Forcing axioms in Pmax extensions
Paul Larson
Department of Mathematics Miami University Oxford, Ohio 45056 [email protected]
March 12, 2014 Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals
Martin’s Maximum with Caicedo, Sargsyan, Schindler, Steel, Zeman. Part of an
Determinacy AIM Square project.
Pmax principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Martin’s For Γ a class of partial orders FA(Γ) is the statement that for all Maximum P ∈ Γ, and for all collections {Dα : α < ω1} consisting of dense Determinacy subsets of P, there is a filter G ⊆ P intersecting each Dα. Pmax principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large Examples. cardinals • FA(c.c.c.) is MAℵ Martin’s 1 Maximum • FA(proper) is PFA Determinacy • FA(preserving stationary subsets of ω ) is MM Pmax 1 principles • FA(σ-closed*c.c.c) : stronger than MAℵ1 , weaker than The Solovay PFA sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Why only ℵ1 many dense sets? Even for Cohen forcing, no Martin’s Maximum filter can meet continuum many dense sets.
Determinacy
Pmax Theorem.[Todorcevic, Velickovic] FA(σ-closed * c.c.c.) implies ℵ0 principles that 2 = ℵ2. The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson • Forcing axioms say that the universe is closed under certain forcing operations (i.e., certain objects that can be Forcing axioms forced to exist exist already). Models of forcing axioms Large can be thought of a maximal, or complete, in contrast to cardinals
Martin’s fine structural models, which are minimal (with respect to Maximum some hypothesis). Determinacy • The consistency of forcing axioms can tell you what the P max absolute objects are in a given class. principles The Solovay • Destroying stationary subsets of ω1 is the only impediment sequence to a forcing axiom. Wadge rank • (Moore) PFA implies that the uncountable linear orders HOD have a five-element basis. MM(c+) • (Velickovic) PFA implies that for all infinite cardinals κ, all automorphisms of P(κ)/Fin are trivial. Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large cardinal hypotheses statements which assert the Large cardinals existence of infinite cardinals with certain properties.
Martin’s Maximum For example, a strongly inaccessible cardinal is a regular Determinacy cardinal closed under cardinal exponentiation. Pmax principles
The Solovay The existence of strongly inaccessible cardinals implies the sequence consistency of ZFC, so cannot be proved in ZFC. Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson • Empirically, large cardinal axioms are linearly ordered by Forcing axioms ϕ < ψ iff ZFC + ψ implies ZFC + Con(ϕ). Large • cardinals Fine structural models have been produced for some initial
Martin’s segment of the hierarchy (roughly a Woodin limit of Maximum Woodins). Determinacy • Below this, we can show that large cardinals are necessary, Pmax principles and, often, show that statements (often having no obvious
The Solovay relation to large cardinals) are equiconsistent with some sequence large cardinal hypothesis. Wadge rank • Forcing axioms may be the most important statements HOD
MM(c+) beyond this level. Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Theorem.[Foreman-Magidor-Shelah] If there exists a Martin’s supercompact cardinal, then there is a forcing extension in Maximum which Martin’s Maximum holds. Determinacy
Pmax principles
The Solovay sequence
Wadge rank
HOD
MM(c+)
Question Is the supercompact cardinal necessary? Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals For a cardinal κ, MM(κ) is the restriction of Martin’s Martin’s Maximum Maximum to partial orders of cardinality at most κ.
Determinacy
Pmax Martin’s Axiom is equivalent to its restriction to partial orders principles of cardinality ℵ1, but MM is not equivalent to its restriction to The Solovay any small cardinal. sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large Theorem.[Woodin] Assuming ADR + “Θ is regular”, there is cardinals a forcing extension in which ZFC + MM(c) holds. Martin’s Maximum
Determinacy MM(c) implies that c = ℵ2. Pmax principles Theorem.[Sargsyan] ADR + “Θ is regular” has consistency The Solovay sequence strength below a Woodin limit of Woodin cardinals.
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals What about MM(c+)? Martin’s Maximum + Determinacy We will show that certain consequences of MM(c ) can be
Pmax produced from hypotheses below a Woodin limit of Woodin principles cardinals.
