Outer Model Satisfiability. M.C. (Mack) Stanley San Jose State

Transcription

1 Outer Model Satisfiability M.C. (Mack) Stanley San Jose State

2 The Universe of Pure Sets V 0 = V α+1 = P(V α ) = { x : x V α } V λ = V α, λ a limit α<λ V = α V α Axiomatized by ZFC incompletely Goal: Investigate first-order extensions of ZFC that are compatible with (1) well-foundedness and (2) inclusiveness/bigness. Difficulty: The semantic counterpart of No feature of V implies φ is impossible is There exists a larger universe in which φ is true. Least restrictive approach: Standard set models of ZFC. 2

3 Use of the Alphabet V a (countable, if need be) standard transitive set model of ZFC, proxy for the real universe M a definable inner model of V W an outer model of V = V OR V. 3

4 Outer Models Let V be a standard transitive model of ZFC. W is a weak outer model of V if W V, W is also a standard transitive model of ZFC, and W OR = V OR. W is a strong outer model of V if also (W ; V ) ZFC. Remarks: Familiar techniques produce strong outer models. We will be concerned with whether statements not containing the symbol V are necessarily false or potentially true. Weak outer models are relevant. 4

5 Generic Absoluteness Two lines of auxiliary axioms compatible with the bigness/inclusiveness of V : Large cardinal axioms strong versions of the Axiom of Infinity Forcing axioms some statements that can be forced already are true in V Remarkable development: Large cardinal axioms imply that statements up to a certain logical complexity are absolute for set forcing. Limit of absoluteness relative to just ZFC: Shoenfield Absoluteness If φ is a Σ 1 2 (Pω) sentence of arithmetic, then φ is true in some outer model of V iff φ is true in V. 5

6 Theorem (Martin, Solovay) Assume that V ZFC + every set has a sharp. If φ is a Σ 1 3 (Pω) sentence of arithmetic, then φ is true in some set generic extension of V iff φ is true in V. Theorem (Beller, Jensen) Assume that V ZFC+ every set has a sharp. There exists a Σ 1 3 ( ) sentence φ such that φ is true in a definably class generic extension of V, but φ is false in V. Suspicious features: The Σ 1 3 sentence is x ω α < ω 1 L α [x] ZFC This is incompatible with the base theory every set has a sharp. The class forcing minimalizes the outer model. 6

7 Minimality and Sufficient Non-minimality A model V is minimal if there exists x V such that each element of V is first-order definable from parameters in x. Notation: HYP(V ) is the smallest admissible set with V as an element. HYP(V ) = L α (V ), where α is least such that this structure satisfies KP. Degrees of non-minimality: is definably regular in HYP(V ) HYP(V ) satisfies that V -Ramsey cardinals are definably stationary in HYP(V ) satisfies that has this or that large cardinal property 7

8 Back to Generic Absoluteness Theorem (Woodin) Assume that V ZFC + CH + there exists a proper class of measurable Woodin cardinals. If φ is a Σ 2 1 (Pω) sentence in the language of arithmetic, then φ is true in a set generic extension of V that satisfies ZFC + CH iff φ is true in V. Best possible theorem: There exist Σ 2 1 (P2 ω) sentences φ and ψ such that φ and ψ are each satisfiable in set generic extensions that add no reals φ ψ implies that ω 1 is collapsed Next best: Determine which Σ 2 1 (P2 ω) statements are necessarily false. Σ 2 1 (P2 ω) Satisfiability Problem: Assume CH. Give a uniform first-order definition of the family of Σ 2 1 (P2 ω) sentences that are false in every outer model having the same reals. 8

9 The Branch Problem at ω 1 Assuming CH, the branch problem at ω 1 is equivalent to the Σ 2 1 (P2 ω) satisfiability problem. Say that a tree T of height and cardinality ω 1 is branchable if it has a cofinal branch in some outer model with the same reals. Branch problem at ω 1 : Give a uniform first-order definition of the set of branchable trees in models of ZFC + CH. 9

10 Approximations to a Solution to the Branch Problem at ω 1 Any regressively special tree cannot get a new branch without collapsing ω 1. Any countably distributive tree can get a new branch without adding reals. Theorem (Baumgartner) It is consistent relative to the existence of an inaccessible cardinal that every tree of height and cardinality ω 1 is either regressively special or countably distributive and 2 ω = ω 2. Theorem (Shelah) It is consistent relative to the existence of an inaccessible cardinal that CH holds and every ω 1 -tree is either regressively special or countably distributive. There are several seemingly plausible strategies for solving the branch problem. 10

11 Anti-characterization Theorem Assume that ZFC + CH has standard models. There is no uniform definition of the family { } T V : T is branchable that works in all standard transitive models V of ZFC + CH. Reasonable features: Uniform definition is ruled out Small parameters are ok parameters in H ω2 Suspicious features: Proof: Jensen coding is used in a model of the form L[x], x ω. The proof is easily broken by large cardinals. Bad models are minimal 11

12 Escape from Anti-characterization Corollary to the Main Theorem There exists a parameter-free formula that defines the family { } T V : T is branchable over any sufficiently non-minimal model of ZFC + Ramsey cardinals are definably stationary. 12

13 Main Theorem There exists a formula GOOD(x) as follows: Let V be a countable standard transitive model of ZFC + = ZFC + Ramsey cardinals are definably stationary. Assume that κ is regular and uncountable in V. Let T Hκ V first-order axioms, perhaps with parameters in Hκ V. be a set of (1) If H V κ GOOD[T ], there exists a weak outer model of V that satisfies T. If also V is sufficiently non-minimal, then V has a strong outer model that satisfies T. (2) If H V κ GOOD[T ] and V is sufficiently non-minimal, then T is not satisfiable in any weak outer model of V. The formula GOOD can be taken to be parameter-free Π 2. If κ > ω 1 and r H κ is uncountable, then GOOD can be taken to be Π 1 in the parameter r. 13

