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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

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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. However, this analogy is not perfect. For instance, AD, if true, holds in LpRq, while I0 is false in LpV`1q; also small forcings can not change the theory of LpRq under AD, while they often alter the theory of LpV`1q under I0. We find another instance against the analogy in our study of another degree theoretic problem in the context of I0. The classical Turing Determinacy states that every subset of the Turing degree poset either contains a cone or is disjoint from a cone. In the context of I0 (at ) and Zermelo degrees, the analogue degree determinacy for Zermelo degrees at is as follows: Every subset of Zermelo degrees at either contains a cone or is disjoint from a cone.WithI0 plus an indestructibility condition on the critical point, we are able to show that the degree determinacy for Zermelo degrees at is false in LpV`1q.

Theorem 1.6 (See §6). Assume ZFC ` I0pq.Letκ “ critpjq and suppose V |ù “the supercompactness of κ is indestructible by κ-directed closed posets”. Then the degree determinacy for Zermelo degrees at is false in LpV`1q. However, unlike in L and Lr¯s, where the failure of degree determinacy is due to the simplicity of relevant degree structures in these models, our proof of the above AXIOM I0 AND HIGHER DEGREE THEORY 973 theorem exploits the sophistication of the degree structure under I0. Although we stated the Posner–Robinson and the degree determinacy results for Zermelo degrees, our argument in fact applies to a wide range of definability degrees, including Δ1-degrees. 1.3. An interesting connection. Our minimal cover and Posner–Robinson results are consequences of large cardinal assumptions, normally it would be natural to study the consistency strength of these degree theoretical properties. However, while studying general structural properties of the degree posets, it makes more sense to consider them in canonical settings such as L, Lr¯s etc. We shall discuss this in §2.3. From L, Lrs to Axiom I0, results mentioned earlier depict a fascinating corre- lation, in canonical models, between the complexity of degree structures at strong limit of countable cofinality and the large cardinal strength of relevant cardinals. The larger the cardinal, the more complicated the associated degree structures; and conversely, the structures of Zermelo degrees at singular cardinals cross over canon- ical models seem to be the same, and the complexity of these structures seems to indicate how high up the associated singular cardinals are in the inner model, rather than which inner model they are in. For example, the Zermelo degree structures at ℵ are well ordered above singularizing degrees cross over fine structure models, at least up to Mitchell fine structure models for sequences of measures, to which our covering argument in §2.2 is applicable. Here we put this as a conjecture. Conjecture 1.7. In any reasonable LrEs-like fine structure inner models, the Zermelo degrees at ℵ are wellordered above singularizing degrees.

A key step is to check whether in M1, the minimal iterable class model for one Woodin cardinal, the Zermelo degrees at ℵ are well ordered above singularizing degrees. On the other hand, it will be remarkable to find out the cardinal at which the Zermelo degree structures in these canonical settings cease to be well ordered (above singularizing degrees) and start to emerge new complexities. If this can be done, it would certainly give some new insight into certain large cardinals. These ideas provide new perspectives, and they are forming a new line of research. It involves multiple major areas in set theory, as well as in recursion theory, endless new questions and problems can be generated. To keep this paper focused, we save further discussion for future articles. Next subsection is a brief account of the organization of the rest of this paper.

1.4. Organization of the paper. The rest of this paper consists of two parts: one on higher degree theory (§2), and the other on the consequences of I0 we mentioned early (§3–§6). §2 discusses the program of higher degree theory. It first introduces the general notions of definability degrees, and proposes to study four degree theoretical prop- erties. In §2.2, we investigate and cross-compare these properties on the structures of Zermelo degrees in various settings, in particular in LrEs-like fine structure models. Aforementioned degree theoretic consequences of I0 are briefly mentioned as part of the global theme. In §2.3 are remarks on the theme of this research program. Since §3, the paper is devoted to those consequences of I0. §3setsupsomeof the background knowledge for later arguments. It introduces basic I0 theory in §3.1. Generic absoluteness theorem for subsets of V`1 is a powerful tool in I0 974 XIANGHUI SHI

theory, our main I0-results in this paper are all applications of this theorem. §3.2 states the projective version of this theorem (which is sufficient for our needs) and discusses its usages in this paper. One can find its proof and applications in Woodin’s monograph [49] and Cramer’s dissertation [3]. In our proofs, we heavily use Prikry type forcings, for the convenience of the readers, we include some basic materials on Prikry forcings in §3.3. A notion of rank is developed in §3.4 for a systematic analysis of Prikry-type forcings, in particular, for proving their -goodness. -goodness is a property introduced by Woodin in [49] for forcing posets to ensure the existence of their M-generics in V . In the proof of the failure of Degree Determinacy, we will use a mixed Prikry tower forcing (§6.4), which is an instance of the tree Prikry forcing discussed in §3.4. In §3.5, we define the analog of perfect tree in the context of ,andthe(large) perfect subset of V`1. Immediately in §4, we show that the analog of perfect set theorem for “projective” subsets of V`1, and the Posner–Robinson Theorem for degrees at is proved in §5 as a corollary. §6 is devoted to the proof of the failure of Degree Determinacy. §6.2 and §6.3 are proofs for a few key lemmas. §6.4 defines and investigates the mixed Prikry tower forcing needed in our argument. In this paper the failure of Degree Determinacy is proved under I0 plus an additional indestructibility condition, §6.5 shows that I0 plus this extra condition is equiconsistent with I0. The indestructibility assumption is independent of I0, we do not know if the same conclusion still holds without it. We conjecture it is the case. Furthermore, in light of Shelah’s theorem that under ZFC, LpV`1q is a model of choice, for strong limit of uncountable cofinality, we conjecture that Conjecture 1.8 (ZFC). For every uncountable cardinal , the degree determinacy for Zermelo degrees at is false in LpV`1q. This will be discussed at the end of this paper (§6.6).

Besides the I0 theory, this paper also involves a great deal of fine structure theory, generalized recursion theory, as well as the theory of Prikry-type forcings. Though this paper is intended to be as much self-contained as possible, there is not enough space here to cover all necessary backgrounds, we have to presume a fair amount of familarity with these subjects from the readers. We will give relevant references at the appropriate places.

§2. Higher Degree Theory. In this section, we give a very sketchy account of gen- eral degree theories at uncountable cardinals (in particular at strong limit singular cardinals), which we would like to call higher degree theory. Higher degree theory extends the study of higher recursion theory in multiple dimensions: it studies more general definability degree notions, its underlying universe is not limited to con- structible universe, and one of its theme is to study the connection between large cardinals and general degree structures. This section involves fine structure theory and generalized notions from recursion theory, familiarity of these areas, though not a ‘must’, will be very helpful. At the end of this section, we will come back to those I0 results and explain their significance in the study of higher degree theory. AXIOM I0 AND HIGHER DEGREE THEORY 975

2.1. Definability degree notions. Given a (definability) degree notion, say a ďΓ b, we write a ”Γ b and a ăΓ b to denote what they normally mean. We say a, b have the same Γ-degree if a ” b. The undertilded symbol a denotes the Γ-degree of Γ r a, i.e. the ” -equivalence class of a.Wesayadegreea is singularizing if there is a Γ r cofinal subset of of ordertype cfpq in a. We often omit the subscript when it’s r clear from context what the current degree notion is. Definition 2.1 (ZFC). Suppose Γ is a fragment of ZFC such that ZFC proves the consistency of Γ. Suppose is an uncountable cardinal satisfying 2ă “ .Let H denote the collection of sets whose transitive closure has cardinality ă .Fixa  Ă which codes a well ordering of H of ordertype . For each a Ď ,letαa be the least ordinal α ą such that Mpaq“def Lαr, as is a model of Γ. If there is a definable wellordering of H (of ordertype )inV , then there is no need to mention  explicitly. For any two subsets a, b Ď ,seta ďΓ b if and only if MpaqĎMpbq.Thisgives rise to a degree notion, which we call Γ-degree.Toeacha Ď , JΓpaq,theΓ-jump of a, is the subset of that codes the structure pMpaq, P,aq. For this section, we work with Γ “ Z, Zermelo set theory, i.e. ZF ´ Replacement. The main reason is that Z is sufficient for proving Covering lemmas for fine structure models and for developing the theory in this section. The exact amount of theory needed for the results in this paper is not important, Z is merely a matter of a personal choice, the reader can replace the occurrences of Zermelo set theory in this section with his favorite fragment of ZFC that suffices for the job. The above definition is given under ZFC. In general, suppose T0 is our working theory, and T1 is a consistently weaker fragment of T0 (i.e. the existence of minimal models of T1 can be derived from T0), then one can define a degree notion for T1 under T0 as above. If Γ is Kripke–Platek (KP) theory, one gets an analogue of hyperarithmetic degrees at uncountable cardinals. Finer degree notions such as the analogue of Turing degree can be defined by putting further limitation on the use of formulas, 0 0 allowing only sets that are both Σ1 and Π1 at , i.e. subsets of V definable via Σ1 and Π1 formulas with quantifier ranging over V. To define degrees beyond ZFC-degree, large cardinals must be assumed, and in that case, one can define degrees associated to higher analogues of inner model operators in Becker [2]. Sacks and his students extensively studied the analogues of Turing degrees and hyperarithmetic degrees at ordinals larger than (see [33], [34]), but mostly limited to degrees in L. Some recent developments indicate that there are some interesting connections between large cardinals and the structure of degrees at uncountable cardinals. In this section, we shall discuss this connection via Zermelo degrees. Remarks about other degrees will be made at the end of the discussion. Given a degree notion for subsets of , uncountable, its degrees form a natural poset, even a lattice-like structure. In this paper, we focus on the structural properties of these posets. The very first question is about the height and width of this poset: Question 2.2. What are the supremum of the sizes of chains of degrees and the supremum of the sizes of antichains of degrees? If there is sufficient choice, these two supremums equal to the sizes of maximal chains and maximal antichains, respectively. The answer for chains is easy: `, 976 XIANGHUI SHI

since every degree has only ď many degrees below it. Below is a list of degree theoretic questions we would like to ask about for the degrees. 1. (Post Problem). Are there two incomparable degrees, i.e. two sets a, b Ď such that a ę b and b ę a? One can also ask a relativized question, i.e. incomparable degrees above a given degree. A set of pairwise incompa- rable degrees form an antichain. A related question is the size(s) of maximal antichains, if they exist. 2. (Minimal Cover). Given a degree a, is there a degree that is minimal above a? r r Such a minimal degree, say c, is called a minimal cover for a, i.e. a ă c,and r r r r there is no b Ď such that a ă b ă c. One can also ask the multi-minimal- r r r cover question, i.e. are there more than one minimal covers? 3. (Posner–Robinson). Is it true that for almost all x Ď , the Posner–Robinson equation for x has a solution, i.e. Dg Ď rpx, gq”Jpgqs?HereJpgq is the jump of g for the given degree notion, and by “almost all”, we mean that there are no more that many exceptions. 4. (Degree Determinacy). Is it true that every degree invariant subset of Ppq either contains or is disjoint from a cone? A set X Ď Ppq is degree invariant if x P X ñ x Ď X . X contains a cone means that there is an x P X such that r ty Ď | y ě xuĎX . Among the four questions, the first one is about antichains, therefore is related to the width of the degree poset, the second is about the organization of degrees, like whether the degrees are dense or discrete, the third is about the internal under- standing of the degrees, such as what information do the degrees carry, and the last is more or less a question about the connection between the members of Ppq and the subsets of Ppq, a bridge between these two types. The first three problems are first order questions regarding Ppq (more frequently pV`1, Pq in practice). However, for the degree determinacy problem, it makes more sense to state it for degree invariant subsets of Ppq in LpPpqq, just like the situation of Turing Determinacy for sets of reals in LpRq—because if it fails then the complexity of the counterexample is the essence of this problem. So it is more appropriate to think the degree determinacy problem as a question quantifying over second order sets in LpPpqq (or LpV`1q in practice). Since V`1 varies in different universes, answers to degree theoretical questions, such as these four questions, often vary in different V ’s. We will give an example on this matter in §2.3.1. These are certainly important degree theoretic questions. This list, however, is by no means meant to be comprehensive, it is merely a list of questions that at this point we are confident to answer. At a strongly inaccessible cardinal (even regular cardinal satisfying ă “ ), the degree notions in general are similar to their counterparts at , since most usual constructions for degrees at , priority argument, local forcing argument, et al, can be carried out at with very few changes. So the degree structures for an analogue degree notion at is very much like its counterpart at , not much new insight is obtained there. At these cardinals, the answers to the first three questions are often “Yes”, as in the case of . However, for the Degree Determinacy question, the answer on the contrary is very likely to be “No”. This will be discussed at the end of §6.6. AXIOM I0 AND HIGHER DEGREE THEORY 977