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Martin’s Traditional consistency proofs for forcing axioms, including the Maximum Foreman-Magidor-Shelah proof, are iterated forcing Determinacy constructions over models of ZFC. Pmax principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large ⊆ ω cardinals Given A ω, the game GA has ω many round, where players
Martin’s I and II alternately choose the members of a sequence Maximum ⟨ni : i ∈ ω⟩, and player I wins if ⟨ni : i ∈ ω⟩ ∈ A. Determinacy P max The set A is determined if either player I or player II has a principles winning strategy. The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large • The Axiom of Determinacy (AD) is the statement that cardinals ⊂ ω Martin’s every A ω is determined. Maximum • The Axiom of Real Determinacy (ADR) is the Determinacy corresponding statement for games where the players play P max elements of ωω. principles + The Solovay • AD (a statement in between, formulated by Woodin) sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals
Martin’s Maximum Θ is the least ordinal which is not a surjective image of ωω. Determinacy
Pmax principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large • P is a partial ordered developed by Woodin in the early cardinals max
Martin’s 1990’s. Maximum • Conditions are elements of H(ℵ1), essentially countable Determinacy transitive models of ZFC with some additional structure. Pmax • principles The order is induced by elementary embeddings with The Solovay critical point ω1. sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals + Martin’s • A Pmax extension of a model of AD satisfies MM(ℵ1). Maximum • P Determinacy A max extension of a model of ADR + “Θ is regular” ℵ Pmax satisfies MM(c) + c = 2. principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms • Pmax is ω-closed, preserves ω2, and makes Θ into ω3. Large cardinals • It forces a wellordering of R of ordertype ω2. Martin’s • ⊆ P + Maximum If G max is a V -generic filter (for V a model of AD ) P V [G] ⊆ R Determinacy then (ω1) L( )[G].
Pmax • Forcing with Pmax over a model of ADR + “Θ is regular” principles does not wellorder P(R), but P(R) can be wellordered The Solovay P sequence over the max extension (without adding subsets of ω2) by
Wadge rank forcing with Add(1, ω3).
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms In Pmax ∗ Add(1, ω3)-extensions of suitable models of Large cardinals determinacy (below a Woodin limit of Woodins) one can Martin’s obtain MM(c+) for partial orders P for which at least one of Maximum the following hold. Determinacy • Pmax Forcing with P does make make ω3 have cofinality ω1. principles • P is stationary set preserving in any outer model with the The Solovay sequence same ω1-sequences of ordinals.
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms The following definition is due to Jensen. Large cardinals For an infinite cardinal κ, asserts the existence of a Martin’s κ + + Maximum sequence ⟨Cα : α < κ ⟩ such that for all α < κ , Determinacy • Cα is a club subset of α P max • ∈ ∩ principles for all β lim(Cα), Cβ = Cα β The Solovay • ot(Cα) ≤ κ sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms • ⊆ Large there cannot exist a club E γ such that for all cardinals α ∈ lim(E), Cα = E ∩ α. Martin’s + Maximum • (κ ) : remove the condition ot(Cα) ≤ κ and assert the Determinacy nonexistence of such an E (so (κ+) is weaker) P max principles principles illustrate why partial orders preserving stationary
The Solovay subsets of ω1 don’t have to have small subalgebras with the sequence same property. Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large Much (possibly all) of the known consistency strength of MM cardinals comes from the following result. Martin’s Maximum
Determinacy Theorem.[Todorcevic] If γ is an ordinal of cofinality greater
Pmax than ω1, there exists a σ-closed*c.c.c. forcing P of cardinality ℵ principles |γ| 0 such that FA({P}) implies ¬(γ). The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals • In the Pmax ∗ Add(1, ω3) extension of a model of ADR + Martin’s ℵ Maximum “Θ is regular”, 2 0 = ℵ2. Determinacy ℵ • Given that 2 0 = ℵ2, MM(c) implies ¬(ω2) and Pmax + MM(c ) implies ¬(ω3). principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Martin’s Theorem.[CLSSSZ] Assuming a certain determinacy Maximum
Determinacy hypothesis below a Woodin limit of Woodin cardinals, the P ∗ ¬ Pmax max Add(1, ω3) extension satisfies MM(c) + ω2 . principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large • The previous upper bound for ¬(ω ) + ¬ was a cardinals 2 ω2
Martin’s quasicompact cardinal, above the current inner model Maximum theory. Determinacy • Lower bound : at least ADL(R) Pmax • principles From a stronger hypothesis (beyond a Woodin limit of The Solovay Woodin cardinals) we get ¬(ω3). sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms The Solovay sequence is the unique continuous sequence Large ⟨θα : α ≤ δ⟩ satisfying the following conditions. cardinals • Martin’s θ0 is the least ordinal γ for which there does not exist an Maximum ordinal definable function from ωω onto γ. Determinacy • if θ < Θ, then θ is the least ordinal γ for which there Pmax α α+1 P principles does not exist an ordinal definable function from (θα)
The Solovay onto γ. sequence • θ = Θ. Wadge rank δ
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals The following gives some indication of the strength of ADR +
Martin’s “Θ is regular”. Maximum
Determinacy Theorem.[Woodin] Assuming AD + V = L(P(R)), ADR holds P max if and only if the Solovay sequence has limit length. principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals HOD is the class of hereditarily ordinal definable sets. Martin’s Maximum
Determinacy Theorem.[Woodin] Assuming AD + DC, all successor
Pmax elements of the Solovay sequence are Woodin in HOD. principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large ⊆ ω ≤ cardinals Given A, B ω, say that A W B (A is Wadge below B) if −1 ω → ω Martin’s A = f [B], for some continuous f : ω ω. Maximum Determinacy Theorem.[Wadge] Under AD, for all A, B ⊆ ωω, either Pmax ω A ≤W B or B ≤W ω \ A. principles
The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals Theorem.[Martin] Under AD, ≤W is a wellfounded relation on Martin’s Maximum the ≤W -equivalence classes. Determinacy ω Pmax So we can associate to each subset of ω its Wadge rank. We ω ω principles let Pα( ω) be the collection of subsets of ω of Wadge rank The Solovay sequence less than α.
