The Stable Core Victoria Gitman vgitman@nylogic.org http://victoriagitman.github.io Reflections on Set Theoretic Reflection Happy Birthday, Joan! November 17, 2018 Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 1 / 21
This is joint work with Sy-David Friedman and Sandra M¨ uller. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 2 / 21
The stable core The universe as a forcing extension of HOD Theorem : (Vopˇ enka) Every set of ordinals is set-generic over HOD : If A is a set of ordinals, then there is a partial order P ∈ HOD and G ⊆ P which is HOD -generic such that HOD [ A ] = HOD [ G ]. Intuition : We can glue all these forcing notions together into a single class partial order making V a class forcing extension of HOD . Question : Is V a class forcing extension of HOD ? Theorem : (Hamkins, Reitz) It is consistent that V is not a class forcing extension of HOD . Theorem : (Friedman) There is a definable class S such that every initial segment of S is in HOD and V is a class forcing extension of ( HOD , S ). ( HOD , S ) | = ZFC . There is a class partial order P definable in ( HOD , S ) and G ⊆ P which is ( HOD , S )-generic such that HOD [ G ] = V . ◮ P has the Ord -cc: every maximal antichain of P definable in ( HOD , S ) is set-sized. ◮ ( V , G ) | = ZFC , but G is not definable over V . Theorem : (Friedman) V is a class forcing extension of stable core ( L [ S ] , S ). Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 3 / 21
The stable core The Stability Predicate S “The stability predicate codes elementarity relations between initial segments H α of V .” Models H α For a cardinal α , H α is the set of all sets a with | tc(a) | < α . (L´ evy) For every cardinal α , H α ≺ Σ 1 V . For every regular cardinal α , H α | = ZFC − . n -good cardinals A cardinal α is n -good for n ≥ 1 if: ◮ α is a strong limit. ◮ H α | = Σ n -Collection. Every strong limit cardinal α is 1-good. For every n -good cardinal α with n ≥ 2, if H β ≺ Σ n H α , then β is n -good. Stability predicate S : triples ( n , α, β ) such that α , β are n -good cardinals. H α ≺ Σ n H β . Let S n = { ( α, β ) | ( n , α, β ) ∈ S } be the n -th slice of S . Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 4 / 21
The stable core The stable core ( L [ S ] , S ) Observation : L [ S ] ⊆ HOD . Proof : All initial segments of S are ordinal definable. � Theorem : (Friedman) It is consistent that L [ S ] � HOD is a proper submodel of HOD . Proof : Use the “coding universe into a real” forcing. � Observation : The stable core knows “something” about V . The collection S 1 of all strong limit cardinals of V is definable in ( L [ S ] , S ). If the GCH holds, then the collection of all limit cardinals of V is definable in ( L [ S ] , S ). Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 5 / 21
The stable core Some forcing absoluteness for the stable core Theorem : Suppose P is a forcing notion of size κ and G ⊆ P is V -generic. If α > κ , then ( n , α, β ) ∈ S iff ( n , α, β ) ∈ S V [ G ] . Proof : Suppose α < β are strong limit cardinals above κ . H α | = Σ n -Collection iff H α [ G ] | = Σ n -Collection. H α ≺ Σ n H β iff H V [ G ] = H α [ G ] ≺ Σ n H β [ G ] = H V [ G ] . α β Forward direction: definability of forcing relation. Backward direction: ground model is ∆ 2 -definable. Corollary : If P has size smaller than the first strong limit cardinal and G ⊆ P is V -generic, then S = S V [ G ] . Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 6 / 21
The stable core Motivating questions Is the stable core a “canonical inner model”? Does the stable core satisfy GCH and other regularity properties? Which large cardinals are compatible with the stable core? Are large cardinals downward absolute to the stable core? Can we code information into the stable core using forcing? Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 7 / 21
Large cardinals in the stable core 0 # in the stable core Lemma : If 0 # exists, then 0 # ∈ L [ S n ] for any n ≥ 1. Proof : Let � α i | i < ω � ∈ L [ S n ] be an increasing sequence of V -cardinals. ϕ ( x 1 , . . . , x n ) ∈ 0 # iff L α n +1 | = ϕ ( α 1 , . . . , α n ) � . Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 8 / 21
