all n, where A B means A \ B is finite. Given A, B X we write A B iff A B is finite. Given any family A P(X) we let A = {B : B A, for all A A} Note that for any A P(X) A is an ideal, but not necessarily a P-ideal. 22

P-ideal dichotomy(pid). Let X be any set and suppose I [X] ℵ 0 is a P-ideal such that all finite subsets of X are in I. Then either there is an uncountable Y X such that or [Y ] ℵ 0 I, there is a countable sequence {X n } n, such that X n I, for all n, and such that X = n X n. The P-ideal dichotomy was introduced by Abraham and Todorcevic and is known to follow from PFA. 23

Theorem 16 (Viale) Assume PID. Then κ ℵ 0 = κ, for all κ 2 ℵ 0. In particular, PID implies SCH. Let κ be a cardinal of countable cofinality. A covering matrix for κ + is a sequence such that C = {K(n, β) : n < ω, β < κ + } 1. K(n, β) < κ, for all n and β 2. K(n, β) K(m, β), for all β and n < m 3. β = n K(n, β), for all β 4. for all α < β and n there is m such that K(n, α) K(m, β). 24

Fact 1 For every singular cardinal κ of countable cofinality there is a covering matrix for κ +. be a cardinal of countable cofi- Now, let κ > 2 ℵ 0 nality and let C = {K(n, β) : n < ω, β < κ + } be a covering matrix for κ +. Let I = C. Fact 2 I is a P-ideal. Fact 3 There is no uncountable set X κ + such that [X] ℵ 0 I. 25

By the P-ideal dichotomy one gets a decomposition κ + = n X n such that X n I, for all n. Fact 4 [X n ] ℵ 0 = m<ω,β<κ + [K(m, β)] ℵ 0. Since there is n such that X n = κ + and K(m, β) is of size < κ, for all m and β it follows that (κ + ) ℵ 0 = κ + sup λ ℵ 0. λ<κ This allows us to show that κ ℵ 0 = κ, for all regular κ 2 ℵ 0 by induction on κ. The only nontrivial case is the successors of singular cardinals of countable cofinality and this is handled by the above facts. 26

Using similar, but more involved ideas, Viale has also shown the following. Theorem 17 (Viale) Assume V W, the P-ideal dichotomy holds in W and V and W have the same reals and cardinals. Let κ be the least such that (κ ω ) W V. Then κ is not a regular cardinal in V. 27

Theorem 18 (Viale) Assume V W are models with the same cardinals and reals. Assume that W satisfies PID and κ is the least such that (κ ω ) W V (so cof(κ) V = ω). 1. For every W -regular λ < κ there is a stationary set in κ + consisting of ordinals of V cofinality λ which is not stationary in W. 2. If κ = ℵ V ω then κ is V -Jónsson, i.e. any algebra on κ which is in V has a proper subalgebra of the same size in W. So, if V and W have the same bounded subsets of κ then κ is Jónsson in W. 28

Open problems Assume If V W, both V and W are models of PFA and have the same cardinals. Is ORD ω 1 V = ORD ω 1 W? Does PFA or MM imply the existence of a wellordering of the reals which is definable without parameters? Asperó has shown that the existence of such a well-ordering is consistent with PFA. Does the P-ideal dichotomy imply 2 ℵ 0 ℵ 2? It is known that most standard forcing for adding a real destroy the P-ideal dichotomy. 29