Forcing axioms in P max extensions Paul Larson Department of - - PowerPoint PPT Presentation

Forcing axioms in Pmax extensions

Paul Larson

Department of Mathematics Miami University Oxford, Ohio 45056 larsonpb@miamioh.edu

March 12, 2014

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

with Caicedo, Sargsyan, Schindler, Steel, Zeman. Part of an AIM Square project.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

For Γ a class of partial orders FA(Γ) is the statement that for all P ∈ Γ, and for all collections {Dα : α < ω1} consisting of dense subsets of P, there is a filter G ⊆ P intersecting each Dα.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Examples.

• FA(c.c.c.) is MAℵ1
• FA(proper) is PFA
• FA(preserving stationary subsets of ω1) is MM
• FA(σ-closed*c.c.c) : stronger than MAℵ1, weaker than

PFA

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Why only ℵ1 many dense sets? Even for Cohen forcing, no filter can meet continuum many dense sets. Theorem.[Todorcevic, Velickovic] FA(σ-closed * c.c.c.) implies that 2ℵ0 = ℵ2.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• Forcing axioms say that the universe is closed under

certain forcing operations (i.e., certain objects that can be forced to exist exist already). Models of forcing axioms can be thought of a maximal, or complete, in contrast to fine structural models, which are minimal (with respect to some hypothesis).

• The consistency of forcing axioms can tell you what the

absolute objects are in a given class.

• Destroying stationary subsets of ω1 is the only impediment

to a forcing axiom.

• (Moore) PFA implies that the uncountable linear orders

have a five-element basis.

• (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 cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Large cardinal hypotheses statements which assert the existence of infinite cardinals with certain properties. For example, a strongly inaccessible cardinal is a regular cardinal closed under cardinal exponentiation. The existence of strongly inaccessible cardinals implies the consistency of ZFC, so cannot be proved in ZFC.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• Empirically, large cardinal axioms are linearly ordered by

φ < ψ iff ZFC + ψ implies ZFC + Con(φ).

• Fine structural models have been produced for some initial

segment of the hierarchy (roughly a Woodin limit of Woodins).

• Below this, we can show that large cardinals are necessary,

and, often, show that statements (often having no obvious relation to large cardinals) are equiconsistent with some large cardinal hypothesis.

• Forcing axioms may be the most important statements

beyond this level.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Theorem.[Foreman-Magidor-Shelah] If there exists a supercompact cardinal, then there is a forcing extension in which Martin’s Maximum holds.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

For a cardinal κ, MM(κ) is the restriction of Martin’s Maximum to partial orders of cardinality at most κ. Martin’s Axiom is equivalent to its restriction to partial orders

• f cardinality ℵ1, but MM is not equivalent to its restriction to

any small cardinal.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Theorem.[Woodin] Assuming ADR + “Θ is regular”, there is a forcing extension in which ZFC + MM(c) holds. MM(c) implies that c = ℵ2. Theorem.[Sargsyan] ADR + “Θ is regular” has consistency strength below a Woodin limit of Woodin cardinals.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

What about MM(c+)? We will show that certain consequences of MM(c+) can be produced from hypotheses below a Woodin limit of Woodin cardinals.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Traditional consistency proofs for forcing axioms, including the Foreman-Magidor-Shelah proof, are iterated forcing constructions over models of ZFC.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Given A ⊆ ωω, the game GA has ω many round, where players I and II alternately choose the members of a sequence ⟨ni : i ∈ ω⟩, and player I wins if ⟨ni : i ∈ ω⟩ ∈ A. The set A is determined if either player I or player II has a winning strategy.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• The Axiom of Determinacy (AD) is the statement that

every A ⊂ ωω is determined.

• The Axiom of Real Determinacy (ADR) is the

corresponding statement for games where the players play elements of ωω.

• AD+ (a statement in between, formulated by Woodin)

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Θ is the least ordinal which is not a surjective image of ωω.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• Pmax is a partial ordered developed by Woodin in the early

1990’s.

• Conditions are elements of H(ℵ1), essentially countable

transitive models of ZFC with some additional structure.

• The order is induced by elementary embeddings with

critical point ω1.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• A Pmax extension of a model of AD+ satisfies MM(ℵ1).
• A Pmax extension of a model of ADR + “Θ is regular”

satisfies MM(c) + c = ℵ2.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• Pmax is ω-closed, preserves ω2, and makes Θ into ω3.
• It forces a wellordering of R of ordertype ω2.
• If G ⊆ Pmax is a V -generic filter (for V a model of AD+)

then P(ω1)V [G] ⊆ L(R)[G].

• Forcing with Pmax over a model of ADR + “Θ is regular”

does not wellorder P(R), but P(R) can be wellordered

• ver the Pmax extension (without adding subsets of ω2) by

forcing with Add(1, ω3).

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

In Pmax ∗ Add(1, ω3)-extensions of suitable models of determinacy (below a Woodin limit of Woodins) one can

• btain MM(c+) for partial orders P for which at least one of

the following hold.

• Forcing with P does make make ω3 have cofinality ω1.
• P is stationary set preserving in any outer model with the

same ω1-sequences of ordinals.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The following definition is due to Jensen. For an infinite cardinal κ, κ asserts the existence of a sequence ⟨Cα : α < κ+⟩ such that for all α < κ+,

• Cα is a club subset of α
• for all β ∈ lim(Cα), Cβ = Cα ∩ β
• ot(Cα) ≤ κ

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• there cannot exist a club E ⊆ γ such that for all

α ∈ lim(E), Cα = E ∩ α.