On the other hand, Sy Friedman showed that Theorem . “ ℵ p 2.3 (Sy Friedman [9]) Assume V L. 1 -degrees the analogue of ℵ q 1 p q Turing Degrees at 1 are well ordered above 0 the analogue of Turing jump ,the ℵr least degree that computes a cofinal subset of 1 . The key element of Friedman’s argument is the analysis of stationary subsets of 1. His method can be applied to general degree notions at strong limit singular cardinals of uncountable cofinality, and produces similar simple structures. As a corollary of his proof, we have Corollary 2.4 (V “ L). Zermelo degrees at singular cardinals of uncountable cofinality are well-ordered above the singularizing degree. In fact, the same argument works in any fine structure models. Therefore there is not much interesting left at singular cardinals of uncountable cofinality. It turns out that the case of countable cofinality is where the treasure is buried – the degree struc- tures at strong limit singular cardinals of countable cofinality carry very valuable information about relevant cardinals. 2.2. Structures of Zermelo degrees in inner models. We study degree structures in some canonical inner models. Unlike the situation of , much less of degree structures at uncountable cardinals can be determined by ZFC, even with large car- dinals. The two main techniques for constructing degrees with certain properties, priority and forcing arguments, in most situation no longer work when we move to degrees at uncountable cardinal ą . Both these techniques use a great deal of combinatorics of , and these very often demand large cardinal properties on the relevant cardinals. They become more useless when degrees at singular cardinals, in particular, of countable cofinalities, are considered. New techniques are in demand for these situations. Besides, forcing posets all preserve but may change things below ą . This is very disturbing, as the basis of studying degree structures at is not to alter the universe V up to . On the other hand, fine structure models provide very complete settings for answering most questions. In these models, objects are constructed in a well organized manner. As we shall see shortly, the rigidity of these models clears out many structural chaos, and makes the impacts of large cardinal axioms to the structure of degrees more evident. Further discussion of the reason for working with these nice models can be found in §2.3.1. In this sec- tion, to simplify the illustration, we shall focus on the structures of Zermelo degrees in fine structure models, in particular at countable cofinality singular cardinals. Our main tool is Covering lemmas. Mitchell’s handbook article [30] is the main reference for this subsection. 2.2.1. Zermelo degrees in L. Let us start with an observation in L about Zermelo degrees at countable cofinality singular cardinals. Recall that a degree x, x Ď ,isa r singularizing degree, if Mpxq contains a cofinal subset of of ordertype cfpq“. Theorem 2.5. Assume V “ L. Suppose ą is a singular cardinal with cfpq“ and x Ď is such that x is a singularizing degree. Then Zermelo degrees at above r x are well ordered. In particular, Zermelo degrees at ℵ are well ordered. r Proof. The argument is a simple application of the Covering Lemma for L. Suppose a Ď and a ěZ x. Consider Mpaq, the minimal Zermelo model con- taining a. Notice that Mpaq has the same reals as V and sharps are absolute 978 XIANGHUI SHI

between transitive models containing 1, therefore Mpaq contains no sharps. In Mpaq,asx P Mpaq,everyz Ă can be identified with a countable subset of . p q P Mpaq “ | |“| |` By Covering in M a , a is covered by a b L Lαa such that b a 1. p q Pp q | |Lαa “ Ñ| |Lαa As M a and Lαa agree on 1 , b 1.Let : b b be a bijection in r sPPp qĎ Lαa .Then a 1 Lαa . It follows that a can be computed from b in Lαa . p q“ (This is essentially the proof of Theorem V.5.4 in Devlin [4].) Thus M a Lαa . This means that the mapping a ÞÑ α is injective. Therefore Zermelo degrees at r a are well ordered above x. Since the sequence tℵ | n ă u is Σ -definable over Lℵ , r n 1 Zermelo degrees at ℵ are fully well ordered. % There are two remarks about this proof. The first one is the idea of applying Covering to the minimal model Mpaq to show that the core model of Mpaq is an initial segment of the core model of V . This type of argument will appear frequently later in this section. The idea of applying Covering Lemma to set models is due to Woodin. As the second remark, we would like to point out a useful fact that is implicit in the proof above, and will be frequently used later in this section. Fact 2.6. Suppose M, N are two Zermelo models agreeing up to a strong limit singular cardinal P M X N, and a P M X N is a cofinal subset of of ordertype cfpqă,thenPpaqXM “ PpaqXN. Theorem 2.5 depicts an extremely simple structure. Here are the answers (in L) to the question list on page 7. Enumerate tαx | x Ă u in increasing order, let α be its -th member. Say x is a successor (resp. limit) degree if α “ α for some r x successor (resp. limit) ordinal . Corollary 2.7. Assume V “ L.Let be a singular cardinal of countable cofinality. Consider Zermelo degrees at above the singularizing degree. (1) There are no incomparable degrees. (2) Every Zermelo degree has a unique minimal cover. (3) Posner–Robinson equations at limit degrees fail to have solutions, i.e. if x Ă has successor degree, then there is no G Ă such that px, Gq”Z JZpGq. Therefore the answer to the Posner–Robinson Problem for Zermelo degrees at is negative. (4) Degree determinacy for Zermelo degrees at is false. Proof. (1) and (2) are immediate from this wellordered structure, so the answers are “No” for the first question and “Yes” for the second. However, for the multi- minimal-cover question, the answer is “No”. In this wellordered structure, Posner–Robinson equation is equivalent to the jump inversion equation, namely, pDGqpx ”Z JZpGqq. Notice that whenever there is a new subset of is constructed, say in Lα`1zLα,wehaveLα`1 |ù “|Lα|“”. Therefore, a successor degree knows that the minimal Zermelo model associated to its (immediate) predecessor degree has size , therefore can compute the jump of its predecessor. So the jump operator in the degree structure coincides with the successor operator in the well-order. Thus limit degrees can not be the jump of any degree. There are ` many limit degrees, therefore the answer for (3), the Posner–Robinson question, is “No”. AXIOM I0 AND HIGHER DEGREE THEORY 979

(4) follows from the fact that there are two disjoint sets of degrees that are unbounded in the degree structure, and thus witness the failure of degree determi- V nacy in V .ButLpV`1q “ L, so the witnesses for the failure of degree determinacy V are also in pLpV`1qq . % Here L is viewed as the core model for the negation of the large cardinal axiom that 07 exists. The core of the argument is the Covering lemma for L.Thesame form of Covering Lemma holds for inner models between L and Lrs (the inner model for one measurable cardinal), which include models like Lp07q, Lp077q etc., and Dodd–Jensen’s core model K DJ, the core model below a measurable cardinal. For instance, Lemma 2.8 (Covering Lemma for K DJ, [7, 30]). Assume that there is no inner model for one measurable cardinal and the Dodd–Jensen core model K DJ exists. Then DJ for every set x of ordinals, there is a y P K such that x Ď y and |x|“|y|`1. These models can be obtained by (proper) partial measures using Steel’s construc- tion. For these models, the same covering argument works. The point is that in these inner models, the minimal model of the form Mpaq, a Ď , a countable cofinality singular cardinal, is always an initial segment of the core model. Thus Zermelo degrees for countable cofinality singular cardinal are always well ordered above the singularizing degree, the same as in L. Therefore the same argument shows that Corollary 2.7 holds for these core models as well. A little extra caution is needed for V arguing the failure of degree determinacy in LpV`1q when LpV`1q ‰ V ,where V is one of the aforementioned core models: as the next large cardinal after in `` the core model is much bigger than PpV`1q“ , the corresponding (partial) `` V measure is inactive before , thus we have PpV`1q Ă LpV`1q, therefore the counterexample is also in LpV`1q. 2.2.2. Zermelo degrees in Lrs. Next is Lrs, the canonical model for one measurable cardinal. The Covering Lemma for Lrs starts to have different form. Lemma 2.9 (Covering Lemma for Lrs, [6,30]). Assume that 0: does not exist but there is an inner model with a measurable cardinal, and that the model Lrs is chosen so that κ “ critpjq, the least ordinal moved by the elementary embedding j given by , is as small as possible. Then one of the following two statements holds:

(1) For every set x of ordinals there is a set y P Lrs with x Ď y and |y|“|x|`1. (2) There is a sequence C Ď κ, which is Prikry generic over Lrs, such that for all sets x of ordinals there is a set y P Lr, C s such that x Ď y and |y|“|x|`1. Furthermore, the sequence C is unique up to finite initial segments. But the difference does not affect the structure of Zermelo degrees at cardinals other than κ. Theorem 2.10 (V “ Lrs). Zermelo degrees at countable cofinality singular car- dinals are well ordered above singularizing degrees. Moreover, any successor degree is the jump of its predecessor. Proof. Reorganize Lrs as LrEs using Steel’sconstruction, where E is a sequence of (possibly partial) measures. For readers who are not familiar with Steel’s construc- tion, the author recommend Schimmerling’s article [37]. Here we omit the predicate for the extender: When we say Lα rEs (α P Ord or α “ Ord), we often refer to the 980 XIANGHUI SHI

structure xLαrEæαs, P,Eæαy or xLαrEæαs, P,Eæα, Epαqy. A crucial point of using Steel construction is the acceptability condition, which says for any ă α,

pLα`1rEszLαrEsq X Pp q‰∅ ùñ Lα`1rEs|ù|α|“ . Here are some benefits of having the acceptability condition:

1. Pp qXLrEs“Pp qXL ` rEs for any cardinal , 2. Suppose is a cardinal, a, b are unbounded subsets of .Ifb ďZ a and αb ă αa ,thenJZpbqPMpaq, hence JZpbqďZ a. Fix an a Ă in LrEs, consider Mpaq in LrEs. First suppose ą κ.AsMpaq and LrEs agree up to , arguing as in L,wegetthatMpaq, a Ď , are initial segments of LrEs. Now suppose ă κ.AsMpaq has the same reals as V , Mpaq does not have 0:,soMpaq could have at most one full measure. Case 1. Mpaq has no full measure. In this case, as the Covering Lemmas for inner models below one measurable have the same form as the Covering Lemma for L (for instance, see the Covering Lemma for K DJ on page 979), applying the corresponding covering lemma, we get that the minimal models Mpaq, a Ă ,are the same as its own core model, namely K Mpaq “ Mpaq. Run the comparison process for Mpaq against K V “ LrEs. Mpaq is iterable and since Mpaq agrees with LrEs up to , iteration maps used during the comparison do not move Mpaq, thus Mpaq is an initial segment of K V “ LrEs. Case 2. Mpaq has one full measure, say Mpaq|ù1 is a measure at some ą . Mpaq satisfies the hypothesis of Covering Lemma for one measurable. Apply the Covering in Mpaq, then there are two cases, either 1 Mpaq 1. a is covered by a set y PpLr sq with |y|“|a|`ℵ1,or 1 Mpaq 2. a is covered by a set y PpLr ,Csq with |y|“|a|`ℵ1,whereC is a Prikry generic over pLr1sqMpaq. Such C is unique up to finite differences. Notice that ă and Prikry generics do not add new bounded subsets. a is a bounded subset of , so it must be case 1 – the covering set y for a is in pLr1sqMpaq. p q“ r 1sMpaq “ r 1s p q V Thus M a L  Lαa  . Run the comparison for M a against K . Arguing as before, Mpaq is an initial segment of K V “ LrEs. So either case, we have that Mpaq, a Ă , are initial segments of LrEs, exactly thesamepictureasinL – Zermelo degrees at in LrEs are well ordered above every singularizing degrees via their Zermelo ordinals. The “moreover” clause follows from the acceptability condition. This completes the proof. % It is not difficult to see that this argument can be adapted to show the results for core models of finitely many measurable cardinals. So Corollary 2.7 should also include large cardinal core models beyond L up to core models for finitely many measurable cardinals. 2.2.3. Zermelo degrees in Mitchell models for an -sequence of measures. Anew picture starts to emerge in the canonical model for many measurable cardinals, Lr¯s,where¯ “xn : n ă y and each n is a measure on κn and κn ă κn`1, n ă . Consider V “ Lr¯s. Again we view Lr¯s as built with (partial) measures using “ Steel’s construction. Let κ supn κn.Let be a countable singular cardinal. AXIOM I0 AND HIGHER DEGREE THEORY 981

It is not difficult to see that, when ą κ or ă κ , arguing as in Lrs,Zermelo degrees at is well ordered above the singularizing degree. The new picture appears at “ κ. The Covering Lemma for Lr¯s is similar to that of Lrs, except that C in the second case now is a system of indiscernibles C “xCn : n ă y with the following property:

