An inner model from Ω-logic. Daisuke Ikegami

Transcription

1 An inner model from Ω-logic Daisuke Ikegami Kobe University 12. November 2014

2 Goal & Result Goal Construct a model of set theory which is close to HOD, but easier to analyze.

3 Goal & Result Goal Construct a model of set theory which is close to HOD, but easier to analyze. Theorem (vague version) Under some conditions on large cardinals and Woodin s Ω-logic, one can find a definable class M such that 1 M is a transitive proper class model of ZFC, M HOD, 2 M is invariant under set generic extensions, 3 M contains all the reals in the mice known to exist so far, 4 M is closed under all the mouse operators known to exist so far, and 5 M satisfies GCH.

4 Motivation 1: HOD Conjecture Theorem (Woodin) Let δ be extendible. Then one of the following holds: 1 for every regular γ > δ, γ is measurable in HOD, or 2 for every singular cardinal γ > δ, γ is singular in HOD and (γ + ) HOD = γ +.

5 Motivation 1: HOD Conjecture Theorem (Woodin) Let δ be extendible. Then one of the following holds: 1 for every regular γ > δ, γ is measurable in HOD, or 2 for every singular cardinal γ > δ, γ is singular in HOD and (γ + ) HOD = γ +. Definition (Woodin) HOD Conjecture states that the latter case in the above theorem holds.

6 Motivation 1: HOD Conjecture ctd. 1 HOD Conjecture is connected to Inner Model Program for a supercompact cardinal. 2 HOD Conjecture has an application to the problem on the existence of Reinhardt cardinals in ZF.

7 Motivation 1: HOD Conjecture ctd. 1 HOD Conjecture is connected to Inner Model Program for a supercompact cardinal. 2 HOD Conjecture has an application to the problem on the existence of Reinhardt cardinals in ZF. To solve HOD Conjecture, one would expect a fine analysis of HOD.

8 Motivation 1: HOD Conjecture ctd. 1 HOD Conjecture is connected to Inner Model Program for a supercompact cardinal. 2 HOD Conjecture has an application to the problem on the existence of Reinhardt cardinals in ZF. To solve HOD Conjecture, one would expect a fine analysis of HOD. But HOD is very non-absolute, e.g., one can add any real to HOD in a set generic extension.

9 Motivation 1: HOD Conjecture ctd. 1 HOD Conjecture is connected to Inner Model Program for a supercompact cardinal. 2 HOD Conjecture has an application to the problem on the existence of Reinhardt cardinals in ZF. To solve HOD Conjecture, one would expect a fine analysis of HOD. But HOD is very non-absolute, e.g., one can add any real to HOD in a set generic extension. Goal Construct a model of set theory which is close to HOD but easier to analyze.

10 Motivation 2: Inner Model Theory Question What is the limitation of the current method of Inner Model Theory?

11 Motivation 2: Inner Model Theory Question What is the limitation of the current method of Inner Model Theory? The keyword is Σ 2 1 (ub).

12 Background: Universally Baire sets Definition A set of reals A is universally Baire if for any continuous function f from a compact Hausdorff space X to the reals, f 1 (A) has the Baire property in the space X.

13 Background: Universally Baire sets Definition A set of reals A is universally Baire if for any continuous function f from a compact Hausdorff space X to the reals, f 1 (A) has the Baire property in the space X. Remark 1 Every universally Baire set is Lebesgue measurable and has the Baire property. 2 The collection of universally Baire sets forms a σ-algebra, and it contains all the open sets. So every Borel set is universally Baire.

14 Background: Universally Baire sets ctd. Remark A set of reals A is universally Baire if and only if for any partial order P, there are a set Y and trees T,U on ω Y such that A = p[t] and P p[ť] = R\p[Ǔ].

15 Background: Universally Baire sets ctd. Remark A set of reals A is universally Baire if and only if for any partial order P, there are a set Y and trees T,U on ω Y such that A = p[t] and P p[ť] = R\p[Ǔ]. Using this fact and the trees, one can canonically interpret a ub set A in a set generic extension V[G] (namely p[t] in V[G]). We write A G for this interpreted set in V[G].