Large cardinals in the stable core Measurable cardinals in the stable core Theorem : (Friedman, G., M¨ uller) If there is a measurable cardinal, then L [ µ ] ⊆ L [ S n ] for every n ≥ 1. Proof : Step 1: Iterate µ to a “simple” measure µ λ ∈ L [ S n ]. Let λ ≫ κ + such that α with ( α, λ ) ∈ S n are unbounded in λ . Let j λ : L [ µ ] → L [ µ λ ] be the λ -th iterated ultrapower by µ . Let � κ α | α ≤ λ � be the critical sequence of the µ -iteration. For large enough cardinals α of V , κ α = α . In L [ µ λ ], µ λ is a normal measure on λ . A ∈ µ λ iff { κ α | ξ < α < λ } ⊆ A for some ξ . Let A ξ = { α | ξ < α < λ with ( α, λ ) ∈ S n } for ξ < λ (tails of α with ( α, λ ) ∈ S n ). Let F be the filter on λ generated by the tails A ξ . F ∈ L [ S n ], so L [ F ] ⊆ L [ S n ]. µ λ ⊆ F . F ∩ L [ µ λ ] = µ λ . ◮ F is a filter and µ λ is an ultrafilter in L [ µ λ ]. L [ µ λ ] = L [ F ] ⊆ L [ S n ]. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 9 / 21
Large cardinals in the stable core Measurable cardinals in the Stable Core (continued) Proof : (continued) θ [ µ ] with ¯ θ ≥ κ + . Step 2: Collapse a well-chosen X ≺ L θ [ µ λ ] to obtain L ¯ In L [ S n ], define a sequence � ν ξ | ξ < κ + � of strong limit cardinals of V above λ such that cf V ( ν ξ ) = κ + . j λ ( ν ξ ) = ν ξ ( ν ξ > λ , length of iteration, is a strong limit of cf greater than κ ). Let θ be above sup � ν ξ | ξ < κ + � . Let X ≺ L θ [ µ λ ] be generated by κ ∪ { ν ξ | ξ < κ + } ⊆ X . ◮ λ ∈ X (the unique measurable cardinal in L θ [ µ λ ]). ◮ X ⊆ j λ " L [ µ ] ( κ ∪ { ν ξ | ξ < κ + } ⊆ j λ " L [ µ ] ≺ L [ µ λ ]). Let N be the Mostowski collapse of X . λ collapses to κ (there is nothing in j λ " L [ µ ] between κ and λ ). θ ≥ κ + and ν a normal measure on κ . θ [ ν ] with ¯ N = L ¯ By uniqueness, ν = µ . � Corollary : The stable core of K DJ is K DJ . The stable core of L [ µ ] is L [ µ ]. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 10 / 21
Large cardinals in the stable core More measurable cardinals in the stable core uller) If κ ( ξ ) for ξ < α < κ (0) are distinct measurable Theorem : (Friedman, G., M¨ cardinals with normal measures µ ( ξ ) , then L [ � µ ( ξ ) | ξ < α � ] ⊆ L [ S n ]. Proof : Generalize the one measurable cardinal argument using Kunen’s generalized uniqueness. � Question : Can the stable core have a measurable limit of measurable cardinals? Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 11 / 21
Large cardinals in the stable core The model L [ Card ] Studied by Kennedy, Magidor, and V¨ a¨ an¨ anen. Theorem : (Kennedy, Magidor, V¨ a¨ an¨ anen) If 0 # exists, then 0 # ∈ L [ Card ]. If there is a measurable cardinal, then L [ µ ] ⊆ L [ Card ]. If κ ( ξ ) for ξ < α < κ (0) are distinct measurable cardinals with normal measures µ ( ξ ) , then L [ � µ ( ξ ) | ξ < α � ] ⊆ L [ Card ]. We generalized their techniques to the stable core using strong limit cardinals. The structure of L [ Card ] becomes regular in the presence of large cardinals. Theorem : (Kennedy, Magidor, V¨ a¨ an¨ anen, Welch) Assume there is (a little more than) a measurable limit of measurables, then in L [ Card ]: There are no measurable cardinals. GCH holds. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 12 / 21
Changing the stable core by forcing Forcing to code into the stability predicate Theorem : (Friedman, G., M¨ uller) Suppose P ∈ L is a forcing notion, G ⊆ P is L -generic, and n ≥ 1. Then there is a further forcing extension L [ G ][ H ] such that G ∈ L [ S L [ G ][ H ] ]. n Proof : Without loss G ⊆ κ for some κ . Above κ , L and L [ G ] agree on the cardinals, GCH , and the stability predicate S . ξ ) ∈ S L We will define a sequence of widely spaced “coding pairs” ( β ξ , β ∗ n for ξ < κ . ξ ) ∈ S L [ G ][ H ] In L [ G ][ H ], we will have ( β ξ , β ∗ iff ξ ∈ G . n Coding forcing: full-support product C = Π ξ<κ C ξ . If ξ ∈ G , then C ξ is trivial. ∈ G , then C ξ = Coll ( δ + If ξ / ξ , β ξ ). δ + δ + δ + δ + δ 0 δ 1 δ 2 δω +1 δω 0 1 2 ω κ β 0 β ∗ β 1 β ∗ βω β ∗ 0 1 ω Suppose H ⊆ C is L [ G ]-generic. ∈ S L [ G ][ H ] ∈ G , then ( β ξ , β ∗ If ξ / ξ ) / . n ξ ) ∈ S L [ G ][ H ] If ξ ∈ G , then ( β ξ , β ∗ . n ◮ Factor C = Π η<ξ C η × Π ξ<η<κ C η . Correspondingly factor H = H 1 × H 2 . ◮ Π η<ξ C η ∈ H L [ G ] β ξ , Π ξ<η<κ C η is ≤ β ∗ ξ -closed. ◮ H L [ G ] β ξ [ H 1 ] = H L [ G ] β ξ [ H ] ≺ Σ n H L [ G ] ξ [ H ] = H L [ G ] ξ [ H 1 ]. � β ∗ β ∗ Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 13 / 21
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