Large Cardinals and Higher Degree Theory

Large Cardinals and Higher Degree Theory SHI Xianghui School of Mathematical Sciences Beijing Normal University May 2013, NJU 1 / 20

Dedicated to Professor DING DeCheng on his 70 th Birthday. 2 / 20

Higher Degree Theory Classical recursion theory studies the structure of Turing degrees. It has been extended/generalized to higher levels of computability (e.g. hyperarithmetic degrees) higher ordinals/cardinals (e.g. α-recursion theory) even both (e.g. generalized hyperarithmetic theory) However these are all done within ZFC, or ZFC + V = L. Recent developments reveal some interesting connections between large cardinals and degree structures at uncountable cardinals, in particular, strong limit singular cardinals of countable cofinality. 3 / 20

Zermelo degrees We will discuss degree structures in canonical inner models. Unlike the situation of ω, not very much of degree structures at uncountable cardinals can be determined by ZFC alone. Fine structure models provide more complete settings. Let Z = Zermelo Set Theory. This talk is limited to Z-degrees. Suppose λ is a limit of at least strongly inaccessible cardinals. Fix a well-ordering w : H(λ) λ. For a, b λ: M[a] denotes the minimal Z-model of the form L α [w, a], α > λ. Let α a, Z-ordinal for a, denote the height of M[a]. a Z b if M[a] M[b]. a Z b if a Z b and b Z a Write ã for the degree of a, the Z -equivalence class of a. J Z (a), Z-jump of G, is the theory of M[a]. It can be coded by a subset of λ. 4 / 20

A list of questions A set A P(λ) is Z-degree invariant if a A ã A. A cone is a set of the form C a = {b a Z b}. Det λ (Z-Deg): Every Z-degree invariant subset of P(λ) either contains a cone or is disjoint from a cone. A list of degree theoretic questions. 1. (Post Problem). Are there incomparable degrees, i.e. (ã b ) (b ã)? 2. (Minimal Cover). Given ã, is there a b minimal above ã, i.e. a < c (ã < < b )? b c 3. (Posner-Robinson). Is it true for co-λ many x λ that ( G)[(x, G) Z J Z (G)]? 4. (Degree Determinacy). Is Det λ (Z-Deg) true? 5 / 20

cf(λ) = λ. Not very interesting. Most degree theoretic constructions at ω can be generalized to strongly inaccessible cardinals. 6 / 20

cf(λ) = λ. Not very interesting. Most degree theoretic constructions at ω can be generalized to strongly inaccessible cardinals. cf(λ) > ω. Theorem (Sy Friedman, 81) (V = L) Nothing interesting left. The analog of Turing degrees at singular cardinals of uncountable cofinality are well-ordered above a singularizing degree. The key to this is the analysis of stationary subsets of cf(λ). Corollary (V = any fine structure model) Z-degrees at singular cardinals of uncountable cofinality are well-ordered above a singularizing degree. 6 / 20

Pictures in L Observation. (V = L) (Woodin) If cf(λ) = ω, then Z-degrees at λ are well-ordered above a singularizing degree. In particular, Z-degrees at ℵ ω is well-ordered. 1 1 In contrast: Δ 1-degree at ℵ ω (i.e. ℵ ω-degree) is well-ordered above 0. 7 / 20

Pictures in L Observation. (V = L) (Woodin) If cf(λ) = ω, then Z-degrees at λ are well-ordered above a singularizing degree. In particular, Z-degrees at ℵ ω is well-ordered. 1 A key ingredient of the argument is Covering Lemma for L. (Jensen, 74) Assume 0. Then every set x Ord is covered by a y L, with y = x + ω 1. 1 In contrast: Δ 1-degree at ℵ ω (i.e. ℵ ω-degree) is well-ordered above 0. 7 / 20