1. Each Cn Ă κn is either finite or a Prikry sequence; 2. C as a whole is a uniform system of indiscernibles, i.e. Ť p@x¯ P Lr¯sq p@n ă qpxn P nqùñ|tCnzxn | n ă u| ă . In fact, for any function f : Ñ Ytu with infinite support, i.e. the set supppfq“def ti P | fpiqą0u is infinite, one can use the following varia- Pf tion of diagonal Prikry forcing ¯ to produce an indiscernible system such that |Cn|“fpnq: • Pf p ¯q Ă | |ď p q The conditions of ¯ are pairs a,¯ AŤ such that each ai κi ,and ai f i , P ă each Ai i ,fori ,moreover, i ai is finite. • The order is defined by pa,¯ A¯qďpa¯ 1, A¯1q iff apiqĚa1piq, A1piqĂApiq,and ´ 1 P ă ai ai Ai for i . These discussion about system of indiscernibles can be found in §4 of Mitchell’s handbook article [30]. The proof of classical Mathias condition for characterizing P Pf diagonal Prikry sequences for ¯ can be easily adapted to show the ¯ -version of fpnq fpnq Mathias condition – simply work with sets A Ă κn such that A P n for each n ă such that fpnqą0, where k denotes the k-dimension product measure given by . Proposition Pf . 2.11 (Mathias Condition for ¯ ) Suppose M is an inner model of ZC, Zermelo set theory plus choice, ¯ P M, f and Pf are defined in M as above. ś ¯ P p qfpnq Pf Suppose G nP κn .ThenG is a generic sequence for ¯ over M if and only if for any sequence A¯ P M such that An P n for n ă , there is an m ă such that GpnqĂAn,forn ě m. To simplify the presentation of our next theorem, we use the standard diagonal P “ Pf p q“ ă Prikry poset, namely ¯ ¯ with the constant function f n 1, for n . With this diagonal Prikry forcing, one can use a single diagonal Prikry sequence C in case (2) of the Covering Lemma for Lr¯s. Lemma 2.12 (Covering Lemma for Lr¯s, [30]). Assume the sharp of Lr¯s does not exist and there is an inner model containing measurable cardinals. Let Lr¯s be “ “ p q such that κ supnă κn,whereeachκn crit jn , is as small as possible. Then one of the following two statements holds:

(1) For every set x of ordinals there is a set y P Lr¯s with x Ď y and |y|“|x|`1. (2) There is a sequence C Ď κ,whichisP¯ -generic over Lr¯s, such that for all sets x of ordinals there is a set y P Lr,¯ C s such that x Ď y and |y|“|x|`1. Furthermore, the sequence C is unique up to finite differences. Using generics for the standard diagonal Prikry forcing as the system of indis- cernible, we describe the structures of Zermelo degrees at countable cofinality singular cardinals in Lr¯s as follows. For this subsubsection, we say an ordinal 982 XIANGHUI SHI

α ą is a Zermelo ordinal for a Ď if Lαr,¯ as|ùZermelo set theory. For each ordinal ,let denote the -th Zermelo ordinal (for ∅).

Theorem 2.13. Assume V “ Lr¯s,where¯ “xn : n ă y is an -sequence of “ p q measures such that κ supn crit jn is as small as possible.

(1) If ‰ κ, then the Zermelo degrees at are wellordered above any singularizing degree; (2) If “ κ , consider only Zermelo degrees at above the degree of ¯, identifying ¯ as a subset of .Then ` (a) tαa | a Ă u“t | ă ^ ą lim ă u. Define a ď b ô α ď α ,fora, b Ă .Thenď is a prewellorder of the r r a b Zermelo degrees at abovethedegreeof¯. ` (b) For each ă ,letα be the -th member of t | ą lim ă u,let A be a subset of that codes the sequence xα : ă y, and C be the set P r s of diagonal Prikry generic sequences for ¯ that are Lα ¯ -generic. Then Zermelo degrees at (above the degree of ¯) whose Zermelo ordinals equal to α are exactly the degrees given by

A ‘ C “tpA ,Cq|C P C Yt∅uu.

For the rest of this subsubsection, whenever we write Mpxq or say the minimal Zermelo model of x Ă ,wemeanitasMp,¯ xq, the minimal Zermelo model for p,¯ xq. Proof. (1). This case was discussed on page 981 immediately before introducing the indiscernible system. (2). First, fix an a Ă “ κ ,sinceMpaq has the same reals as V “ Lr¯s,the real coding the theory of Lr¯s is not in Mpaq, so one can apply the Covering for Lr¯s within Mpaq.Therearetwocases: Mpaq 1. a is covered by a set y PpLr¯sq with |y|“|a|`ℵ1, Mpaq 2. a is covered by a set y PpLr¯srCa sq with |y|“|a|`ℵ1,whereCa is P r s ¯ -generic over Lαa ¯ . Such Ca is unique up to finite differences. p q“p r sqMpaq “ r s Case 1 gives M a L ¯ Lαa ¯ , and case 2 gives p q“p r sr sqMpaq “ r s M a L ¯ Ca Lαa ,¯ Ca p r s P q for some Lαa ¯ , ¯ -generic sequence Ca . The second equality is because adjoining diagonal Prikry generics does not change the heights of Zermelo models. Therefore ` every αa , a Ă ,isa for some ă . But notice that in either case, can not be a limit of Zermelo ordinals, i.e. it must be that ą lim ă α : because P r s ă otherwise, for instance in case 2, it would be that a L ,¯ Ca for some ; r s r s but then since Ca is also L ¯ -generic, L ,¯ Ca is a Zermelo model containing a, contradicting to the minimality of Mpaq. For the other direction, notice that for any , lim ă “ lim ă α .Sogiven ă ` P r s r s“ r sĚ p q any ,wehaveA L ¯ ,soL ¯ L ,¯ A M A ; on the other R r s hand, A Llim ă ¯ and is the least Zermelo ordinal above lim ă , thus it p q“ r s ă ` must be that M A L ¯ . Thus for every Zermelo ordinal α that is not a limit of Zermelo ordinals, there is an a Ě such that αa “ α.Thisproves(2.a). AXIOM I0 AND HIGHER DEGREE THEORY 983

Moreover, • p q“ r s“ r s“ p q ă ` in case 1, M a Lα ¯ Lα ,¯ A M A for some ; • p q“ r s“ r s“ p q ă ` in case 2, M a Lα ,¯ Ca Lα ,¯ A ,Ca M A ,Ca for some . ` So every a Ă is Zermelo equivalent to a set of the form pA ,Cq for an ă and a C P C Yt∅u. This proves (2.b). % Although we use the standard diagonal Prikry sequence to state this theorem, the Pf argument works for every ¯ . So for each f as on page 981, every Zermelo degree Pf can be represented by a diagonal Prikry sequence for ¯ . Compared with previous pictures, though not eventually well ordered, this is still a rather simple structure. We have definite answers to the four questions. Corollary 2.14. Assume V “ Lr¯s, and ,¯ κ,¯ be as in Theorem 2.13. Let “ supn κn. Consider the Zermelo degrees at above the degree of ¯. (1) There are incomparable Zermelo degrees. (2) No Zermelo degree has a minimal cover. (3) Posner–Robinson Theorem for Zermelo degrees at is false. (4) Degree determinacy for Zermelo degrees at is false. Proof. (1). We prove something slightly stronger. Let C be a system of indis- Pf p q“ ă | p q| “ cernibles associated to ¯ ,wheref n 2foralln . Thus C n 2, f f for n ă . Suppose C P C ,whereC is the set C defined relative to f.LetC0,C1 be such that C0pnq“min C pnq and C1pnq“max C pnq for n ă .ByMathias Pf r s ¯ “x ă y P ă condition for ¯ over Lα ¯ ,foreveryA An : n such that An n, n , there is an m ă such that Ci pnqPAn, for each i Pt0, 1u and for all n ě m. P r s p r s P q By Mathias condition for ¯ over Lα ¯ , both C0 and C1 are Lα0 ¯ , ¯ -generic sequences. An important fact about diagonal Prikry sequences for P¯ is that if M C0,C1 are pM, pP¯ q q-generic, where M is a ZC model, and |tn | C0pnq‰ p qu| “ R r s R r s R r s C1 n ,thenC0 M C1 and C1 M C0 . Thus we have C0 Lα ,¯ C1 R r s p q p q ď and C1 Lα ,¯ C0 . This implies that A ,C0 and A ,C1 are Z-incomparable. (2). Consider two degrees, represented by pA ,Cq and pA ,Dq,withC P C Yt∅u and D P C Yt∅u, respectively. Suppose pA ,CqăZ pA ,Dq. This implies that ď , as adjoining indiscernible systems does not increase the heights of Zermelo models. We want to find a Zermelo degree that lies strictly in-between. ă P p q ă r s First, suppose that .InthiscaseC M A ,D and α α ,soLα ,¯ C can be “computed” in MpA ,Dq, thus MpA ,CqPMpA ,Dq and MpA ,Dq|ù “|MpA ,Cq| “ ”. Therefore MpA ,Dq “knows” the theory of MpA ,Cq,so 1 one can find a C P MpA ,Dq that is MpA ,Cq-generic for P¯ .Buttheheightof 1 1 MpA ,C,C q is still α .Asα ă α , MpA ,C,C q is again a member of MpA ,Dq. Therefore we have 1 pA ,CqăZ pA ,C,C qăZ pA ,Dq.

Now we assume that “ .SincepA ,CqăZ pA ,Dq, pA ,Dq “computes” pA ,C Y Dq.HerewetreatC Y D as the sequence xtC pnq,Dpnqu : n ă y. So MpA ,DqĚMpA ,CYDq. Notice that the two minimal models MpA ,CYDq and MpA ,Dq have the same infinite subsets of , D as a subset of a countable set C Y D is in MpA ,C Y Dq. Therefore MpA ,DqĎMpA ,C Y Dq and hence 984 XIANGHUI SHI

pA ,C Y Dq”Z pA ,Dq.Letf : Ñ be such that fpnq“|tC pnq,Dpnqu| 2 for n ă .SincepA ,Cq does not compute pA ,Dq, the set supp pfq“def tn P | fpnq“2u must be infinite. Take an a Ď supp2pfq such that both a and supp2pfqza are infinite, let g : Ñ be such that gpnq“2forn P a, gpnq“1 P z P Pf Pg r s for n a. Using Mathias condition for ¯ , ¯ ,and ¯ over Lα ¯ , one can Yp æ q r s Yp æ q see that C D a is an indiscernible system for Lα ¯ .SinceC D a differs with C, C Y D at infinitely many places, we have C YpDæaqRMpA ,Cq and C Y D R MpA ,C YpDæaqq. Therefore, we have

pA ,CqăZ MpA ,C YpDæaqq ăZ pA ,C Y Dq”Z pA ,Dq.

This shows that no Zermelo degree has a minimal cover. (3). Suppose ă ` is a limit ordinal. We claim that Posner–Robinson equation 1 for A has no solution. Suppose it does, let g Ă be such that pA ,gq”Z g .Here 1 ` we write g for the Zermelo jump of g. Suppose αg “ α ,forsome ă .Note 1 “ “ ` “ that αg α `1 and αA α . Then we have an equation: α α α `1.This ` can’t happen if is a limit ordinal. Since there are many A ’s, Posner–Robinson theorem is false for Zermelo degree at . (4). Let I Ă ` be such that it is unbounded and co-unbounded in `.Then ` the two sets tA : P I u and tA : P zI u induce two disjoint unbounded sets of degrees. Therefore Degree Determinacy at is false. This counterexample exists V in both V and LpV`1q ,astheyarethesame. % For (1), a further question is whether there is a size antichain of Zermelo degrees. As there are no minimal degrees, the usual way of getting 2 many incom- parable degree at by constructing 2 many minimal degrees no longer works here. Note that all the models of the form MpA ,Cq, C P C havethesamereals. Let C0,C1 P C be such that C0 Ă C1 and C1zC0 is infinite, then C0 ăZ C1.Forany Ei , i “ 0, 1, such that C0 Ă Ei Ă C1, E0 ıZ E1 iff E0 Δ E1 is infinite, and E0 ďZ E1 ˚ ˚ iff E0zE1 is finite. Thus the poset pPpq{Fin, Ď q,whererasĎ rbs iff azn Ď bzn for some n ă , can be embedded into the Zermelo degree poset between the degrees of C0 and C1. Therefore, there are infinite descending sequences of Zermelo degrees. However, as Zermelo-jump increases associated Zermelo ordinals, there is no Harrison-type ([14], see also [33] III.3.6) descending sequence, i.e. Corollary 2.15. Assume V “ Lr¯s, and ,¯ κ,¯ be as in Theorem 2.13. At ,there are infinite descending sequences of Zermelo degrees, but there is no infinite sequence xa : i ă y abovethedegreeof¯ such that JZpa ` qďZ a . ri ri 1 ri Let pD, ďq denote the poset of Zermelo degrees at above the degree of ¯. The theorem below says that in Lr¯s, the set of degrees represented by the sets coding the Zermelo ordinals, and the relation that two sets share the same Zermelo ordinal, are definable over the structure pD, ďq. Theorem 2.16. Assume V “ Lr¯s, and ,¯ κ,¯ be as in Theorem 2.13. The following are definable over the structure pD, ăq: (1) I “tA | A Ă codes xα : i ă y, ă `u. r i (2) R “tpa,bq|a, b Ă , α “ α u r r a b AXIOM I0 AND HIGHER DEGREE THEORY 985

Proof. The definability of R follows immediately from that of I: Rpa,bq iff @c P I pc ď a Ø c ď bq. r r r r r r r For the definability of I over pD, ăq, we claim that I “ta |@b,c ă a Dd pb ď d ^ c ď d ^ d ă aqu. r r r r r r r r r r r We showed earlier that given an appropriate indiscernible system C , one can par- tition it into two disjoint indiscernible systems C0,C1, which have Zermelo degrees strictly below that of C .Inthiscase,C clearly has the least degree above both degrees of C0 and C1 (see the proof of Corollary 2.14-1). This shows that every degree of the form pA ,Cq, C P C , is not in the set (on the right side) in the last display. ` Suppose a “ A for some ă ,soαa “ α . Suppose b,c ă a are degrees with ď ă r r ă r r r ” p q αb αc α . By Theorem 2.13-2.b, there are b, c such that b Z A b ,B ” p q P C Yt∅u P C Yt∅u “ and c Z A c ,C ,whereB b and C c . Thus αb α b and “ αc α c . First suppose that b “ c .Letf : Ñ be such that fpnq“|tBpnq,Cpnqu| ă P Pf r s for n . As argued before, using Mathias condition for ¯ and ¯ over Lαc ¯ , Y p r s Pf q r Y s one can see that B C is a Lαc ¯ , ¯ -generic sequence. Therefore Lαc ,¯ B C p Y q is the smallest Zermelo model containing A c ,B C .SinceB, C are subsets of a countable set B Y C and all the Zermelo models have the same reals, we have r Y s“ p q“ p q “ ă Lαc ,¯ B C M A ,B,C M b, c and αpb,cq αc .Asb, c Z a and ă “ r Y s r s αc αa α , Lαc ,¯ B C can be computed in Lα ¯ , thus p qY p qĎ p q“ r Y sP p q M b M c M b, c Lαc ,¯ B C M a .