14 Weak vs Strong Outer Models Main Theorem There exists a formula GOOD(x) as follows: Let V be a countable standard transitive model of ZFC + and let κ be regular and uncountable in V. Let T H V κ be a set of first-order axioms, perhaps with parameters in H V κ. (1) If H V κ GOOD[T ], there exists a weak outer model of V that satisfies T. If also V is sufficiently non-minimal, then V has a strong outer model that satisfies T. (2) If H V κ GOOD[T ] and V is sufficiently non-minimal, then T is not satisfiable in any weak outer model of V. sufficiently non-minimal = HYP(V ) is definably regular sufficiently non-minimal = HYP(V ) Ramseys are definably stationary in satisfiable in a weak outer model satisfiable in a strong outer model, provided that V is sufficiently non-minimal 14

15 First-order Extensions of ZFC are not Enough Fixed-Point Limitation Assume that ZFC ZFC is recursive and has countable standard transitive models V ZFC + PROMISING[T ] V has an outer model that satisfies T V ZFC + UNPROMISING[T ] No outer model of V satisfies T There exists φ such that neither PROMISING[ZFC +φ] nor UNPROMISING[ZFC +φ] holds in any countable standard transitive model of ZFC. Let φ be such that ZFC φ UNPROMISING(ZFC + φ). Fix some V ZFC. V UNPROMISING(ZFC + φ); otherwise V ZFC + φ contradiction! V PROMISING(ZFC + φ); otherwise let W be an outer model, W ZFC + φ. Then W UNPROMISING(ZFC + φ) contradiction! Rough conclusion: The quality of W satisfying a PROMISING φ may decline. 15

16 Proof of the Main Theorem Questions to answer: What is the formula GOOD? How is the first-order large cardinal hypothesis used? How is sufficient non-minimality used? 16

17 Main Theorem, almost There exists a formula GOOD(x) as follows: Let V be a countable standard transitive model of ZFC + = ZFC + measurable cardinals are definably stationary. Let T V be a set of first-order axioms in the language of set theory with parameters in V. (1) If V GOOD[T ], there exists a weak outer model of V that satisfies T. (2) If V GOOD[T ] and V is sufficiently non-minimal, then T is not satisfiable in any weak outer model of V. 17

18 Outer Model Theories Language for weak outer models: {, F} { a : a V } Weak outer model theory of T over V womth(v, T ): T ZFC in the language of set theory x ( x a b a x = b), for each a V y ( y F" OR ) α F(α) F"α Equivalent: V has a weak outer model satisfying T womth(v, T ) has a model omitting Θ (x) = {x OR} { x = α : α < } Can this be reduced to simple consistency? 18

19 Define: Operation Γ on X V consisting of sentences in the weak outer model language: φ Γ(X) iff X φ or φ is x OR ψ(x) where X ψ(α) for all α < Notation: Let X + = the smallest Z X such that Γ(Z) = Z Then: (1) womth(v, T ) + locally omits Θ (x) = {x OR} { x = α : α < } (2) womth(v, T ) + is Σ 1 definable over HYP(V ) (3) Equivalent: V has a weak outer model satisfying T womth(v, T ) + is consistent Is womth(v, T ) + definable over V? 19

20 Define: womth(v, T ) = womth(v κ, T ) + measurable κ with T V κ Using that V ZFC + measurables are definably stationary, Then: womth(v, T ) is a fixed-point of Γ So womth(v, T ) + womth(v, T ) So V womth(v, T ) consistent T satisfied in some weak outer model of V Let GOOD(T ) be womth(v, T ) is consistent 20

21 For D a measure on κ: Then M = lim α< Ult (α)( V κ, D ) HYP(M) = lim α< Ult (α)( HYP(V κ ), D ), assuming V is sufficiently non-minimal (= is definably regular in HYP(V ) ). So womth(v κ, T ) + womth(m, T ) + womth(v, T ) + So V womth(v, T ) inconsistent no weak outer model of V satisfies T 21

22 Back to Generic Absoluteness, Again Looks good: Assume that V ZFC + CH, V is rich in large cardinals, and V is sufficiently non-minimal. Then any Σ 2 1 (Pω) sentence that holds in some set generic extension satisfying ZFC + CH already holds in V, and there is a parameter-free formula BAD (= GOOD) such that, for any set of sentences T ZFC + CH: V BAD(T ) if and only if T is unsatisfiable in any weak outer model of V. Looks suspicious: Restricted to T H V ω 1, BAD(x) is Σ

23 Corollary (of the Main Theorem) There exists a parameter-free Σ 1 3 sentence φ of arithmetic such that if V ZFC + is sufficiently non-minimal, then φ is false in V, but φ is true in some outer model satisfying ZFC +. Remarks: φ does not hold in any set generic extension of V. Possible: Drop non-minimality hypothesis or get a non-minimal outer model. Conclusion: Even if we require V and its outer models (1) to satisfy some large cardinal extension of ZFC and (2) to be sufficiently non-minimal, ψ holds in some outer model ψ holds in a set generic extension cannot be equivalent, even for Σ 1 3 sentences of arithmetic. Point: This cannot be waived away like the Beller-Jensen example. 23

24 Degrees of Goodness and Badness Assume V ZFC + is sufficiently non-minimal. Outer ZFC + -Model Satisfiable Sentences Sentences Satisfiable in a Sufficiently Non-minimal Outer Model of ZFC + Set Forcible Sentences Outer ZFC + -Model Unsatisfiable Sentences End Extension Unsatisfiable Sentences Sentences Refuted by ZFC + GOOD BAD Each family is a proper subset of those above it.