The Stable Core Victoria Gitman vgitman@nylogic.org - - PowerPoint PPT Presentation

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

• rdinals, 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 Hα of V .” Models Hα For a cardinal α, Hα is the set of all sets a with |tc(a)| < α. (L´ evy) For every cardinal α, Hα ≺Σ1 V . For every regular cardinal α, Hα | = ZFC−. n-good cardinals A cardinal α is n-good for n ≥ 1 if:

◮ α is a strong limit. ◮ Hα |

= Σn-Collection.

Every strong limit cardinal α is 1-good. For every n-good cardinal α with n ≥ 2, if Hβ ≺Σn Hα, then β is n-good. Stability predicate S: triples (n, α, β) such that α, β are n-good cardinals. Hα ≺Σn Hβ. Let Sn = {(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 S1 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) ∈ SV [G]. Proof: Suppose α < β are strong limit cardinals above κ. Hα | = Σn-Collection iff Hα[G] | = Σn-Collection. Hα ≺Σn Hβ iff HV [G]

α

= Hα[G] ≺Σn Hβ[G] = HV [G]

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 = SV [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[Sn] for any n ≥ 1. Proof: Let αi | i < ω ∈ L[Sn] be an increasing sequence of V -cardinals. ϕ(x1, . . . , xn) ∈ 0# iff Lαn+1 | = ϕ(α1, . . . , αn) .

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[Sn] for every n ≥ 1. Proof: Step 1: Iterate µ to a “simple” measure µλ ∈ L[Sn]. Let λ ≫ κ+ such that α with (α, λ) ∈ Sn are unbounded in λ. Let jλ : L[µ] → L[µλ] be the λ-th iterated ultrapower by µ. Let κα | α ≤ λ be the critical sequence of the µ-iteration. For large enough cardinals α of V , κα = α. In L[µλ], µλ is a normal measure on λ. A ∈ µλ iff {κα | ξ < α < λ} ⊆ A for some ξ. Let Aξ = {α | ξ < α < λ with (α, λ) ∈ Sn} for ξ < λ (tails of α with (α, λ) ∈ Sn). Let F be the filter on λ generated by the tails Aξ. F ∈ L[Sn], so L[F] ⊆ L[Sn]. µλ ⊆ F. F ∩ L[µλ] = µλ.

◮ F is a filter and µλ is an ultrafilter in L[µλ].

L[µλ] = L[F] ⊆ L[Sn].

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) Step 2: Collapse a well-chosen X ≺ Lθ[µλ] to obtain L¯

θ[µ] with ¯

θ ≥ κ+. In L[Sn], define a sequence νξ | ξ < κ+ of strong limit cardinals of V above λ such that cfV (νξ) = κ+. jλ(νξ) = νξ (νξ > λ, length of iteration, is a strong limit of cf greater than κ). Let θ be above supνξ | ξ < κ+. Let X ≺ Lθ[µλ] be generated by κ ∪ {νξ | ξ < κ+} ⊆ X.

◮ λ ∈ X (the unique measurable cardinal in Lθ[µλ]). ◮ X ⊆ jλ " L[µ] (κ ∪ {νξ | ξ < κ+} ⊆ jλ " L[µ] ≺ L[µλ]).

Let N be the Mostowski collapse of X. λ collapses to κ (there is nothing in jλ " L[µ] between κ and λ). N = L¯

θ[ν] with ¯

θ ≥ κ+ and ν a normal measure on κ. 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

Theorem: (Friedman, G., M¨ uller) If κ(ξ) for ξ < α < κ(0) are distinct measurable cardinals with normal measures µ(ξ), then L[µ(ξ) | ξ < α] ⊆ L[Sn]. 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[SL[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. We will define a sequence of widely spaced “coding pairs” (βξ, β∗

ξ ) ∈ SL n for ξ < κ.

In L[G][H], we will have (βξ, β∗

ξ ) ∈ SL[G][H] n

iff ξ ∈ G. Coding forcing: full-support product C = Πξ<κCξ. If ξ ∈ G, then Cξ is trivial. If ξ / ∈ G, then Cξ = Coll(δ+

ξ , βξ).

κ δ0 δ+ β0 β∗ δ1 δ+ 1 β1 β∗ 1 δ2 δ+ 2 δω δ+ ω βω β∗ ω δω+1

Suppose H ⊆ C is L[G]-generic. If ξ / ∈ G, then (βξ, β∗

ξ ) /

∈ SL[G][H]

n

. If ξ ∈ G, then (βξ, β∗

ξ ) ∈ SL[G][H] n

.

◮ Factor C = Πη<ξCη × Πξ<η<κCη. Correspondingly factor H = H1 × H2. ◮ Πη<ξCη ∈ HL[G]

βξ , Πξ<η<κCη is ≤ β∗ ξ -closed.

◮ HL[G]

βξ [H1] = HL[G] βξ [H] ≺Σn HL[G] β∗

ξ [H] = HL[G]

β∗

ξ [H1]. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 13 / 21

Changing the stable core by forcing

Coding generic sets into the stable core over L

Corollary: The following can consistently happen in the stable core: The GCH fails on a large initial segment of the cardinals. An arbitrarily large cardinal of L is countable. Martin’s Axiom holds. Theorem: (Friedman, G., M¨ uller) It is consistent that the GCH fails at all regular cardinals in the stable core. Proof: Start in L. Let L[G] be the class forcing extension in which the GCH fails at all regular cardinals. Let A ⊆ Ord code all subsets of cardinals added by G. Define a sequence of “coding pairs” (βξ, β∗

ξ ) ∈ SL n for ξ ∈ Ord.