So b, c ďZ pb, cqăZ a. For the general case b ď c , we prove the following general statement: ` 1 Claim 2.17. If ď ă , then for every B P C ,thereisaB P C Yt∅u such 1 that pA ,BqďZ pA ,B q. 1 Let z “pA ,Bq. Applying the Covering for Lr¯s in Mpzq,thereisaB P C Yt∅u 1 1 such that Mpzq“MpA ,B q. Thus pA ,BqďZ pA ,Bq”Z pA ,B q. 1 P C Yt∅u Back to the general case. By this claim, there is a B c such that ď p 1q p 1q b Z A c ,B . Replacing b by A c ,B , we reduce the general case to the case that b “ c . This completes our proof. % 2.2.4. Zermelo degrees in models beyond many measurable cardinals. Let us look at Mitchell models with more measurable cardinals. According to Mitchell’s Theorem 4.1 in [29], if there is no inner model with an inaccessible limit of measurable cardinals then, as in the Dodd–Jensen covering lemma, for each minimal Zermelo model Mpaq,thereisasingle maximal system of indiscernibles C which can be used to cover any set x Ă in Mpaq. A fair amount of analyses above can be carried out at -limits of measurable cardinals below the least inaccessible limit of measurable cardinals, if there exists one. Therefore, the pictures at those places are rather similar to the one at κ in Lr¯s. Once we go past core models for inaccessible limit of measurable cardinals, the systems of indiscernibles are no longer unique – may depend on the set of ordinals to be covered (see [30], p.1555) – and are extremely difficult to analyze. However, Yang proved the existence of minimal covers at -limit of certain measurable cardinals. 986 XIANGHUI SHI

Theorem 2.18 (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 .

Yang’s proof works for Δ1-degrees and any coarser degree notions at ,herewe only state it for Zermelo degrees. This result can be relativized to any degrees above that of W . This implies that for instance, in the Mitchell model for opκq“κ,2 a new picture appears at the in the hypothesis – there are minimal covers (for almost every degree). Yang’s forcing in fact produces a -perfect set (has size ) of subsets of that are minimal above W . As every degree contains only at most many of them, thus the size of antichains of Zermelo degrees at this can be as large as possible. So we have “Yes” to the first two questions. We do not know the answers to Posner–Robinson and Degree Determinacy at this , but speculate “No” for both of them.

2.2.5. The picture from I0. The analyses above rely heavily on the fine structure theory, especially covering and comparison. Once past Mitchell models, we are out of the comfort zone. Though there are still some variations of covering lemmas for inner models past Mitchell models, very little have we derived from them for the structures of Zermelo degrees in those models. But the emerging new pictures suggest that larger cardinals give us more power to create rich degree structures. In the later part of this paper, we consider the degree structures at countable cofinality singular cardinals from the other extreme – looking at the strongest kind of large cardinal, Axiom I0. I0 asserts the existence of an elementary embedding j : LpV`1qÑLpV`1q at some and with critical point below .This is an - limit of very strong large cardinals (for instance, it is an -limit of ă-supercompact cardinals), it satisfies Yang’s hypothesis. Therefore at this ,thereisa-perfect set of minimal covers for every degree (above the degree of Yang’s W set).

Corollary 2.19. Assume ZFC ` I0pq.LetW be as in Theorem 2.18. Consider the Zermelo degrees at above the Zermelo degree of W . 1. There are incomparable Zermelo degrees. In fact, there are antichains (of Zer- melo degrees) of size . 2. Every Zermelo degree above the Zermelo degree of W has a minimal cover. In fact, every such Zermelo degree has many minimal covers.

Moreover, as applications of Generic Absoluteness Theorem in I0 theory (see [48, 49]) we have the following following results regarding Posner Robinson problem and Degree determinacy at for Zermelo degrees.

Theorem 2.20 (See Theorem 5.1). Assume ZFC`I0pq. Then for every A P V`1, and for almost all (i.e. except at most many) B ěZ A, the Posner–Robinson equation 1 1 for B has a solution, i.e. there exists a G P V`1 such that pB, Gq”Z G ,whereG denote the Zermelo jump of G.

The proof in fact shows something stronger, ďZ here can be replaced by Δ1- reducibility for subsets of . The argument works if G 1 is replaced by any reasonable

2This is not the minimal inner model for Yang’s hypothesis, however it has the “shortest” o-expression. AXIOM I0 AND HIGHER DEGREE THEORY 987 jump operator at . This theorem says that Posner–Robinson holds for Zermelo degrees of subsets of above any A Ă . For the degree determinacy problem, we have an almost negative answer.

Theorem 2.21 (See Theorem 6.2). Assume ZFC and j is an I0pq-embedding. Let κ “ critpjq and suppose V |ù “the supercompactness of κ is indestructible by κ-directed closed posets”. Then LpV`1q|ùDegree Determinacy fails for Zermelo degrees at . Although the theorem is stated with an additional indestructibility require- ment (see [21]), the hypothesis of this theorem is equiconsistent with ZFC ` I0 (see Proposition 6.14). In fact, in light of Shelah’s result that LpPpqq is a model of choice if is a strong limit singular cardinal and cfpqą, together with evidences for degree structures at other cardinals, we conjecture that the failure of Degree Determinacy for Zermelo degrees in LpPpqq at any uncountable cardinal is a theorem of ZFC. We shall discuss more about this conjecture at the end of this paper. 2.3. Some remarks. In this subsection, we make a few remarks. 2.3.1. Why inner models?. The first remark is regarding the question why we focus on degree structures in inner models. Yang’s theorem and those I0 results are stated under large cardinal assumption, it seems to be natural to study the consistency strength of those degree theoretical properties. For instance, • What is the consistency strength of having minimal covers as in Yang’s Theorem (see page 17)? • What is the consistency strength of having Posner–Robinson result as in Theorem 5.1? These are interesting questions in its own, especially for set theorists. Properties regarding generalized recursive degree are subjects of α-recursion theory, which concerns only degrees in L (see Sacks [33]). While we investigate structural prop- erties of higher level degree notions, it also makes more sense to consider them in canonical settings such as fine structure extender models. This is because ZFC alone, even plus large cardinal assumption, though may decide certain individual proper- ties, can hardly determine the structure of degree posets. For instance, consider the structure of Zermelo degrees at ℵ. Example 2.22. Assume ZFC`GCH and plus some large cardinal assumption, say a measurable cardinal κ of Mitchell order opκq“κ`` plus a measurable cardinal 1 κ ą κ. With a small forcing, one can arrange that in the generic extension κ “ ℵ, ℵ 1 GCH remains true below ℵ,2 “ ℵ`2 while the measurability of κ is preserved (This combines results of Woodin and Gitik, see [11]). But then the Zermelo degree posets at ℵ can not be well ordered in the generic extension, as every degree has only ℵ many predecessors in the degree partial ordering. This is in contrast to the pictures in Lrs (see Theorem 2.10, p.10). It is the well organized structure of Lrs (organized using Steel’s construction) that forces the degrees to line up in a well ordered fashion. The existence of mea- surable cardinals alone (more precisely, without appealing to forcings) is not strong 988 XIANGHUI SHI

enough to create “untamed” degrees – incomparable degrees, unless we go up to the -limit of measurable cardinals (see Corollary §2.14) and beyond.

2.3.2. Degree structures in canonical models. In §2, we analyzed Zermelo degree structures in several canonical models or under some stronger large cardinals. An immediate conclusion is that larger cardinals create more complicated Zermelo degree structures at some critical cardinals (more precisely, -limit of certain large cardinals). In other words, in these models the complexity of the Zermelo degree structure at these critical cardinals reflects the strength of the relevant cardinals. The next natural step is to look into larger cardinal axioms and hope to find more complicated degree structures. For instance, what degree structures can one see at an -limit of strong cardinals, or Woodin cardinals, or supercompact cardinals, etc.? During the process, it would be interesting in itself to extend the question list on page 7 to differentiate these degree structures, in a way that the natural order of large cardinals sorts these structural properties into layers. At the same time, the pictures of Zermelo degree structures in L and through up to the core models for finitely many measurable cardinals strongly suggest that in any reasonable inner model, at every singular cardinal with cfpq“ and below the least measurable, the Zermelo degrees are wellordered above some degree. In particular, Conjecture 2.23. In all fine structure extender models the Zermelo degree struc- tures at (their) ℵ are all (eventually) wellordered and the immediate successor is given by Zermelo jump. Combining these remarks, one can see that the complexity of a particular degree structure does not necessarily give the large cardinal strength of the core model, but it does indicate the levels of the associated cardinals that the structure resides. In other words, from the variety of the types of degree structures that appear in a core model one can tell the lower bound of the large cardinals the given core model carries. This is a complete new perspective for looking into large cardinal axioms.

2.3.3. New techniques are needed. Our proofs for L and up to Lr¯s use heavily one particular form of covering lemmas, we expect that that analysis will work as far as that form of Covering Lemma holds, namely at least up to Mitchell models for sequences of measures. Next key step is to check whether in M1, the minimal iterable class model for one Woodin cardinal, the scenarios described above continue.

Conjecture 2.24. In M1, the Zermelo degrees at ℵ are well ordered. Moreover, we expect this to be true for singular countable cofinality ’s that are above or in-between critical large cardinals, as this fits well with the intuition that universes of small large cardinals are initial segment of universes of larger cardinals. Climbing up the cardinal ladder, although new pictures may appear at certain cardinals, as well as degree structures in between these special cardinals, as what we have just discussed about ℵ, it seems reasonable to conjecture that the structure at a particular cardinal once appear in a core model for certain large cardinal axiom, will stay unchanged as we move up to core models for larger cardinals, assuming they exist. AXIOM I0 AND HIGHER DEGREE THEORY 989

However, as the classical form of Covering is not available for M1 and larger core models, deeper understanding of their structures3 and new techniques are necessary for the investigation of degree structures in these models. Besides the classical fine structure models, recent developments in descriptive inner model theory (see for example Sargsyan’s survey paper [36]) suggest a much advanced and daring path of investigation – looking into higher degrees in the HODs of determinacy models, as determinacy gives a whole family of canonical ` models – the ones given by Solovay hierarchy. Assume AD ` V “ LpPpRqq, it’s believed that HOD is a canonical model. Although it’s still an open question ` whether AD ` V “ LpPpRqq proves that HOD is a fine structure model, HOD of ` AD ` V “ LpPpRqq models are believed to be fine structure models at least up onto to Θ “def suptα |Df : R ÝÝÑ αu (see Steel [45] for LpRq and Sargsyan [36] §3for ` a discussion on general AD models). Based on this understanding, the first test question would be ` Question 2.25. Assume AD ` V “ LpPpRqq. Look at Zermelo degrees within HOD HOD at pℵq , are they (eventually) well ordered? This is a great question! One cannot expect to solve this problem with only Covering, ` one would need mouse analysis for arbitrary AD models. But the mouse analysis technique is still in its development, there is very little on this matter that is valuable to say at this point. 2.3.4. Evidences of the impact of large cardinals on structures of degrees. Next we leave the canonical models, look at the impact of large cardinal alone on the structures of degrees. The theme is that stronger large cardinal yields more com- plicated degree structures at certain strong limit, countable cofinality, and singular cardinals. We have seen two evidences, one is the prewellordered degree structure in Theorem 2.13, where you can find incomparable degrees (see Corollary 2.14), the other is Yang Sen’s minimal cover result quoted on page 17. In the second part of the paper, we will prove that I0, one of the strongest large cardinal hypotheses, entails a richer degree structures at a certain strong limit with countable cofinality – it gives positive answers to Post problem, minimal cover prob- lem and Posner–Robinson problem. Furthermore, we prove in §6thatI0 together with a mild indestructibility assumption imply the failure of degree determinacy in LpPpqq for Zermelo degrees at a particular strong limit, countable cofinality, singular cardinal by exploiting the richness of the degree structure provided by the large cardinal axiom. As part of our global conjecture (see page 51), we conjecture that the failure of degree determinacy in LpPpqq for Zermelo degrees at ,for countable cofinality ,isatheoremofZFC. But as our analysis indicates, if one wants to prove this conjecture, the proof has to be very subtle: in early stages of canonical inner models, the degree structures are very simple, the degree determi- nacy fails due to that simplicity and our approach for proving the failure of degree determinacy by exploiting the richness of degree structures does not work there. In this paper, our discussion focus on Zermelo degrees, we only compare struc- tures of Zermelo degrees crossing over inner models, or at different cardinals in one

3So far the best result on constructing core models is due to Neeman [31], who produces a core model for a Woodin limit of Woodin cardinals. 990 XIANGHUI SHI

inner model, and study the impact of large cardinals on the structure of Zermelo degrees. There is a whole spectrum of degrees one can look into – degrees asso- ciated different theories, fragments or extensions of Zermelo–Fraenkel set theory. Certainly there are many more interesting things one can say if structures of different degree notions are compared. These questions lead to a different research direction and need a different set of tools. They are beyond the scope of this paper, we shall not elaborate them here. Without further ado, we proceed to the second part of this paper, consequences of I0 in higher degree theory. The key to the proofs of the three I0 theorems in this paper is the Generic Absoluteness Theorem (Theorem 135, [49]), which is a fundamental tool in I0 theory. In the next section, we shall give a brief account of the part of I0 theory involved in our arguments.