16 Background: Universally Baire sets ctd. Remark A set of reals A is universally Baire if and only if for any partial order P, there are a set Y and trees T,U on ω Y such that A = p[t] and P p[ť] = R\p[Ǔ]. Using this fact and the trees, one can canonically interpret a ub set A in a set generic extension V[G] (namely p[t] in V[G]). We write A G for this interpreted set in V[G]. Example 1 Every Π 1 1-set of reals is universally Baire.

17 Background: Universally Baire sets ctd. Remark A set of reals A is universally Baire if and only if for any partial order P, there are a set Y and trees T,U on ω Y such that A = p[t] and P p[ť] = R\p[Ǔ]. Using this fact and the trees, one can canonically interpret a ub set A in a set generic extension V[G] (namely p[t] in V[G]). We write A G for this interpreted set in V[G]. Example 1 Every Π 1 1-set of reals is universally Baire. 2 The following are equivalent: 1 every Π 1 2-set of reals is universally Baire, 2 every set has a sharp.

18 Motivation 2: Inner Model Theory; Σ 2 1 (ub) Definition 1 A formula φ is Σ 2 1 (ub) if it is of the form ( A: universally Baire) (H ω1,,a) ψ, where ψ is a first order formula. 2 A set A H ω1 is Σ 2 1 (ub) if it is defined by a Σ2 1 (ub) formula. 3 A set A H ω1 is 2 1 (ub) in a countable ordinal if there is a countable ordinal α such that both A and H ω1 \A are Σ 2 1 (ub) with parameter α.

19 Motivation 2: Inner Model Theory; Σ 2 1 (ub) Definition 1 A formula φ is Σ 2 1 (ub) if it is of the form ( A: universally Baire) (H ω1,,a) ψ, where ψ is a first order formula. 2 A set A H ω1 is Σ 2 1 (ub) if it is defined by a Σ2 1 (ub) formula. 3 A set A H ω1 is 2 1 (ub) in a countable ordinal if there is a countable ordinal α such that both A and H ω1 \A are Σ 2 1 (ub) with parameter α. Remark 1 All the reals in the mice known to exist so far are 2 1 (ub) in a countable ordinal.

20 Motivation 2: Inner Model Theory; Σ 2 1 (ub) Definition 1 A formula φ is Σ 2 1 (ub) if it is of the form ( A: universally Baire) (H ω1,,a) ψ, where ψ is a first order formula. 2 A set A H ω1 is Σ 2 1 (ub) if it is defined by a Σ2 1 (ub) formula. 3 A set A H ω1 is 2 1 (ub) in a countable ordinal if there is a countable ordinal α such that both A and H ω1 \A are Σ 2 1 (ub) with parameter α. Remark 1 All the reals in the mice known to exist so far are 2 1 (ub) in a countable ordinal. 2 If M is A-closed for every A which is universally Baire and 2 1 (ub), then M is closed under all the mouse operators known to exist so far.

21 Motivation 2: Inner Model Theory; A-closure Definition (A-closure) Let A be universally Baire. An ω-model M of ZFC is A-closed if for any V-generic filter G on a partial order in M, M[G] A G M[G], where A G is the canonical interpretation of A in V[G].

22 Motivation 2: Inner Model Theory; A-closure Definition (A-closure) Let A be universally Baire. An ω-model M of ZFC is A-closed if for any V-generic filter G on a partial order in M, M[G] A G M[G], where A G is the canonical interpretation of A in V[G]. Example 1 For an ω-model M of ZFC, the following are equivalent: M is A-closed for any Π 1 1 -set A, and M is well-founded.

23 Motivation 2: Inner Model Theory; A-closure Definition (A-closure) Let A be universally Baire. An ω-model M of ZFC is A-closed if for any V-generic filter G on a partial order in M, M[G] A G M[G], where A G is the canonical interpretation of A in V[G]. Example 1 For an ω-model M of ZFC, the following are equivalent: M is A-closed for any Π 1 1-set A, and M is well-founded. 2 For an ω-model M of ZFC, the following are equivalent: 1 M is A-closed for every Π 1 2 -set A, and 2 M every set has a sharp.