Proof of Observation. Suppose a λ, a Z b, and b singularizes λ. Then a computes a cutoff function. Work in M[a]. Every x λ is identified as a member of [λ] ω. M[a] has no sharps, by Covering, b L M[a] P(λ) s.t. a b b ω 1. Then a b z ω 1, for some z ω 1. M[a] and L M[a] have the same P(ω 1 ). Thus a L M[a] = L αa. In other word, M[a] = L αa. Z-degrees at λ are well-ordered above d. 8 / 20

Answers to the list. (above the singularizing degree) Post Problem Minimal Cover Posner-Robinson Degree Determinacy No. Yes. No for > 1 minimal covers. No. Fail to have solution at the limit. No. Remark. A bit unusual: using Covering within L. As for inner models between L and L[μ], such as L(0 ), the same argument applies, since their Covering Lemmas are of the same form. A little wrinkle in L[μ], but still the same picture. 9 / 20

Pictures in L[μ] Let κ be the measurable, λ strong limit and cf(λ) = ω. Reorganize L[μ] as L[E], by Steel s construction, using partial measures. The point is the acceptability condition, i.e. γ < α, (L α+1 [E] L α [E]) P(γ) L α [E] = α = γ. Two cases: λ > κ. Argue as in L. λ < κ. Fix a λ above the least L α+1 [E] that singularizes λ. M[a] contains no 0. The most K M[a], the core model for M[a], could be is either L[μ ] or there is no measurable. If no measurable, then M[a] = K M[a], by Covering as before. By Comparison, M[a] K = L[E]. If K M[a] = L[μ ], then there are two cases. 10 / 20

Covering Lemma for L[μ]. (Dodd-Jensen, 82) Assume 0, but there is an inner model L[μ]. Let κ = crit(μ). Then for every set x Ord, one of the following holds: 1. Every set x Ord is covered by a y L[μ], with y = x + ω 1. 2. C, Prikry generic over L[μ], s.t. every set x Ord is covered by a y L[μ][C], with y = x + ω 1. Such C is unique up to finite difference. Case 1. M[a] = V = L[μ ], as before. Case 2. Note that λ < κ = crit(μ ), and C κ adds no new bounded subsets of κ. Some y L[μ ] P(λ) covers a. So M[a] = V = L[μ ] again. By Comparison, M[a] K = L[E]. 11 / 20

Pictures in L[ˉμ] Consider L[ˉμ], ˉμ = μ i : i < ω is a sequence of measures, and κ n = crit(μ n ). The case λ sup n κ n can be argued as in L[μ]. The Z-degrees at λ = sup n κ n present a new structure. C in Case 2 of the Covering for L[ˉμ] can be chosen to be an ω-sequence, essentially a diagonal Prikry sequence for L[ˉμ]. Fix a λ. K M[a], by Covering, is either of the form L η [ˉμ] or L η [ˉμ][C a ]. C a is Prikry, so L η [ˉμ][C a ] is a Z-model. Therefore Z-degrees at λ is pre-wellordered by the associated Z-ordinals. Let α 0 be the least Z-ordinal past λ. Note that a Z C a, if α a = α 0. Z-degrees associated to α 0 are exactly the ones induced by C 0 = {diagonal Prikry sequences for L α0 [ˉμ]}. Let α η, η > 0, be the η-th Z-ordinal above λ. Let a <η λ codes α i : i < η. Some of C 0 remain to be L αη [ˉμ]-generic. The Z-degrees associated to α η are the ones induced by a <η C 0 = def {(a <η, C) C C 0 }. 12 / 20

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singularizing degree :-) 13 / 20

α 0 singularizing degree :-) Deg[C 0 ]. 13 / 20

α η Deg[C η ] = Deg[a <η C 0 ] α 0 singularizing degree :-) Deg[C 0 ]. 13 / 20

Answers to the list. (at λ = sup n κ n ) Post Problem Minimal Cover Posner-Robinson Degree Determinacy Yes. a LPS 2 of pairwise incomp. degrees. No. No. Fail to have solution at the limit. No. Moreover, there are infinite descending chains of degrees. Meta-Conjecture In any reasonable inner model, at every singular λ, cf(λ) = ω, below the least measurable, the Z-degrees are well ordered above some degree. 2 Abbreviation: LPS = Large Perfect Set 14 / 20