§3. Preliminaries. This section consists of six subsections, the first three give a brief account of the part of I0 theory needed for this paper, and the last three discuss notions such as supercompact Prikry forcing, -good forcings, and -perfect subsets of V`1.

3.1. Basic I0 theory. Basic theory of I0 axiom can be found in [47] and [17], the latest developments and more advanced materials are in [49]. Laver wrote a series of papers on elementary embedding axioms, [26–28] etc., which cover a fair amount of structural I0 theory. He also investigated the algebra associated to these elementary embeddings in [22–25]. These are fun to read but not much related to the results in this paper. Let EpMq denote the class of nontrivial elementary embeddings from M to M. By a celebrated theorem of Kunen [20], ZFC implies that EpV q“∅,infactthe argument gives EpV`2q“∅,forany P Ord. However, it is not known whether ZFC rejects weaker propositions, such as

1. for some , EpV`1q‰∅, 2. for some , EpVq‰∅. These assertions and their variants were first discussed in [10] and [44]. In §24 of [17], these two are called axiom I1 and I3 respectively. The assertion that

“For some , there is an elementary embedding j : LpV`1qÑLpV`1q with critical point ă .”

is called I0 in [17]. In this paper we write I0pq to make this explicit. It was first proposed and investigated by Woodin in the early 1980’s, and later by Laver. The strength of these axioms are as follows: I0 implies I1,andI1 implies I3.They are on the top of large cardinal hierarchy and are not known to be inconsistent with ZFC. Fix a I0pq-embedding j, i.e. a j P EpLpV`1qq with critpjqă.Let

Zj “tjpF qpjæVq|F is a function and F P LpV`1qu.

Let Mj denote the transitive collapse of Zj ,andletk be the inverse of the collapsing map. Mj is the ultrapower UltpLpV`1q,j q, where the measure j is defined as: for any a Ă V, a P j ðñ jæV P jpaq. On the other hand, by Lemma 1.B.17 AXIOM I0 AND HIGHER DEGREE THEORY 991 of [47], Zj ă LpV`1q and V`1 Ď Zj . Therefore as the transitive collapse of Zj , p q “ p q p qą L V`1 Mj L V`1 . So the embedding k is either an identity or has crit k Θ , where p q L V`1 “ t |D P p qp Ñ qu Θ def α  L V`1  : V`1 α is onto .

The case k “ identity is the following proper version of I0 axiom.

Definition 3.1. Suppose j is an I0pq-embedding. We say j is proper if Zj “ LpV`1q. The proper version of I0pq is the assertion that there exists a proper I0pq-embedding.

Note that we have just shown that I0pq-embedding can be factored as j “ k ˝ j0, where j0 : LpV`1qÑMj – LpV`1q is the embedding given by j ,andj0 is clearly proper. So the proper version of I0 is actually equivalent to I0. Properness is used to define iterates of I0 embedding. Let M0 “ LpV`1q, j0,1 “ j be as above and 0 “ j . Inductively, for successor ordinal α “ ` 1, define jα,α`1 : Mα Ñ Mα`1 “def UltpMα ,α q to be the map associated to Ť α “def j , `1p q“ tj , `1pranpqq |  P M ^  : V`1 Ñ  u and j , `1 “ j , `1 ˝ j , for ď . For limit ordinal α,letpMα,j ,α q, ă α,be the direct limit of xM ,j , : ď ď ă αy. j is said to be iterable if for all iterate pMα ,j0,α q of pLpV`1q,jq, Mα is wellfounded and j0,α : M0 Ñ Mα is elementary. Lemma 21 of [49] says that if j is proper then j is iterable. A sophisticated structural theory of LpV`1q has been developed by Woodin under I0 and its variants (see [47] and [17]). This theory exhibits a great deal of resemblance to the structural properties of LpRq under the Axiom of Determinacy. One instance of this resemblance relevant to us (see Theorem 5.1, page 40) is that 7 I0 (in fact I1 suffices) implies that A exists for every A Ă V (see [26,43]). In this paper, we shall work under this proper version of I0 axiom, discuss three of its consequences, two of which support this resemblance and the third opposes to it.

3.2. Generic Absoluteness. A common ingredient of the arguments for all the three I0 results in this paper is the Generic Absoluteness Theorem, which is a consequence of proper I0 axiom and plays a fundamental role in the theory of I0. Woodin [49] proves it (see Theorem 136) for subsets of V`1 in LpV`1q. Here we only state its “projective” version, or equivalently the version for sets in PpV`1qXL1pV`1q, which suffices for our need.

Theorem 3.2 (Generic Absoluteness). Assume ZFC and j : LpV`1qÑLpV`1q is a proper I0pq-embedding. Let pM,j0, q be the -th iterate of pLpV`1q,jq. Suppose that Mrg0srg1s is a generic extension of M such that pg0,g1qPV , g0 is M-generic for a partial order P P M and such that cfpq“ in Mrg0s, g1 is Mrg0s-generic for a partial order Q in M rg0s,whereP and Q are such that 9 P ‹ Q P j0,pVq,then

Mrg0sXV`1 ă Mrg0srg1sXV`1 ă V`1. 992 XIANGHUI SHI

Let ϕ be a Σ0 formula in the language of set theory. By Generic Absoluteness, if one wants to prove that V`1 |ù @ uDvϕpu, vq, just forces over M to get a g0 P V such that Mrg0s|ùcfpq“ and

V`1 |ù D vϕpa, vq, for every a P Mrg0sXV`1. (:)

To get (:), one often needs to do another forcing over Mrg0s to get g1 P V such that M rg0srg1s|ùϕpa, bq,forsomeb P Mrg0srg1sXV`1. Our proofs of the three I0 results run very much like this (a bit subtle in the proof of the third result). 3.3. Supercompact Prikry forcing. Prikry type forcings are used in the proofs of all three theorems. In this subsection, we give a brief account of the supercompact Prikry forcing, as well as other relevant materials that will be used in our proof of the -Perfect Set Theorem. Fix cardinals κ ď .LetPκpq“tX Ď ||X |ăκu.LetU be a κ-complete ultrafilter over Pκpq.WesaythatU is normal iff for every X P U and every f : X Ñ satisfying fpxqPx for every x P X ,thereareX 1 P U and α1 ă such that for every x P X 1, fpxq“α1. U is fine iff for every α ă ,theset tx P Pκpq|α P xuPU . κ is -supercompact iff there exists a fine normal measure over Pκpq. Next is a notion needed in the definition of the supercompact Prikry forcing. If X, Y P P pq,thenX is strongly included in Y , denoted as X Ă Y ,ifX Ď Y and κ r otppX qăotppY X κq.ForanyX P P pq,thesettY P P pq|X Ă Y uP. κ κ r Let  be a fine normal measure over Pκpq. The supercompact Prikry forcing P is defined as follows.

Definition 3.3. P consists of conditions of the form ps, Aq,where 1. s is a finite Ă-increasing sequence of elements in P pq, r κ 2. A P ,and 3. for every a P A,andeveryi P dompsq, spiqĂa. r For every two conditions ps, Aq, pt, BqPPκpq, ps, Aqďpt, Bq iff 1. domptqďdompsq, 2. s extends t, i.e. sæ domptq“t, 3. A Ď B,and 4. for every i P dompsq´domptq, spiqPA. In particular, we write ps, Aqď˚ pt, Bq for the case dompsq“domptq. We often identify s with its range. The following are some standard facts about the supercompact Prikry forcing:

Proposition 3.4. Let P be P as defined above. 1. pPrikry Conditionq.Letps, Aq be a condition in P and ϕ be a sentence in the forcing language (of xP, ďy). Then there is a B P  such that the condition p “ps, Bqď˚ ps, Aq and decides ϕ, i.e. either p , ϕ or p , ϕ. 2. pMathias Conditionq. Suppose M is an inner model of Zermelo set theory plus Axiom of Choice and  P M.Thenan-sequence G is P-generic over M if and only if for any A P M such that A P , there is an m ă such that GpnqPA for n ą m. 3. P adds no new bounded subsets of . AXIOM I0 AND HIGHER DEGREE THEORY 993

Let P ˆ P denote the set of pairs of conditions in P whose first components p 1 1qď p q are finite sequences of the same length, and ordered by p ,q P ˆP p, q iff 1 1 ˚ 4 p ďP p and q ďP q. The direct extension ď is defined in the same manner.   P ˆP P2 Let  denote the following supercompact Prikry forcing for producing a pair of P2 p q supercompact Prikry sequences: conditions in  are triples of the form r, s, A such 1 1 1 1 1 that pr, Aq, ps, AqPP and |r|“|s|,andpr, s, AqďP2 pr ,s ,Aq iff pr, AqďP pr ,Aq    p qď p 1 1q P2 and s, A P s ,A , the direct extension is defined similarly. Notice that  is isomorphic (via pr, s, Aq ÞÑppr, Aq, ps, Aqq) to a dense subset of P ˆ P (both in the sense of ď and ď˚), it is not difficult to derive the following Prikry condition for P ˆ P P2   (from the one for ). This property will be used in our proof of -Perfect Set Theorem.

Proposition 3.5 (Prikry Condition for P ˆP). Let ppr, Aq, ps, Bqq be a condition in P ˆ P and ϕ be a sentence in the forcing language of xP ˆ P, ďy. Then there is a C P  such that p “ppr, C q, ps, C qq ď˚ ppr, Aq, ps, Bqq and p decides ϕ. As pointed out earlier, the essence of this proposition is the Prikry condition for P2  . We leave it to the reader to work out the details. A key ingredient is the following notion of diagonal intersection of the system tAa,b | a, b P Pκpqu: tc |@a@b ra Ă c ^ b Ă c Ñ c P A su. r r a,b This is a member of  if Aa,b P  for every pa, bqPPκpqˆPκpq. The fine normal measure  over Pκpq induces a normal measure  on κ, due to the fact that A P  iff t | α P AuP,whereα is the least ordinal not in . Let P be the basic Prikry forcing conditioned on  (see [12] §1.1). The following proposition connects P-generics and P-generics. It is relevant to the proof of -Perfect Set Theorem.

Proposition 3.6. Suppose ¯ “xi : i ă y is a P-generic sequence. For each i ă ,letαi be the least ordinal that is not in i . Then the sequence α¯ “xαi : i ă y is P-generic.

Proof. In fact, letting P “ P and R “ P , the mapping  ÞÑ α , the least ordinal that is not in , gives rise to a function  : R Ñ P,namelypps, Aqq “ ps˚,A˚q for ps, AqPR,wheres˚ “tαspiq | i ă|s|u and A˚ “tαa | a P Au.This satisfies the following properties: 1.  is onto and order-preserving, 1 2. for any p P P and r P R such that p ďP prq,thereisar P R such that 1 1 r ďR r and pr q“p. Verifying 1 is routine. To see 2, suppose r “ps, AqPR,thenprq“ps˚,A˚q. 1 1 Suppose p “pt, BqďP prq,thenletr “ps ,AXta P Pκpq|αa P Buq,where 1 1 s extends s and is such that s˚ “ t. It is easy to check that if D Ă P is dense then tr | prqPDu is dense in R. Thus whenever H is a pV, R q-generic filter, “H generates a pV, P q-generic filter. % A function like  : R Ñ P in the proof above, according to Abraham [1], is called a projection. In this case, the relation between P and R is denoted as P Ÿ R.

