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

Forcing axioms in P max extensions Paul Larson Department of Mathematics Miami University Oxford, Ohio 45056 larsonpb@miamioh.edu March 12, 2014

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals Martin’s with Caicedo, Sargsyan, Schindler, Steel, Zeman. Part of an Maximum AIM Square project. Determinacy P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals For Γ a class of partial orders FA(Γ) is the statement that for all Martin’s Maximum P ∈ Γ, and for all collections { D α : α < ω 1 } consisting of dense Determinacy subsets of P , there is a filter G ⊆ P intersecting each D α . P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Examples. Large cardinals • FA(c.c.c.) is MA ℵ 1 Martin’s Maximum • FA(proper) is PFA Determinacy • FA(preserving stationary subsets of ω 1 ) is MM P max • FA( σ -closed*c.c.c) : stronger than MA ℵ 1 , weaker than � principles PFA The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large Why only ℵ 1 many dense sets? Even for Cohen forcing, no cardinals Martin’s filter can meet continuum many dense sets. Maximum Determinacy Theorem. [Todorcevic, Velickovic] FA( σ -closed * c.c.c.) implies P max that 2 ℵ 0 = ℵ 2 . � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions • Forcing axioms say that the universe is closed under P.B. Larson certain forcing operations (i.e., certain objects that can be Forcing forced to exist exist already). Models of forcing axioms axioms can be thought of a maximal, or complete, in contrast to Large cardinals fine structural models, which are minimal (with respect to Martin’s Maximum some hypothesis). Determinacy • The consistency of forcing axioms can tell you what the P max absolute objects are in a given class. � principles • Destroying stationary subsets of ω 1 is the only impediment The Solovay 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 P max extensions P.B. Larson Forcing axioms Large cardinal hypotheses statements which assert the Large existence of infinite cardinals with certain properties. cardinals Martin’s Maximum For example, a strongly inaccessible cardinal is a regular Determinacy cardinal closed under cardinal exponentiation. P max � principles The existence of strongly inaccessible cardinals implies the The Solovay sequence consistency of ZFC, so cannot be proved in ZFC. Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson • Empirically, large cardinal axioms are linearly ordered by Forcing φ < ψ iff ZFC + ψ implies ZFC + Con( φ ). axioms Large • Fine structural models have been produced for some initial cardinals segment of the hierarchy (roughly a Woodin limit of Martin’s Maximum Woodins). Determinacy • Below this, we can show that large cardinals are necessary, P max and, often, show that statements (often having no obvious � principles relation to large cardinals) are equiconsistent with some The Solovay sequence large cardinal hypothesis. Wadge rank • Forcing axioms may be the most important statements HOD beyond this level. MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large Theorem. [Foreman-Magidor-Shelah] If there exists a cardinals supercompact cardinal, then there is a forcing extension in Martin’s Maximum which Martin’s Maximum holds. Determinacy P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals For a cardinal κ , MM( κ ) is the restriction of Martin’s Martin’s Maximum to partial orders of cardinality at most κ . Maximum Determinacy Martin’s Axiom is equivalent to its restriction to partial orders P max of cardinality ℵ 1 , but MM is not equivalent to its restriction to � principles any small cardinal. The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Theorem. [Woodin] Assuming AD R + “Θ is regular”, there is Large cardinals a forcing extension in which ZFC + MM( c ) holds. Martin’s Maximum MM( c ) implies that c = ℵ 2 . Determinacy P max � principles Theorem. [Sargsyan] AD R + “Θ is regular” has consistency The Solovay strength below a Woodin limit of Woodin cardinals. sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large What about MM( c + )? cardinals Martin’s Maximum We will show that certain consequences of MM( c + ) can be Determinacy produced from hypotheses below a Woodin limit of Woodin P max cardinals. � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals Traditional consistency proofs for forcing axioms, including the Martin’s Maximum Foreman-Magidor-Shelah proof, are iterated forcing Determinacy constructions over models of ZFC. P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Given A ⊆ ω ω , the game G A has ω many round, where players Large cardinals I and II alternately choose the members of a sequence Martin’s Maximum ⟨ n i : i ∈ ω ⟩ , and player I wins if ⟨ n i : 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 P max extensions P.B. Larson Forcing axioms • The Axiom of Determinacy ( AD ) is the statement that Large cardinals every A ⊂ ω ω is determined. Martin’s Maximum • The Axiom of Real Determinacy ( AD R ) is the Determinacy corresponding statement for games where the players play P max elements of ω ω . � principles • AD + (a statement in between, formulated by Woodin) The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals Martin’s Θ is the least ordinal which is not a surjective image of ω ω . Maximum Determinacy P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms • P max is a partial ordered developed by Woodin in the early Large cardinals 1990’s. Martin’s Maximum • Conditions are elements of H ( ℵ 1 ), essentially countable Determinacy transitive models of ZFC with some additional structure. P max • The order is induced by elementary embeddings with � principles critical point ω 1 . The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms Large cardinals • A P max extension of a model of AD + satisfies MM( ℵ 1 ). Martin’s Maximum • A P max extension of a model of AD R + “Θ is regular” Determinacy satisfies MM( c ) + c = ℵ 2 . P max � principles The Solovay sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing • P max is ω -closed, preserves ω 2 , and makes Θ into ω 3 . axioms Large • It forces a wellordering of R of ordertype ω 2 . cardinals • If G ⊆ P max is a V -generic filter (for V a model of AD + ) Martin’s Maximum then P ( ω 1 ) V [ G ] ⊆ L ( R )[ G ]. Determinacy • Forcing with P max over a model of AD R + “Θ is regular” P max does not wellorder P ( R ), but P ( R ) can be wellordered � principles The Solovay over the P max extension (without adding subsets of ω 2 ) by sequence forcing with Add(1 , ω 3 ). Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing axioms In P max ∗ Add (1 , ω 3 )-extensions of suitable models of Large determinacy (below a Woodin limit of Woodins) one can cardinals obtain MM( c + ) for partial orders P for which at least one of Martin’s Maximum the following hold. Determinacy • Forcing with P does make make ω 3 have cofinality ω 1 . P max � principles • P is stationary set preserving in any outer model with the The Solovay same ω 1 -sequences of ordinals. sequence Wadge rank HOD MM ( c + )

Forcing axioms in P max extensions P.B. Larson Forcing The following definition is due to Jensen. axioms Large cardinals For an infinite cardinal κ , � κ asserts the existence of a Martin’s sequence ⟨ C α : α < κ + ⟩ such that for all α < κ + , Maximum Determinacy • C α is a club subset of α P max • for all β ∈ lim( C α ), C β = C α ∩ β � principles • ot ( C α ) ≤ κ The Solovay sequence Wadge rank HOD MM ( c + )