The Journal of Symbolic Logic http://journals.cambridge.org/JSL Additional services for The Journal of Symbolic Logic: Email alerts: Click here Subscriptions: Click here Commercial reprints: Click here Terms of use : Click here AXIOM I 0 AND HIGHER DEGREE THEORY XIANGHUI SHI The Journal of Symbolic Logic / Volume 80 / Issue 03 / September 2015, pp 970 - 1021 DOI: 10.1017/jsl.2015.15, Published online: 22 July 2015 Link to this article: http://journals.cambridge.org/abstract_S0022481215000158 How to cite this article: XIANGHUI SHI (2015). AXIOM I 0 AND HIGHER DEGREE THEORY. The Journal of Symbolic Logic, 80, pp 970-1021 doi:10.1017/jsl.2015.15 Request Permissions : Click here Downloaded from http://journals.cambridge.org/JSL, by Username: ASLXShi, IP address: 212.93.9.222 on 23 Jul 2015 The Journal of Symbolic Logic Volume 80, Number 3, September 2015 AXIOM I0 AND HIGHER DEGREE THEORY XIANGHUI SHI Abstract. In this paper, we analyze structures of Zermelo degrees via a list of four degree theoretic questions (see §2) in various fine structure extender models, or under large cardinal assumptions. In particular we give a detailed analysis of the structures of Zermelo degrees in the Mitchell model for many measurable cardinals. It turns out that there is a profound correlation between the complexity of the degree structures at countable cofinality singular cardinals and the large cardinal strength of the relevant cardinals. The analysis applies to general degree notions, Zermelo degree is merely the author’s choice for illustrating the idea. I0pq is the assertion that there is an elementary embedding j : LpV`1qÑLpV`1q with critical point ă . We show that under I0pq, the structure of Zermelo degrees at is very complicated: it has incomparable degrees, is not dense, satisfies Posner–Robinson theorem etc. In addition, we show that I0 together with a mild condition on the critical point of the embedding implies that the degree determinacy for Zermelo degrees at is false in LpV`1q. The key tool in this paper is a generic absoluteness theorem in the theory of I0, from which we obtain an analogue of Perfect Set Theorem for “projective” subsets of V`1, and the Posner–Robinson follows as a corollary. Perfect Set Theorem and Posner–Robinson provide evidences supporting the analogy between AD over LpRq and I0 over LpV`1q, while the failure of degree determinacy is one for disanalogy. Furthermore, we conjecture that the failure of degree determinacy for Zermelo degrees at any uncountable cardinal is a theorem of ZFC. §1. Introduction. At a given uncountable cardinal α, one can define the degree of Δ1-reducibility for subsets of α in analogy to Turing degree for subsets of . This is so called α-degree (see [33] and [8]). In [8], Sy Friedman showed that if “ ℵ α is a singular cardinal of uncountable cardinality, in particular, α 1 , then the α-degrees in L are well ordered above any singularizing degrees (degrees that can compute a sequence cofinal in α). Very little is known about the case cfpαq“,and very little attention has been given to degree structures in inner models other than L. In this paper we will investigate the structures of generalized definability degrees at singular cardinals of countable cofinality and their connection with large cardinal hypotheses. To make our points, we only work with strong limit cardinals1 and focus our attention on the structure of Zermelo degrees. Received November 9, 2013. Key words and phrases. Axiom I0, Prikry type forcing, generic absoluteness, -Perfect Set Theorem, Posner–Robinson Theorem, degree determinacy, indestructibly supercompact, higher degree theory, Covering Lemma During this project, the author has been supported by NSFC (No. 11171031), SRF for ROCS of SEM (No. 105213006), RFDP, and the Fundamental Research Funds for the Central Universities (No. 2014KJJCB20). 1In this paper, we focus on canonical models which are all GCH models, thus working with strong limit cardinals does not lose anything essential. c 2015, Association for Symbolic Logic 0022-4812/15/8003-0011 DOI:10.1017/jsl.2015.15 970 AXIOM I0 AND HIGHER DEGREE THEORY 971 By comparing the structures of Zermelo degrees in canonical models for various large cardinals, we discover that at strong limit cardinals of countable cofinality there is a much more interesting and more profound connection between the complexity of the degree structures and the large cardinal strength that the canonical model carries. 