4 Note that our definition of P ˆ P literally is not the product of two copies of P ,howeversince the former poset is dense inside the latter, as far as the generic extension is concerned, they are the same forcing notions. 994 XIANGHUI SHI

By the theory of iterated forcing, R can be viewed as a two-step iteration P ‹ R{G9 , where R{G9 is the pV, P q-name for the quotient poset as defined on page 336 of [1]: Suppose G Ă P is V -generic, then R{G “def tr P R | prqPGu. Suppose R “ P ‹ Q9 ,whereQ9 “ R{G9 .LetG be a pV, P q-generic filter. Consider R{GˆR{G in V rGs. Identify its pV, P q-name with Q9 ˆQ9 . The following observation will be used in the proof of -Perfect Set Theorem (see page 37). Fact 3.7. Let D9 be a pV, P q-name such that , 9 9 ˆ 9 ½P “D is a dense and open subset of Q Q”. ˚ 9 Then the set D “def tppp0, q90q, pp1, q91qq P R ˆ R | p0 ,P “pq90, q91qPD”u is dense and open in R ˆ R.

3.4. -good forcing. In order to apply the Generic Absoluteness Theorem, we need to ensure that their generics (over M)existinV . For that, we introduce a notion of -goodness for forcing posets. This notion was first introduced by Woodin in his [49]. However, that version is not quite accurate – not exactly what he has in mind. Here is our modified version. Definition 3.8. Let be an infinite cardinal. We say a partially ordered set P is -good (in V ) if it adds no bounded subsets of and for every generic filter G and for every A Ă Ord in V rGs andŤ of size ă , there is a non-Ă-decreasing -sequence x ă y “ ă Ai : i such that A i Ai and each Ai , i ,isinV . Two facts are immediate. Proposition 3.9. 1. If poset P is -distributive (i.e. the intersection of ă many open dense subsets of P is dense), then P is -good. 2. Suppose P is -good in V , G Ď P is pV, Pq-generic, and further Q P V rGs is -good in V rGs,thenP ‹ Q9 is -good in V . The first one is because -distributive forcing adds no new sequences of ordinals of length ă . The second one simply follows from the definition. The poset P to be seen in §6.2 (page 43) is more than -closed (in M,inwhich “ κ is regular), therefore is -good (in M). Our later discussions will involve a number of Prikry-type forcings, we are going to show that they are all -good. A useful sufficient condition for showing -goodness is the following modified version of -goodness in [49], page 404: For all D ĎtD Ď P | D is open dense in Pu ˝ such that |D|ă,foranyp P P,therearep ďP p and a nondecreasing sequence xD : i ă y of subsets of D such that the following hold p,i Ť D “ tD | ă u 1. p,i i , Ş ˝ ˝ 2. for all i ă such that Dp,i ‰ ∅, Dp,i is dense below p , i.e. for all r ďP p , 1 1 there exists r ďP r such that r P D for every D P Dp,i .

3.4.1. P and P are -good. The notion of -goodness is mainly intended for Prikry-type forcings. Here we give two examples to illustrate the basic ingredients in showing the -goodness of Prikry type forcings. These examples are implicit in the AXIOM I0 AND HIGHER DEGREE THEORY 995 proof of -perfect theorem (see Theorem 4.1) – the idea for proving their -goodness is directly used to find their generics. Let  be a normal measure over ,andP be the associated classical Prikry forcing. Then

Proposition 3.10. P is -good.

Proof. Suppose D is a collection of dense open subset of P,and|D|ă.Weuse the following result from Gitik’s handbook article ([12], Lemma 1.13). For A Ď ` and s Ă A,setA psq“def tα P A | α ą maxpsqu.

Lemma 3.11. Let p “ps, AqPP and D Ă P be dense open. Then there are ps, A qď˚ ps, Aq and m ă such that for every n ě m and every p,D P p,D p,D Pr sn p Y ` p qq P t Ap,D , s t, Ap,D t D. “p qPP ă D “t | ď u Fix a p s, A . For each i ,let p,i D mŞp,D i . We may assume D ‰ ∅ ă ă “ t | P D uP that p,i for all i . For each i ,letAp,i Ap,D DŞ p,i . ă P ˝ “ P By the -completeness of , for each i , Ap,i , hence A i Ap,i .So ˝ ˝ ˝ ˚ p “ps, A q is in P and clearly p ďP p. We claim that Ş  ˝ Claim 3.12. For each i ă , Dp,i is dense below p . ˝ Fix an i ă and a r “pt, BqďP p .Let be the first i elements of B.SinceB Ď ˝ 1 A Ď Ap,i and mp,D ď i for all D P Dp,i , the condition r “pt Y , B zpmaxpq`1qq is in every D P Dp,i . %

Let  be a fine normal measure over Pp q for some cardinal ą ,andP be the associated supercompact Prikry forcing. Then

Proposition 3.13. P is -good.

Lemma 3.11 is the dense-open-set version of the Prikry condition for P and it follows from the Rowbottom-ness of the normal measure . In [12], one can find the analog of Rowbottom theorem for P on page 1362. Although Gitik includes the formula version of the Prikry condition for P in [12] (Lemma 1.46), for the proof of -goodness of P, we restate it using dense open sets. For A Ă Pp q and s Ă A,let A`psq“ ta P A |@b P s pb Ă aqu. def r Lemma 3.14. Let p “ps, AqPP and D Ă P be dense open. Then there are ps, A qď˚ ps, Aq and m ă such that for every n with m ď n ă and p,D P p,D p,D every t PrA sn that is Ă-increasing, ps Y t, A` ptqq P D. p,D r p,D 3.4.2. A notion of rank. Next we show that certain tree Prikry forcings described in §1.2 of [12] are also -good. The poset Q to appear on page 45 (see §6.4 for the definition) is such a tree Prikry forcing. Let X be a set of size ě . Suppose  is an ultrafilter on X . Suppose ď is a prewellordering on X such that for every x P X , tu P X | x ă uuP,whereă is the strict part of ď.LetΞbethesetofallă-increasing finite sequences in X . For s, t P Ξ, we say t is above s, s  t if s is a ă-initial segments of t. pΞ, q is a tree order. A subtree T Ď Ξ is called a -tree if it has a trunk t (i.e. the longest node in T such that T “ Tt ,whereTt “def t P T | t   _   tu) and for every node  s P T above t,succT psq“def tx P X | s xxyPT u is in . We are interested in pairs of the form pt, T q,whereT is a -tree with trunk t. 996 XIANGHUI SHI

Let Q be the tree Prikry forcing defined as in Gitik’s handbook article (see §1.2 of [12]): the conditions in Q are pairs of the form pt, T q,whereT is a -tree with trunk t; and the forcing order is given by pt0,T0qďpt1,T1q if and only if ˚ T0 Ď T1,andpt0,T0qď pt1,T1q if further t0 “ t1. To show the -goodness of Q, we introduce a notion of rank. Similar notion can be defined for many Prikry type forcings. Ď Q O P Suppose O  is dense open. Define Rα on t Ξ by induction on α: • O P p qP R0 is the set of all s Ξ such that there is a Ť-subtree T such that s, T O. • ą O “t P |t P | x yP OuP u For α 0, Rα s Ξ a X s a ăα R  . O Ď O ď p q Since O is open, R Rα if α.LetrankO s ,theO-rank of s,betheleast P O p q“8 ordinal α such that s Rα , if it exists; otherwise rankO s . Proposition 3.15. Suppose D Ď Q is dense open. Then rankD ptqă8for every t P Ξ. If in addition  is -complete, i.e. closed under countable intersection. Then for every t P Ξ, rankD ptqă.

Proof. Without lost of generality, we may assume t “ ∅ and rankD p∅q“8. We claim that for any s P Ξ, if rankD psq“8, then there is a -measure one set A   such that s xayPΞfora P A and rankD ps xayq “ 8. Because otherwise, there   is a -measure one set B such that s xbyPΞforb P B and rankD ps xbyq ă  8; and then rankD psq“suptrankD ps xbyq ` 1 | b P Bu wouldbeanordinal. Inductively, one can build a -subtree T Ă Ξ (of height ) such that for every P p q“8 “p∅ q “p1 1qď P s T ,rankD s .Letp ,T . Suppose q t ,T Q p and q D. 1 1 Then t P T and rankD pt q“0. Contradiction! Next, suppose rankD p∅qě. As the function rankD p¨q is always chosen to be the least possible ordinal, for any α ď rankD p∅q,thereisans P T such that rankD psq“α.Pickans P T such that rankD psq“.By-completeness, there is a unique n ă such that   tx P X | s xxyPT ^ rankD ps xxyq “ nuP,

but then rankD psq“n ` 1 ă , contradiction! %

Next lemma is the dense-open-set version of the Prikry condition for Q,which is the key ingredient for proving -goodness of Q. A proof of the formula version of the Prikry condition for Q can be found in [12] (Lemma 1.23). The formulation below contains slightly more information than the formula version.

Lemma 3.16. Suppose D Ď Q is dense open. For any condition p “pt, T qPQ, there is a -subtree T ˚ Ď T such that pt, T ˚qď˚ p and for s P T ˚ such that t  s,

1. rankD psq“maxt0, rankD ptq`|t|´|s|u; and moreover | |ě p q`| | p ˚qP 2. if s rankD t t ,then s, Ts D. Proof. The subtree T ˚ Ď T is obtained by

(i) removing all subtrees of the form Ts for s P T above t andsuchthat|s|ď rankDptq`|t| and rankD psq`|s|‰rankD ptq`|t|,ifexist,andthen (ii) at s P T above t such that |s|“rankD ptq`|t| (hence rankD psq“0), 1 Ď p 1qP replacing Ts by a -subtree Ts Ts such that s, Ts D. It is not difficult to see that T ˚ is a -tree and works as desired. % We would like to remark that the property of T ˚ given in Lemma 3.16 holds for any -subtree of T ˚.WithT ˚, we are ready to prove the -goodness result. AXIOM I0 AND HIGHER DEGREE THEORY 997

˚ T plays the role of Ap,D in Lemma 3.11 and in the proof of Proposition 3.10. P ˚ | |ě p q`| | p ˚qP The point is that for any s T above t such that s rankD t t , s, Ts D. Proposition 3.17. Suppose  is a -complete measure on X , ě 1,thenQ is -good.

Proof. Suppose D is a collection of dense open subset of Q,and|D|ă. “p qPQ ˚ ˚ Let p t, T .ForanyD,letTp,D be the T asŞ in Lemma 3.16. For each ă D “t P D | p qď u ˚ “ t ˚ | P D u i ,let p,i D rankD t i and Tp,i Tp,DŞ D p,i .Bythe ˚ ă ˝ “ ˚ -completeness of ,eachTp,i is a -tree, i ; hence T i Tp,i is also a -tree. These trees all have the same trunk t.Sop˝ “pt, T ˝qPQ and p˝ ď˚ p.Fixan  Q ˝ i ă such that Dp,i ‰ ∅. Suppose p, Sqďp .ForeveryD P Dp,i and every s P S | |ě||` | |ě||` p q above  (hence above t) such that s t i,wehave s t rankD t ,thenbyŞ Lemma 3.16 and the remark following its proof, ps, Ss qPD. This shows that Dp,i is dense below p˝. % Note that given an -measure one set A Ă X ,thesetofallă-increasing finite sequences in A forms a -tree (with trunk ∅). Thus P and P can be viewed as tree Prikry forcings: Ξ is the set of all ă-increasing finite sequences of ordinals ă in the case of P , and is the set of all Ă-increasing finite sequences in P p q in  r thecaseofP. The notion of rank and the above analysis can be adapted for them. Next lemma is the reformulation of Lemma 3.11 and Lemma 3.14 in terms of rank. 1 Lemma 3.18. Let P “ P or P. Suppose D Ď P is dense open. Then rankD pt qă 1 P “p qPP P Ă t p q“ for every t Ξ. Suppose p t, A .Fors Ξ such that s A,letrankD s ˚ rankD pt Ysq. Then there is an Ap,D such that pt, Ap,D qďP p and for every finite s P Ξ such that s Ă Ap,D , s p q“ t t p∅q´| |u 1. rankD s max 0, rankD s ; | |ě t p∅q p Y ` p qq P 2. if s rankD ,then t s, Ap,D s D. Besides the classical proof(s) (see [12]), as  (or ) is a normal measure with respect to ă (or Ă, respectively), one can also arrive Lemma 3.18 immediately from r Lemma 3.16 for P as follows: view P (or P) as a tree Prikry forcing, then you ˚ have the corresponding T (with trunk t), Ap,D is obtained by taking the diagonal ˚ intersection with respect to ă (or Ă, respectively) of A “ succ ˚ psq,fors P T r s T p Ap,D qď˚ p ˚q Ap,D above the finite part of p. The point is that t, Tt t, T ,whereTt is the  tree ptt s | s P Ξ ^ s Ă Ap,D u, q. Note that this property of Ap,D holds also for any -measure one subset of Ap,D . The dense-open-set version of Prikry condition for the product P ˆ P can be formulated in the same manner (cf. Proposition 3.5).