24 Back to Result Theorem (precise version) Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then there is a definable class M such that 1 M is a transitive proper class model of ZFC, M HOD, 2 M is invariant under set generic extensions, 3 the reals in M are exactly those which are 2 1 (ub) in a countable ordinal, 4 M is A-closed for all universally Baire A which are 2 1 (ub), and 5 M satisfies GCH.

25 Inner models from logics Definition Given a logic L with a definability notion, L 0 (L) =, L α+1 (L) = Def L ( (Lα (L), ) ), L γ (L) = α<γl α (L) (γ is limit), L(L) = α On L α (L).

26 Inner models from logics Definition Given a logic L with a definability notion, Example L 0 (L) =, L α+1 (L) = Def L ( (Lα (L), ) ), L γ (L) = α<γl α (L) (γ is limit), L(L) = α On L α (L). When L is first order logic, L(L) is L. When L is full second (or higher) order logic, L(L) is HOD.

27 Inner models from logics Definition Given a logic L with a definability notion, Example L 0 (L) =, L α+1 (L) = Def L ( (Lα (L), ) ), L γ (L) = α<γl α (L) (γ is limit), L(L) = α On L α (L). When L is first order logic, L(L) is L. When L is full second (or higher) order logic, L(L) is HOD. In this we talk, we will use Woodin s Ω-logic for a desired inner model M.

28 Background 2; Ω-logic Definition Let φ[a] be a Π 2 sentence with a set parameter a in the language of set theory. 1 We say φ[a] is Ω-valid if φ[a] holds.

29 Background 2; Ω-logic Definition Let φ[a] be a Π 2 sentence with a set parameter a in the language of set theory. 1 We say φ[a] is Ω-valid if φ[a] holds. 2 Suppose that a is a real. Then we say φ is Ω-provable if there is a universally Baire set A such that if M is a countable transitive model of ZFC with a M and M is A-closed, then M φ.

30 Background 2; Ω-logic Definition Let φ[a] be a Π 2 sentence with a set parameter a in the language of set theory. 1 We say φ[a] is Ω-valid if φ[a] holds. 2 Suppose that a is a real. Then we say φ is Ω-provable if there is a universally Baire set A such that if M is a countable transitive model of ZFC with a M and M is A-closed, then M φ. Definition The Ω-Conjecture with real parameters states that φ is Ω-valid if and only if φ is Ω-provable for all φ[a]. Point: One can reduce the complexity of Ω-valid Π 2 sentences to Σ 2 1 (ub)

31 Inner models from logics: the model L Ω Definition Let φ be a Σ 2 formula and ψ be a Π 2 formula in the language of set theory. We say (φ,ψ) is a ZFC 2 -pair if ZFC ( x) φ( x) ψ( x).

32 Inner models from logics: the model L Ω Definition Let φ be a Σ 2 formula and ψ be a Π 2 formula in the language of set theory. We say (φ,ψ) is a ZFC 2 -pair if ZFC ( x) φ( x) ψ( x). Definition Let A be a set, a A <ω, and (φ,ψ) be a ZFC 2 -pair. Then the triple (φ,ψ, a) is suitable to A if for any element x of A, either φ[x, a,a] or φ[x, a,a] holds.

33 Inner models from logics: the model L Ω ctd. Definition 1 Let (φ,ψ, a) be suitable to A. Then a set X A is Ω-definable via (φ,ψ, a) if X = {x A φ[x, a,a]}.

34 Inner models from logics: the model L Ω ctd. Definition 1 Let (φ,ψ, a) be suitable to A. Then a set X A is Ω-definable via (φ,ψ, a) if X = {x A φ[x, a,a]}. 2 Def Ω (A) is the collection of Ω-definable subsets of A via some (φ,ψ, a) suitable to A.

35 Inner models from logics: the model L Ω ctd. Definition 1 Let (φ,ψ, a) be suitable to A. Then a set X A is Ω-definable via (φ,ψ, a) if X = {x A φ[x, a,a]}. 2 Def Ω (A) is the collection of Ω-definable subsets of A via some (φ,ψ, a) suitable to A. Definition L Ω 0 =, L Ω ( α+1 = Def Ω (L Ω α, ) ), L Ω γ = α<γl Ω α (γ is limit), L Ω = L Ω α. α On

36 The model L Ω ; basic properties Observation L Ω HOD.

37 The model L Ω ; basic properties Observation L Ω HOD. Proposition L Ω is an inner model of ZF. Point: Def Ω (A) Def FOL (A).