1.1. Post problem and minimal cover problem in L-like models. Suppose is a strong limit cardinal of countable cofinality. Fix a Ă coding a wellordering of H “ta | the transitive closure of a has size ă u of ordertype .LetMpxq be the inner model operator assigning to each x Ď the smallest Zermelo model containing x and .Fora, b Ď , define a ď b iff MpaqĎMpbq. The induced equivalence classes are called Zermelo degrees. Write a to denote the Zermelo degree r of a.Wesaya is a singularizing degree if there is a cofinal subset of of ordertype r cfpq in a. r First, Woodin observed that in V “ L, the structure of Zermelo degrees at ℵ is ℵ thesameasat 1 (see Corollary on page 8): Theorem 1.1 (See §2.2.1). Assume V “ L. Zermelo degrees at every singular cardinal of countable cofinality are well ordered above every singularizing degree. The pictures (at all singular cardinals) remain the same – all well ordered above singularizing degrees – in larger fine structure models such as Lr07s andevenupto Lrs (see Theorem 2.10), the canonical inner model for one measurable. Until in Lr¯s, the Mitchell model for an -sequence of measurable cardinals, a new picture start to emerge: letting be the supremum of these many measurable cardinals, the Zermelo degrees at are not well ordered above singularizing degrees. Theorem 1.2 (See §2.2.3). Assume V “ Lr¯s. There are incomparable degrees in the Zermelo degree structure at above the degree of ¯, viewing it as a subset of . Therefore we have a positive answer to Post problem (i.e., there are incomparable degrees, see page 976 for more details) in this setting. Though not well ordered, the Zermelo degree structure at in Lr¯s is still rather nice – it is dense above the degree of ¯, and there is a natural prewellorder structure over the Zermelo degrees above thedegreeof¯ (see Theorem 2.13-(2)). For two degrees a ď b, b is a minimal cover of a if there is no degrees c strictly in r r r r r between a and b. A recent work ([51]) shows that assuming stronger large cardinals r r one can find minimal covers (unboundedly often) in the structures of Zermelo degrees, therefore the structures of Zermelo degrees in this situation are not dense. More precisely, Theorem 1.3 (Yang [51]). Suppose xκn : n ă y is an increasing sequence of measurable cardinals such that each κn`1 carries κn different normal measures, n P , “ U and supn κn.Let denote this matrix of normal measures, and let W be any subset of that codes xV, P,,tκi | i ă u, Uy,where Ă and codes a well ordering of V of ordertype . Then there is a minimal cover for the Zermelo degree of W . 1.2. Posner–Robinson Theorem, degree determinacy and Axiom I0. Besides Post problem and minimal cover problem, we also look into other classical results from classical recursion theory—Posner–Robinson Theorem and Turing determinacy— in the context of our generalized degree structures. In L and Lr¯s, answers to these 972 XIANGHUI SHI two questions are negative. It is not known whether Posner–Robinson is true in an inner model for the above situation. To obtain the Posner–Robinson phenomena at a countable cofinality singular cardinal, we climbed much further up the large cardinal hierarchy. Axiom I0 asserts that for some , there is an elementary embedding j : LpV`1qÑLpV`1q with critpjqă. Here critpjq denotes the critical point of j. We often say “I0 holds at ”orwrite I0pq to make this explicit. This j is called an I0pq-embedding. I0 is among the large cardinals by far not known to be inconsistent with ZFC. For two degrees a, b, r r let a _ b denote the least upper bound of a and b in the degree poset, and Jpaq be r r r r r the Zermelo degree of the theory (as a subset of ) of the smallest Zermelo model containing a (and the fixed wellordering ). Theorem 1.4 (See §5). Assume I holds at . Then for every Zermelo degree a 0 r at , for almost every Zermelo degree b above a (only ď many exceptions), the r r ´ ¯ Posner–Robinson equation for b has a solution, i.e. Dg b _ g “ Jpgq . r r r r r We proved this result as an easy corollary of the analog of Perfect Set Property for “projective” subsets of V`1. Theorem 1.5 (-Perfect Set Theorem for projective sets, see §4). Assume I0pq. If X Ď V`1 is definable (with parameters) in pV`1, Pq and |X |ą,thenX contains a -perfect set (see §3.5 for definitions). Axiom I0 was first formulated and investigated by Woodin in early 80s (see [47, 49] and [17]-§24), and was used to prove the relative consistency of Axiom of Determinacy (AD). In his recent monographs [48–50], the theory of I0 is extensively studied in connection with the notion of ultimate L. It turns out that structural properties of subsets of V`1 in LpV`1q under ZFC ` I0 have strong resemblance to those of R in LpRq under ZF ` DC ` AD.Theabove-Perfect Set Theorem and Posner–Robinson Theorem support this resemblance.