Lemma 3.19. Let P “ P ˆ P. Suppose D is a dense open subset of P.Then 1 1 1 1 1 1 rankD pr ,s qă for every r ,s P Ξ with |r |“|s |. Suppose p “ppr, Aq, ps, BqqPP. For r1,s1 P Ξ such that r1 Ă A, s1 Ă B,let r,s p 1 1q“ p Y 1 Y 1q rankD r ,s rankD r r ,s s . ˚ 1 1 Then there is a Cp,D such that ppr, Cp,D q, ps, Cp,D qq ďP p and for every r ,s P Ξ such 1 1 1 1 that r ,s Ă Cp,D and |r |“|s |, r,s p 1 1q“ t r,s p∅ ∅q´| 1|u 1. rankD r ,s max 0, rankD , r ; | 1|“| 1|ě r,s p∅ ∅q 2. if r s rankD , ,then pp Y 1 ` p 1 Y 1qq p Y 1 ` p 1 Y 1qqq P r r ,Cp,D r s , s s ,Cp,D r s D. 998 XIANGHUI SHI

Similarly, the property of Cp,D holds for any -measure one subset of Cp,D . We will apply these two lemmas repeatedly in our proof of -Perfect Set Theorem (see page 37). The notion of rank is a nice tool for systematically analyzing Prikry type forcings. A more sophisticated instance of the rank analysis is included in a recent joint work with Nam Trang, [39]. 3.4.3. Generics for -good forcings. At last, we show that generics for forcings that are -good in M exist in V.

Proposition 3.20. Suppose j is an I0pq-embedding. Let pM,j0, q be the -th iterate of pLpV`1q,jq. Suppose P P j0,pVq and P is -good in M. Then there exists G Ď P in V such that G is M-generic.

Proof. Let xi : i ă y be the critical sequence associated to j. For each 0 ă i ă ,set Z “ j ` pj “V q. Ť i 2i 1, i,2i 2i p q“ P ă P j0, V năjn,“V and ji,2i “V2i V, so for each i , Zi M.Andas p ` q“ ` ą| | |ù | |“| |ă crit j2i 1, 2i 1 Ť V2i ,wehaveM “ Zi V2i ”. Moreover, we p q“ claim that j0, V iăZŤi . p qĚ It is clear that j0, V iă Zi . For the other direction, we use the following calculation: for any 0 ă m ă n ă and for any α ă ,

jm,npαq“pj0,mpj0,n´mqqpαqďpj0,npj0,n´mqqpαq“jn,2n´mpαqďjn,2npαq. Therefore for every 0 ă m ă n ă , P ùñ p qP Ď “ x Vn jm,n x Vjm,n pnq Vjn,2n pnq V2n . “ p q P ą Thus if y jm, x for some x Vn , n m,then

y “ j2n`1,pjn,2npjm,n pxqqq P Zn. Ť p q“ This shows that j0, V iă Zi . Since V |ù “|j0,pVq| “ ”, we can enumerate the set of all maximal antichains 1 1 of P in M, which are all in j0,pVq,asA “ta | ă u for some ď . 1 For each i ă ,letIi “t P | a P A X Zi u.Let be the following P-name for a p Pq txx y y| ă 1 ^ P u ă “ æ M, -generic filter: , p ˇ,p p a ,andforeachŤ i , i  Ii ,

, “ ^@ ă p| |ă q txx y y| P ^ P u ½ i.e. , p ˇ,p Ii p a .Wehave P “ i i i i ”, P x ă y where ½P is the largest element of . We will construct a sequence pn : n in V such that it generates a filter that is M-generic. ă P x ă y By -goodness,Ť for each i , there is a sequence of -names i,j : j such , “ ^@ ă p P q P P that ½P “i j i,j j i,j V ”. The point is that for each p and i, j ă ,thereisaq ďP p that determines the value of i,j , i.e. for some zi,j P V , p¨q Ñ ˆ q , i,j “ zˇi,j .Letp0 “ ½P.Let : be a bijection. Given pn,let pn`1 be a condition q ďP pn that determines the value of pnq. By our construction, xpn : n ă y determines all the values of i,j , i, j ă , thus meets every member of A exactly once. Therefore xpn : n ă y generates an M-generic filter. %

3.5. -perfect subset of V`1. For the analogue of perfect subset in the context ¯ of V`1, we introduce the notion of -splitting tree. For this subsection, is a fixed singular cardinal of countable cofinality. Let T “pă, q,wheres  t iff s is an initial segment of t. AXIOM I0 AND HIGHER DEGREE THEORY 999

3.5.1. ¯-splitting trees. Suppose ¯ “xi : i ă y is an increasing sequence of “ “t Ñ |p@ ă qp p qP qu Ă cardinals with limi i .LetT¯ s : n i n s i i . T¯ . p q ¯ p ď q ¯ The tree T¯, is called a standard -splitting tree.Atree T, T is -splitting if “ p q T is an isomorphic copy of T¯, i.e. T ran  for some injective mapping  with p q“  Ø p qď p q P dom  T¯ and such that     T   .Foreveryt T¯ and every  P domptq, pt x yq is called an immediate successor of ptq in T . Intuitively, a ¯ -splitting tree T “ ranpq is a tree such that every node ptq in T has domptq many immediate successors in T . We are only interested in the cases that pT, ďT q is a ă subtree of T, i.e. T Ď and ďT “  XpT ˆ T q. Note that our splitting trees are not necessarily closed under initial segments. For a ¯-splitting tree T “ ranpqĎă and n ă ,let “ t p q| P ^ p q“ u Tn def  t t T¯ dom t n denote the n-th level of T ,andrT s“def tf |@mDn ą m pfæn P T qu denote the set of cofinal branches through T .AsetX Ď contains a -perfect set if there exists an increasing sequence of cardinals ¯ “xi : i ă y with “ limi i and a ¯-splitting tree T such that X ĚrT s. The definition for a set to contain -perfect subset does not depend on the choice of the sequence ¯.Thisisbecauseiflimi i “ limi κi , then T¯ contains an isomorphic copy of Tκ¯ , and vice versa. A useful equivalent definition for an X Ď to contain a -perfect subset is that there is a subtree T of T such that for every s P T and for every cardinal ă ,thereare -many pairwise -incomparable successors (not necessary immediate successors) of s in T . In this paper, we only work with that is strong limit. Fix an absolute well P ordering of V that extends and has ordertype .Ť As each subset of V is naturally “p X qYp p Xp z qqq partitioned into many pieces: X X V0 ś iă X Vi`1 Vi , subsets “ | |Ă of V are always treated as members of Y¯ iă Vi `1 . So every set Ď ˚ Ďr sĂ Ď X V`1 corresponds to an X Y¯ .WesayX V`1 is a -perfect ˚ subset of V`1 if the corresponding X is a -perfect subset of . As commented “ above, Y¯ can be embedded into Tκ¯ ,forany¯κ such that limi κi .Letf be such an embedding, then it is equivalent to say that X is a -perfect subset of V`1 if and only if the set f“X ˚ is a -perfect subset of . 3.5.2. Uniform ¯-splitting tree. Furthermore, in our situation (working under I0), if a subset of V`1 contains a -perfect set, the witness can be chosen to be a very nice one, a uniform ¯-splitting tree. “ Ñ T Let T T¯. We say a function  : T is uniform if there is a sequence of x ă y ă Ď T positive integers ni : i such that for every i , “Ti ni ,theni -th level of T.A¯-splitting tree S is uniform if its ¯-splitting-ness is witnessed by a uniform mapping. Intuitively, a uniform ¯-splitting tree is a ¯-splitting tree that for each n, all the n-splitting nodes lie at the same level. The following proposition gives a situation in which uniform ¯-splitting subtrees exist inside every ¯-splitting subtree of T. It will be used in the proof of our Posner– Robinson result (Theorem 5.1).

Proposition 3.21. Suppose ¯ “xi : i ă y is an increasing sequence of measur- “x ă y ` able cardinals, and ¯ i : i is a sequence such that each i is a i´1-complete ultrafilter over i (let ´1 “ ). Then every ¯-splitting subtree of T contains a uniform ¯-splitting subtree. 1000 XIANGHUI SHI

Proof. Suppose  : T Ñ T witnesses that S is a ¯-splitting tree. To show that S contains a uniform ¯-splitting subtree, it suffices to produceś two sequences x ă y x ă y ă P Ď T Xi : i and ni : i such that for i , Xi i and “ jăi Xi ni . The construction proceeds by induction. At step k, for each pair p,¯ q,we x k ă y construct a positive integer nk and a sequence Xi : i such that ă k Ď k´1 k P 1. for each i , Xi Xi śand Xi i , k T 2. theś-image of the product iăk Xi falls within the nk-th level of ,namely k Ď T “ iăk Xi nk . We suppress the use of parameters ¯ and  for the default case, i.e. when the objects p q p 1 1q k p 1 1q are associated to the given ,¯  . We will write explicitly like nk ¯ , , Xi ¯ , when it is not. Ş “ k ă At the end, let Xi kXi for each i . By the countable completeness of i ,eachXi is in i , i ă . Therefore these two sequences, xXi : i ă y and xni : i ă y, work as required. ă 0 “ “ Now we describe the construction. Initially, for i ,letXi i and n0 0 (since p∅q“∅). When k “ 1, by the countable completeness of 0,therearea set X P 0 and a positive integer n such that “tx y| P X uĎTn. These give us 1 1 “ 0 ą X0 and n1 respectively. At the same time, let Xi Xi for i 0. ą x k´1p 1 1q ă y Now suppose k 1 and that so far we have constructed Xi ¯ , : i 1 1 1 1 and xni p¯ , q : i ă ky for every p¯ , q, including the default pair p,¯ q. For each P k´1 X0 , consider the following objects: • ă Ñ k´1 For each i ,leti : i`1 Xi`1 be the order preserving bijection, and let ˚ i be the measure on i`1 induced by i , i.e. ˚p q“ p q“ i X 1iffi`1 i “X 1. ˚ “x ˚ ă y Let ¯ i : i . • The present subtree above px yq, ś i “ t∅uYt px y q| P k´1 ^| |ą u S def  s s iă|s|Xi`1 s 0 . • k x ă y k kp∅q“ The function  witnessing the i`1 : i -splitting-ness of S :Set ∅ k p q“ px y q P zt∅u p q“ p p qq ă| | and  s  s for s T ,wheres i i s i for i s . P k´1 Then by the inductive hypothesis, for each X0 , there exist a positive integer “ p ˚ k q x “ k´1p ˚ k q ă y n nk´1 ¯ , and a sequence Xi Xi ¯ , : i such that Ď “ k´1 P ă 1. i “`Xi śi “i`1 Xi`˘ 1 and i “Xi i`1 for each i ,and t uˆ Ď T 2. “ iăk i “Xi n . Ď k´1 P By the countable completeness of 0,thereisanX X0 such that X 0 and ÞÑ k the function n is constant on X .Letnk be the common value of n and X0 be this X and Ş k “ t | P k u ă Xi`1 i “Xi X0 , for i . ` ą k By the 0 -completeness of i (i 0), each Xi is in i . This completes the construction. %

§4. -Perfect Set Theorem. Classical Perfect Set Theorem says that every closed set of reals either is countable or contains a perfect subset, which is often identified AXIOM I0 AND HIGHER DEGREE THEORY 1001 as the set of cofinal branches of a perfect binary tree. Under Projective Determinacy (PD), this conclusion holds for projective subsets of reals, namely those are definable over pV`1, Pq.WithAD, it holds for sets of reals that are in LpRq.Thesecanbe found in Jech [15] and Kanamori [17]. In the context of V`1, the notion of -perfect set is defined via so-called ¯-splitting trees (see §3.5.1). In this section, we prove a -Perfect Set Theorem for subsets of V`1.Thekey tool in our argument is the Generic Absoluteness Theorem on page 23. Although our argument can be adapted to prove -perfect set property for sets as much as the Generic Absoluteness is applicable, we only present a proof for “projective” subsets of V`1, i.e. sets definable (with parameters) over pV`1, Pq, which “simplifies” the presentation and yet suffices for our later applications. One can find a different approach implicit in the proof of Theorem 175 of [49]. In his doctoral dissertation, Cramer [3] improves the -Perfect Set Theorem to all subsets in LpV`1q using a new technique – inverse limit reflection.

Theorem 4.1 (-Perfect Set Theorem for projective sets). Assume I0 holds at . If X Ď V`1 is definable (with parameters) in pV`1, Pq and |X |ą,thenX contains a -perfect set. Before proving this theorem, we would like to prepare some notations and basic facts about various Prikry generics related to the I0 embedding j and the associated -iterate M.Let

• 0 be the critical point of j, • 0 be the least strongly inaccessible cardinal above 0, • 0 be the normal measure at 0 induced by j, i.e.

a P 0 iff 0 P jpaq, • P p q 0 be the fine normal measure on 0 0 induced by j, i.e.

a P 0 iff j“ 0 P jpaq.

For every 1 ď n ď ,letpMn,j0,nq be the n-th iterate of pLpV`1q,jq and

pn, n,n,nq“j0,np0, 0,0,0q.