38 The model L Ω ; basic properties Observation L Ω HOD. Proposition L Ω is an inner model of ZF. Point: Def Ω (A) Def FOL (A). Observation L Ω = (L Ω ) VP for any partial order P assuming that there are proper class many Woodin cardinals.

39 The model L Ω ; basic properties Observation L Ω HOD. Proposition L Ω is an inner model of ZF. Point: Def Ω (A) Def FOL (A). Observation L Ω = (L Ω ) VP for any partial order P assuming that there are proper class many Woodin cardinals. To show: Def Ω (A) = ( Def Ω (A) ) V P for any set A. : Easy. : Use the stationary tower forcing Q <δ.

40 The model L Ω ; basic analysis Lemma (Maximality) Suppose that there are proper class many Woodin cardinals. Then 1 L Ω contains all the reals which are 2 1 (ub) in a countable ordinal, and 2 L Ω is A-closed for any set of reals A which is universally Baire and 2 1 (ub).

41 The model L Ω ; basic analysis Lemma (Maximality) Suppose that there are proper class many Woodin cardinals. Then 1 L Ω contains all the reals which are 2 1 (ub) in a countable ordinal, and 2 L Ω is A-closed for any set of reals A which is universally Baire and 2 1 (ub). Points: One can express a 2 1 (ub) statement in (H δ + 0, ) where δ 0 is the least Woodin cardinal, in all set generic extensions in a uniform way.

42 The model L Ω ; basic analysis ctd. Lemma (Choice) Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω is a model of AC.

43 The model L Ω ; basic analysis ctd. Lemma (Choice) Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω is a model of AC. Points: 1 Given an ordinal α, let < α be the canonical well-order on L Ω α. Then < α is 2 1 (ub) with the parameter LΩ α in any sufficiently large set generic extension. 2 Using Lemma (Maximality), one can show that < α is in L(Ω) for all α.

44 The model L Ω ; GCH Theorem Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω satisfies GCH.

45 The model L Ω ; GCH Theorem Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω satisfies GCH. For CH in L Ω : Lemma The reals in L Ω are exactly those which are 2 1 (ub) in a countable ordinal. Moreover, in V, there is a good Σ 2 1 (ub) well-order of the reals in LΩ.

46 The model L Ω ; GCH Theorem Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω satisfies GCH. For CH in L Ω : Lemma The reals in L Ω are exactly those which are 2 1 (ub) in a countable ordinal. Moreover, in V, there is a good Σ 2 1 (ub) well-order of the reals in LΩ. Then using the Basis Theorem for 2 1 (ub), for a given real x in LΩ, one can pick a real in L Ω which codes the initial segment of x w.r.t. this well-order.

47 The model L Ω ; GCH Theorem Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then L Ω satisfies GCH. For CH in L Ω : Lemma The reals in L Ω are exactly those which are 2 1 (ub) in a countable ordinal. Moreover, in V, there is a good Σ 2 1 (ub) well-order of the reals in LΩ. Then using the Basis Theorem for 2 1 (ub), for a given real x in LΩ, one can pick a real in L Ω which codes the initial segment of x w.r.t. this well-order. For 2 κ = κ + in L Ω : Prove CH in (L Ω ) Coll(ω,κ) in the same way.

48 Summary Theorem Suppose that there are proper class many Woodin cardinals and assume that the Ω-Conjecture with real parameters holds in any set generic extension. Then there is a definable class M such that 1 M is a transitive proper class model of ZFC, M HOD, 2 M is invariant under set generic extensions, 3 the reals in M are exactly those which are 2 1 (ub) in a countable ordinal, 4 M is A-closed for all universally Baire A which are 2 1 (ub), and 5 M satisfies GCH.

49 Questions 1 Is there any measurable cardinal in L Ω? 2 Does the hierarchy (L Ω α α Ord) have some weak condensation property?