Picture in L[U] for o(κ) = κ Theorem (Yang, 2011) Assume ˉκ = κ n : n < ω is a sequence of measurable cardinals s.t. each κ n+1 carries κ n many normal measures. Let λ = sup n κ n. Then there is a minimal Z-degree above D, where D λ codes relevant information, in particular, the above system of measures. This can be relativized to degrees above D. Yang s argument can produce a LPS of minimal covers for D, which are automatically pairwise incomparable. This picture appears in Mitchell s model for o(κ) = κ. 3 YES to Post and Minimal Cover questions at sup n κ n. However, the system of indiscernibles for this inner model is very difficult to analyze. We conjecture NO to the other two questions. 3 Not minimal, but it is the shortest with o-expression. 15 / 20

Picture from I 0 Definition I 0 is the following assertion: There exists an elementary embedding j : L(V λ+1 ) L(V λ+1 ) such that crit(j) < λ. Using Laver s notation, E(L(V λ+1 )) 16 / 20

Picture from I 0 Definition I 0 is the following assertion: There exists an elementary embedding j : L(V λ+1 ) L(V λ+1 ) such that crit(j) < λ. Using Laver s notation, E(L(V λ+1 )) By Kunen, ZFC E(V λ+2 ) =. I 0 is among the strongest large cardinal axioms not known to be inconsistent with ZFC. There is a strong resemblance between structural properties of subsets of V λ+1 under ZFC + I 0 and those of subsets of R = V ω+1 under ZF + DC + AD: AD L(R) I 0 L(V λ+1 ). Next are two recent results in I 0 -theory, one support this analogy, and the other against it. 16 / 20

Two results in I 0 -theory Theorem 1 (S-Woodin) Assume ZFC + I 0. Then for almost all (λ many exceptions) X λ, ( G λ ) [(X, G) Γ J Γ (G)]. 4 4 True for finer equivalence as well, e.g. (x, G) Σ 0 1 (V λ ) G. 17 / 20

Two results in I 0 -theory Theorem 1 (S-Woodin) Assume ZFC + I 0. Then for almost all (λ many exceptions) X λ, ( G λ ) [(X, G) Γ J Γ (G)]. 4 Theorem 2 Assume ZFC + I 0. Suppose j E(L(V λ+1 )) and κ 0 = crit(j). Suppose in V λ, κ 0 is supercompact, and it supercompactness is indestructible by κ 0 -directed posets. L(V λ+1 ) = Det λ (Z-Deg). 4 True for finer equivalence as well, e.g. (x, G) Σ 0 1 (V λ ) G. 17 / 20

Suppose j E(L(V λ+1 )). Then λ is an ω-limit of measurable cardinals λ also satisfies the condition in Yang s Theorem 18 / 20

Suppose j E(L(V λ+1 )). Then λ is an ω-limit of measurable cardinals λ also satisfies the condition in Yang s Theorem So answers to the list are as follows: Post Problem Minimal Cover Posner-Robinson Degree Determinacy Yes. a LPS of pairwise incomp. degrees. Yes. a LPS of minimal covers. Yes. very plausible to be No. 18 / 20

Remarks The complexity of degree structures at certain cardinal reflects the strength of large cardinals carried in the model. Among (fine structure) inner models, the richness of the degree structures seem to be correlated to where λ is in the inner model, rather than to the strength of the inner model. The basic method of exploiting the complex degree structures to prove the failure of the degree determinacy (as in our proof of Theorem 2) does NOT work in general. This means that the proof of the conjecture, i.e. L(P(λ)) = Det λ (Z-Deg), from ZFC is going to be very subtle. 19 / 20

Thank You! 20 / 20