• (κ+) : remove the condition ot(Cα) ≤ κ and assert the

nonexistence of such an E (so (κ+) is weaker) principles illustrate why partial orders preserving stationary subsets of ω1 don’t have to have small subalgebras with the same property.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Much (possibly all) of the known consistency strength of MM comes from the following result. Theorem.[Todorcevic] If γ is an ordinal of cofinality greater than ω1, there exists a σ-closed*c.c.c. forcing P of cardinality |γ|ℵ0 such that FA({P}) implies ¬(γ).

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• In the Pmax ∗ Add(1, ω3) extension of a model of ADR +

“Θ is regular”, 2ℵ0 = ℵ2.

• Given that 2ℵ0 = ℵ2, MM(c) implies ¬(ω2) and

MM(c+) implies ¬(ω3).

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Theorem.[CLSSSZ] Assuming a certain determinacy hypothesis below a Woodin limit of Woodin cardinals, the Pmax ∗ Add(1, ω3) extension satisfies MM(c) + ¬ω2.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

• The previous upper bound for ¬(ω2) + ¬ω2 was a

quasicompact cardinal, above the current inner model theory.

• Lower bound : at least ADL(R)
• From a stronger hypothesis (beyond a Woodin limit of

Woodin cardinals) we get ¬(ω3).

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The Solovay sequence is the unique continuous sequence ⟨θα : α ≤ δ⟩ satisfying the following conditions.

• θ0 is the least ordinal γ for which there does not exist an
• rdinal definable function from ωω onto γ.
• if θα < Θ, then θα+1 is the least ordinal γ for which there

does not exist an ordinal definable function from P(θα)

• nto γ.
• θδ = Θ.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The following gives some indication of the strength of ADR + “Θ is regular”. Theorem.[Woodin] Assuming AD + V = L(P(R)), ADR holds if and only if the Solovay sequence has limit length.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

HOD is the class of hereditarily ordinal definable sets. Theorem.[Woodin] Assuming AD + DC, all successor elements of the Solovay sequence are Woodin in HOD.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Given A, B ⊆ ωω, say that A ≤W B (A is Wadge below B) if A = f −1[B], for some continuous f : ωω → ωω. Theorem.[Wadge] Under AD, for all A, B ⊆ ωω, either A ≤W B or B ≤W ωω \ A.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Theorem.[Martin] Under AD, ≤W is a wellfounded relation on the ≤W -equivalence classes. So we can associate to each subset of ωω its Wadge rank. We let Pα(ωω) be the collection of subsets of ωω of Wadge rank less than α.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The hypotheses for our theorems on the failure of square are derived from the following theorem, plus Sargsyan’s analysis of HOD. Theorem.[Woodin] It is consistent relative to a Woodin limit

• f Woodin cardinals that there exist Wadge-incomparable

A, B ⊆ ωω such that L(A, R) and L(B, R) both satisfy AD.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

HODX is the class of set hereditarily ordinal definable from parameters in X. Theorem.[Woodin] Suppose that AD+ + V = L(P(R)) holds θα is a member of the Solovay sequence. Then HODPθα(ωω) is a model of AD whose Θ is θα and whose P(ωω) is Pθα(ωω). Furthermore, if θα is regular in HOD then it is regular in HODPθα(ωω), and stationary sets are preserved as well.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Theorem.[CLSSSZ] Assume that AD+ + V = L(P(R)) holds, and the cofinalities of the members of the Solovay sequence are unbounded below Θ. Then ω2 fails in the Pmax extension of HODPθ(ωω).

• Proof. A name for such a sequence would have to be definable

from a subset of ωω of Wadge rank below θ. Fixing a θ on the Solovay sequence above this Wadge rank, the entire ω2-sequence would exist in HODPθ(ωω)[G], where G is the Pmax-generic filter. However, ordinals of cofinality greater than θ must remain so in HODPθ(ωω)[G], and a ω2-sequence would witness that every ordinal below Θ has cofinality at most 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 cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The converse also holds. Theorem.[CLSSSZ] Assume that AD+ + V = L(P(R)) holds, and the cofinalities of the members of the Solovay sequence are bounded below Θ. Then ω2 holds in the Pmax extension of HODPθ(ωω).

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The hypothesis for the following theorem is below a Woodin limit of Woodin cardinals. Theorem.[CLSSSZ] Assume that ADR + V = L(P(R)) holds, and that stationarily many elements of the Solovay sequence are regular in HOD. Then in the Pmax ∗ Add(1, ω3)-extension there is no partial ω2-sequence defined on all points of cofinality at most ω1.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

The following theorem gives a failure of (ω3).

• Theorem. Suppose that M0 ⊆ M1 are models of ZF + ADR

with the same reals such that, letting Γ0 = P(R) ∩ M0, the following hold:

• M0 = HODM1

Γ0 ;

• M0 |

= “Θ is regular”;

• ΘM0 < ΘM1;
• ΘM0 has cofinality at least ω2 in M1.

Let G ⊂ Pmax be M1-generic, and let H ⊂ Add(ω3, 1)M0[G] be M1[G]-generic. Then (ω3) fails in M0[G][H].

• Proof. M1[G] will satisfy MM(c), and this still holds after

forcing with Add(1, ω3)M0[G]. 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 Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

We have improved the hypothesis for this to just ADR + “Θ is regular” plus the assertion that a certain form of Π2

1-reflection

holds for all subsets of ωω. Again, the failure of this hypothesis implies that (ω3) holds in the Pmax extension.

Forcing axioms in Pmax extensions P.B. Larson Forcing axioms Large cardinals Martin’s Maximum Determinacy Pmax principles The Solovay sequence Wadge rank HOD MM(c+)

Woodin’s proof of MM(c) in the Pmax extension of a model of ADR + “Θ is regular” uses tree representations for subsets of

ωω.

One plan for obtaining MM(c+) in a Pmax extension involves developing a theory of Hom2

∞ subsets of P(ωω).