Define the coding forcing Easton-support product C = Πξ∈OrdCξ. ξ ∈ A if and only if (βξ, β∗

ξ ) ∈ SL[G][H] n

.

Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 14 / 21

Changing the stable core by forcing

Coding generic sets into the stable core over L[µ]

Theorem: (Friedman, G., M¨ uller) Suppose P ∈ L[µ] is a forcing notion and G ⊆ P is L[µ]-generic. Then there is a further forcing extension L[µ][G][H] such that G ∈ L[SL[µ][G][H]]. Proof: L[µ] ⊆ L[SL[µ][G][H]]. Theorem: (Friedman, G., M¨ uller) It is consistent that the stable core has a measurable cardinal and the GCH fails on on a tail of regular cardinals. Big Open Question: Does the structure of the stable core become regular in the presence of stronger large cardinals?

Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 15 / 21

Changing the stable core by forcing

Measurable cardinals are not downward absolute to the stable core

Theorem: (Kunen) Weakly compact cardinals are not downward absolute. Proof: Suppose κ is weakly compact. Let Pκ be the Easton-support iteration of length κ forcing with Add(ξ, 1) at every inaccessible cardinal ξ ∈ V Pξ. Let G ⊆ Pκ be V -generic. In V [G], let Q be the forcing to add a homogeneous κ-Suslin tree. Let T ⊆ Q be V [G]-generic. In V [G][T], κ is not weakly compact. Let b ⊆ T be a V [G][T]-generic branch through T. Q ∗ ˙ T is forcing equivalent to Add(κ, 1). κ is again weakly compact in V [G][T][b] (Pκ ∗ Add(κ, 1) preserves weak compactness of κ).

Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 16 / 21

Changing the stable core by forcing

Measurable cardinals are not downward absolute to the stable core (continued)

Theorem: (Friedman, G., M¨ uller) It is consistent that κ is measurable in V , but not even weakly compact in the stable core. Proof: Start in V = L[µ] where κ is measurable. Let G ∗ T be V -generic for Pκ ∗ ˙ Q. κ is not weakly compact in V [G][T]. Let C be the coding forcing (high above κ) to code T into the stable core. Let H ⊆ C be V [G][T]-generic. κ is not weakly compact in L[SV [G][T][H]].

◮ T ∈ L[SV [G][T][H]]. ◮ V [G][T][H] does not have a branch through T (C is highly closed).

Let b ⊆ T be V [G][T][H]-generic. κ is measurable in V [G][T][H][b].

◮ κ is measurable in V [G][T][b] (Pκ ∗ Add(κ, 1) preserves measurability of κ). ◮ V [G][T][b] and V [G][T][H][b] have the same subsets of κ.

SV [G][T][H][b] = SV [G][T][H] (not obvious). κ is not weakly compact in L[SV [G][T][H][b]]. Big Open Question: Are measurable cardinals downward absolute to the stable core in the presence of large cardinals?

Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 17 / 21

Changing the stable core by forcing

Another generic coding

Start in L. Let P be a forcing notion and G ⊆ P be L-generic. Assume G ⊆ κ. In L, high above κ, choose a sequence (βξ, β∗

ξ ) for ξ < κ of widely spaced “coding pairs”.

Let C be the Easton-support product forcing: GCH fails cofinally often in βξ iff ξ / ∈ G. Let H ⊆ C be L[G]-generic. Fix some large n. In L[G][H], we will have ξ ∈ G iff (βξ, β∗

ξ ) ∈ SL[G][H] n

. If ξ / ∈ G, then (βξ, β∗

ξ ) /

∈ SL[G][H]

n

because GCH fails cofinally in βξ, but not in β∗

ξ .

If ξ ∈ G, then (βξ, β∗

ξ ) ∈ SL[G][H] n

.

◮ Factor C = Csmall × Ctail with Csmall ∈ Hβξ and Ctail is ≤ β∗

ξ -closed.

◮ Factor H = Hsmall × Htail correspondingly. ◮ HL[G]

βξ [Hsmall] = HL[G] βξ [H] ≺Σn HL[G] β∗

ξ [H] = HL[G]

β∗

ξ [Hsmall]. Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 18 / 21

Changing the stable core by forcing

Separating L[Card] and L[S]

Theorem: (Friedman, G., M¨ uller) It is consistent that L[Card] L[S]. Proof: Start in L. Let g ⊆ ω be L-generic for Cohen forcing. Let C be the forcing to code G into the stability predicate using GCH failure. Let H ⊆ C be L[g]-generic. L[CardL[g][H]] = L (L and L[g][H] have same cardinals). g ∈ L[SL[G][H]]. Note: Same argument works over L[µ].

Victoria Gitman The Stable Core Reflections on Set Theoretic Reflection 19 / 21

Changing the stable core by forcing

Open Questions

Question: Is it consistent that the stable core of the stable core is smaller than the stable core? The analogous result for HOD uses generic coding. Question: Is the stable core of every canonical inner model the inner model itself? L[SKDJ ] = K DJ. L[SL[µ]] = L[µ]. Is L[SM1] = M1? (M1 is the canonical model for one Woodin cardinal.) Question: What does the stable core look like in the presence of large cardinals? Is there a bound on the large cardinals the stable core can have? Or: Are large cardinals downward absolute to the stable core? Does the GCH hold?

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Changing the stable core by forcing

Happy Birthday, Joan!

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