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms The hypotheses for our theorems on the failure of square are Large cardinals derived from the following theorem, plus Sargsyan’s analysis of Martin’s HOD. Maximum
Determinacy Theorem.[Woodin] It is consistent relative to a Woodin limit Pmax of Woodin cardinals that there exist Wadge-incomparable principles ⊆ ω R R The Solovay A, B ω such that L(A, ) and L(B, ) both satisfy AD. sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms HODX is the class of set hereditarily ordinal definable from Large parameters in X . cardinals
Martin’s + Maximum Theorem.[Woodin] Suppose that AD + V = L(P(R)) holds Determinacy θ is a member of the Solovay sequence. Then HODP ω is α θα ( ω) Pmax P ω P ω a model of AD whose Θ is θα and whose ( ω) is θα ( ω). principles
The Solovay sequence Furthermore, if θα is regular in HOD then it is regular in HODP ω , and stationary sets are preserved as well. Wadge rank θα ( ω) HOD
MM(c+) Forcing axioms in P max + extensions Theorem.[CLSSSZ] Assume that AD + V = L(P(R)) holds, P.B. Larson and the cofinalities of the members of the Solovay sequence are P Forcing unbounded below Θ. Then ω2 fails in the max extension of axioms ω HODPθ( ω). Large cardinals
Martin’s Maximum Proof. A name for such a sequence would have to be definable Determinacy from a subset of ωω of Wadge rank below θ. Fixing a θ on the Pmax Solovay sequence above this Wadge rank, the entire principles ω The Solovay ω2 -sequence would exist in HODPθ( ω)[G], where G is the sequence Pmax-generic filter. However, ordinals of cofinality greater than Wadge rank ω θ must remain so in HODPθ( ω)[G], and a ω2 -sequence would HOD witness that every ordinal below Θ has cofinality at most MM(c+) ω2.
This leaves open the issue of whether a ω2 -sequence can be added by Add(1, ω3). Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large The converse also holds. cardinals Martin’s + P R Maximum Theorem.[CLSSSZ] Assume that AD + V = L( ( )) holds,
Determinacy and the cofinalities of the members of the Solovay sequence are P P max bounded below Θ. Then ω2 holds in the max extension of principles ω HODPθ( ω). The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms The hypothesis for the following theorem is below a Woodin Large limit of Woodin cardinals. cardinals
Martin’s Maximum Theorem.[CLSSSZ] Assume that ADR + V = L(P(R)) holds, Determinacy and that stationarily many elements of the Solovay sequence P max are regular in HOD. Then in the Pmax ∗ Add(1, ω3)-extension principles there is no partial ω2 -sequence defined on all points of The Solovay sequence cofinality at most ω1.
Wadge rank
HOD
MM(c+) Forcing The following theorem gives a failure of (ω ). axioms in 3 Pmax extensions ⊆ P.B. Larson Theorem. Suppose that M0 M1 are models of ZF + ADR with the same reals such that, letting Γ0 = P(R) ∩ M0, the Forcing axioms following hold: Large • M = HODM1 ; cardinals 0 Γ0 Martin’s • M0 |= “Θ is regular”; Maximum • M0 M1 Determinacy Θ < Θ ; P • M0 max Θ has cofinality at least ω2 in M1. principles M [G] Let G ⊂ Pmax be M1-generic, and let H ⊂ Add(ω3, 1) 0 be The Solovay sequence M1[G]-generic. Then (ω3) fails in M0[G][H]. Wadge rank
HOD + MM(c ) Proof. M1[G] will satisfy MM(c), and this still holds after M [G] forcing with Add(1, ω3) 0 . So M1[G][H] will see a thread through any (Θ0)-sequence in M0[G][H]. The thread is unique, however, so a definable name for it exists in M0. Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms
Large cardinals We have improved the hypothesis for this to just ADR + “Θ is 2 Martin’s regular” plus the assertion that a certain form of Π1-reflection Maximum holds for all subsets of ωω. Determinacy
Pmax Again, the failure of this hypothesis implies that (ω3) holds in principles the Pmax extension. The Solovay sequence
Wadge rank
HOD
MM(c+) Forcing axioms in Pmax extensions
P.B. Larson
Forcing axioms Large P cardinals Woodin’s proof of MM(c) in the max extension of a model of
Martin’s ADR + “Θ is regular” uses tree representations for subsets of Maximum ωω. Determinacy
Pmax + One plan for obtaining MM(c ) in a Pmax extension involves principles 2 ω developing a theory of Hom∞ subsets of P( ω). The Solovay sequence
Wadge rank
HOD
MM(c+)