In particular, “ “ j0,p0q. Thus for every n ď , |ù P p q Mn “n is a normal fine measure on n n ”. Below is a key fact in our proof of the theorem. Proposition . “x ă y P 4.2 Suppose that ¯ i : i is  -generic over M.For each i ă ,letαi be the least ordinal that is not in i .Letα¯ “xαi : i ă y.Then M r¯ s|ù“ |Mrα¯ sXV`1|“”. Proof. This follows from a standard fact in the theory of supercompact Prikry forcing (see [12] §1.4): In Mr¯ s, is collapsed to . By Proposition 3.6,α ¯ p P q P is M ,  -generic. Note that the Prikry forcing  is a small forcing (of size ă )inM, remains strongly inaccessible in M rα¯ s,soMrα¯ s|ù |V`1|ă .SinceMrα¯ sĂMr¯ s and ¯ collapses to ,wehaveMr¯ s|ù “|Mrα¯ sXV`1|“”. %

Our proof of Theorem 4.1 uses the fact that the critical sequence ¯ “xi : i ă y P is  -generic over M. Here instead of proving this fact directly, we give a more 1002 XIANGHUI SHI

informative argument – derive it from the sequence of typical objects associated to j0, for the -supercompactness of in M.

Proposition 4.3. For each i ă ,leti “ ji`1, pji,i`1“ i q.Then “x ă y P (1) ¯ i : i is  -generic over M, (2) ¯ “xi X : i ă yPMr¯ s, and ¯ “x ă y P (3) i : i is  -generic over M. Proof. p qPP X (1). Fix a condition s, A  M, by Mathias Condition, it suffices to find a k such that xn : n ě kyĎA. Note that there is a k such that for each ě p qPP X p q“ pp qq n k,thereisapair sn,An n Mn such that s, A jn, sn,An .As0 is induced by j, we have for each n ě k,

An P n iff jn,n`1“ n P jn,n`1pAnq,

iff n “ jn`1,pjn,n`1“ nqPjn,pAnq“A.

This shows that xn : n ě kyĎA “ jn,pAnq. (2). Notice that as critpji,i`1q“i for each i ă ,wehave

ji,i`1“ i X i`1 “ ji,i`1“ i X ji,i`1pi q“i ,

for every i ă . And since the critical point of ji`1, is i`1,

i X “ ji`1, pji,i`1“ i X i`1q“ji`1, pi q“i . (3) follows from Proposition 3.6 and (2). % We need a lemma. Lemma 4.4. Assume a sufficient fragment of ZFC. Suppose j : V Ñ M Ă V is a nontrivial elementary embedding. Let “ critpjq and suppose ą a cardinal. Let  be the induced normal measure on κ and  be the induced -supercompact fine normal measure over Pp q from j. Suppose A P . Then for any ă , there are at least jp q many distinct elements in jpAq,whereB “def ta P B | a X jpq“u for B Ď Pjpqpjp qq. Proof. As j“ X jpq“ P jpq, we may assume that A has the property that  P A ñ  X P .LetA| “def t X |  P Au. First we claim that A| P . This is because jpA|q“t X jpq| P jpAqu and “ j“ X jpqPjpA|q, thus A| P . Without loss of generality, we may assume that A| “ . By the theory of elementary embeddings, the embedding j : V Ñ M can be factored as j “ k ˝ i, where i : V Ñ Q – UltpV,  q is the ultrapower embedding associated to  and k : Q Ñ M is an elementary embedding such that kprfsq“jpfqpq for f : Ñ V . For every α ă ,letAα “ta P A | a X “ αu.Takeany ă .

Claim 4.5. There are -measure one many α ă such that |Aα|ą .

Let B Ď be such that α P B iff |Aα |ď . Assume otherwise, i.e. B P .Then A0 “ A Xta P Pp q|a X P BuP, as the second set is in .So|A0|ď ¨ ă . But by the fineness of ,every-measure one set has a union equaling to , thus has at least many members. Contradiction! So we may assume that |Aα |ą for every α ă .Lete : Ñ be the identity function. It represents in Q. Consider the collection Z of functions f : Ñ A such that for each α ă , fpαqPAα. Choose tfg | g : Ñ uĎZ such that for AXIOM I0 AND HIGHER DEGREE THEORY 1003 Ñ ă p q‰ p q p q‰ p q any g0,g1 : and for any α , g0 α g1 α iff fg0 α fg1 α .Thiscan be done by the above claim. By the definition of Z, ipAq contains at least ip q many distinct elements. Note that every f P Z represents a member of ipAqĂPipqpip qq such that rfs X ipq“res,whereepαq“α for α ă . Thus every member of kpZq,sayz, has the property that z X jpq“kpresq“jpeqpq“.SojpAq Ě kpZq,andthe latter by the elementarity of k contains at least jp q many distinct elements. % This gives more than what is needed in our proof of the theorem. Only “ many distinct elements” is needed every time it is used. Now we prove the theorem. Proof of Theorem 4.1. As remarked on page 23, we may assume that j is proper. “x ă y p P q ă Suppose ¯ i : i is M ,  -generic. For each i ,letαi be the least “x ă y p P q ordinal not in i . By Proposition 3.6,α ¯ αi : i is M ,  -generic. 9 9 p P q P We identify ¯ , α¯ as their canonical M ,  -names. We will focus on the  - generics ¯ such that the correspondingα ¯ is the critical sequence associated to the elementary embedding j, ¯ “xi : i ă y, which according to Proposition 4.3 is P a  -generic sequence over M. As remarked on page 31, we identify members of P V`1 as members of , and thus subsets of V`1 as subsets of .Since  and P §  are -good in M (see 3.4.1), by Generic Absoluteness, we have

Mr¯sXV`1 ă Mr¯ sXV`1 ă V`1.

It suffices to prove the theorem for definable subsets of V`1 that are parameterized over Mr¯sXV`1:Givenaformulaϕ and a P M X V`1,let M “t P X | X |ù p qu Xϕ,a x M V`1 M V`1 ϕ a, x “ V P and Xϕ,a Xϕ,a.IfXϕ,a has the -perfect set property in V`1 for every a r ¯sX P M V`1, then by the absoluteness for  , r ¯s ¯ M M rsXV`1 |ù @ a, Xϕ,a has the -perfect set property. P Using the Generic Absoluteness for  again, we have

V`1 |ù @ a, Xϕ,a has the -perfect set property.

Thus every “projective” subset of V`1 has the -perfect set property. Now let X be Xϕ,a for some formula ϕ and a P Mr¯sXV`1 and suppose that X has cardinality ą in V . By elementarity and Proposition 4.2, Mr¯ s|ù 9 ¯ 9 “|X |ą ^|MrsXV`1|“”, where X is an pM, P q-name for X .Sothere 9 9 p P q 9 ,P 9 P z r s 9 is a M ,  -name  such that   X V α¯ . We treat the interpretation of  P (via a  -generic) as a member of . Without loss of generality, if necessary, by replacing j “ j0,1 with jn,n`1 for a sufficiently large n ă ,wemayassumethat p P q “ p q there is a M0, 0 -nameŤ0 such that  j0, 0 . “ “t∅u ď ă Work in V .LetS nSn,whereS0 and each Sn,1 n , consists of the sequences s satisfying the following conditions: 1. dompsq“n, ă p qPP p q p qX P 2. for every i n, s i i`1 i`1 and s i i`1 i`1, 3. for every k ă l ă n, j ` ` pspkqq Ă splq. k 1,l 1 r 1004 XIANGHUI SHI

Order S by extension, i.e. for any s, s1 P S,sets ď s1 iff s “ s1æ dompsq. Note that for every s P S, the set of its immediate successors in S  succps, Sq“def tu | s xuyPSdompsq`1uPdompsq`1. Let rSs denote the set of all infinite paths through S.Foreveryx P S YrSs,letx˚ denote the -lift of s, i.e. the -sequence obtained by setting ˚ x piq“ji`1, pxpiqq, for i P dompxq. ˚,n For n ą 0ands P Sn,lets denote the n-lift of s, i.e. the sequence defined by: ˚,n s piq“ji`1,npspiqq, for i P dompsq. We shall build a ¯-splitting tree in V such that each of its cofinal path produces a different element of X . More precisely, we claim that Claim 4.6. There exists a ¯-splitting tree T Ď S such that

(i) for every t P T and for every i P domptq, tpiqXi`1 “ i , Pr s ˚ P (ii) for every x T , x is  -generic over M. ˚ ˚ ¯ (iii) for x, y PrT s,ifx ‰ y then Mrx sXMry sXV`1 “ M rsXV`1. ˚ Assume this can be done. (i) implies that Mr¯sĎM rx s for every x PrT s. (ii) ˚ r ˚s r s Pr s P M x P says that every M x , x T is a  -generic extension of M. Thus  r ˚s r ˚s M x z r ¯s Pr s P M x “ X M for every x T . By the Generic Absoluteness for  , X ˚ ˚ M rx s X X Mrx s.So P X for every x PrT s. 4.6 implies that the interpretations of  by x˚ and y˚,forx, y PrT s such that x ‰ y, are distinct. Therefore X contains a -perfect set. Before the construction of T , let usŤ remind the readers of the following fact. For every definable class X Ă M, X “ jn,“Xn, where each Xn is the class defined in Mn thesamewayasX defined in M. This is because if x P X and x “ jn,pyq for some n ă and y P Xn,thenx P jn,“Xn. “ P “ P P Yt u Let Rn n and Pn n , n Ť . Then the collection of dense subsets of R, denoted as D, is the union jn,“Dn, where each Dn is the collection of the dense subsets of Rn in Mn.Clearly,n ď|Dn|ăn`1 for n ă ,andjm,n“Dm Ď Dn for m ď n ď . For n ď ,letEn be the collection of dense open set (in Mn)oftheform ˚ Ă ˆ p q 9 D Rn Rn that are obtained from Mn,Pn -names D for dense openŤ subsets 9 9 of Rn{Gn ˆ Rn{Gn,asinFact3.7.Thesituationisthesame:E “ jn,“En, n ď|En|ăn`1 for n ă ,andjm,n “Em Ď En for m ď n ď . “t∅u Now we start the construction of T . First letŤ T0 . We shallś construct a ¯ “x ă y “ “ sequence C Cn`1 : n so that T nTn,whereTn iăn Ci`1 for n ą 0, satisfies the three requirements. : “t P P p q| X P u p p :qq First, let C0 a 0 0 a 0 0 .LetC1 be a subset of j0,1 C0 0 | |“ “tx y| P u such that C1 0. SuchŤ C1 existsŤ by Lemma 4.4. Clearly, T1 a a C1 is | |ă P p q P contained in S1.As C1 1, C1 is a member of 1 1 .Foreveryb C1, Ť Ť : Ť b Ď C and hence otppbqďotpp C q.LetC 1 “ta P j pC q| C Ă au. 1 1 1 0,1 0 1 r Then C 1 P  and for every b P C and every a P C 1,wehaveb Ă a. 1 1 1 1 r Consider the partial order R1 and the collection D0 of all dense subsets of R0. px y 1q P Apply Lemma 3.18 in R1 to conditions of the form a ,C1 , a C1,andevery D Ď 1 P member of j0,1“ 0. One gets an Aa,D C1 as in Lemma 3.18 for every a C1 and AXIOM I0 AND HIGHER DEGREE THEORY 1005 Ş D P D0.Since1 is 1-complete and |C1|ˆ|D0|ă1,thesetA1 “ tAa,D | D P D0u is in 1.ThisA1 satisfies the following properties,

1 a xay a ‘ pA1q.Foreverya P C1 and every D P D0,letk “ rank p∅q,thenk ă 1 D j0,1pDq D and for every finite Ă-increasing s Ă A1, x y r 1. rank a psq“maxt0,ka ´|s|u; j0,1pDq D | |ě a px y `p qq P p q 2. if s kD ,then a s, A1 s j0,1 D . ‘1p q It is not difficult to see that 1 A1 remains to be true if A1 is replaced by any 1-measure one subset of A1. Next, apply Lemma 3.19 in R1ˆR1 to ppxay,A1q, pxby,A1qq for every two a, b P C1 E a,b P and every member of j0,1“ 0.OnegetsaŞ CE 1 as in Lemma 3.19 for each p qP ˆ P E : “ t a,b | P ^ P E u : P a, b C1 C1 and E 0.LetC1 CE a, b C1 E 0 .ThenC1 1, | ˆ |¨|E |ă : as C1 C1 0 1. C1 satisfies the following properties, ‘2p :q P P E 1 C1 .Foranytwoa, b C1 and for every E 0,let x y x y ka,b “ rank a , b p∅, ∅q, E j0,1pEq : then ka,b is finite and for any two finite Ă-increasing r, s Ă C with E r 1 |r|“|s|, x y x y 1. rank a , b pr, s q“maxt0,ka,b ´|r|u; j0,1pEq E | |“| |ě a,b 2. if r s kE ,then ppx y p :q`p Y qq px y p :q`p Y qqq P p q a r, C1 r s , b s, C1 r s j0,1 E . ‘2p :q : : 1 C1 remainstobetrueifC1 is replaced by any 1-measure one subset of C1 . : Ď : P ‘1p :q Meanwhile, as C1 A1 and C1 1, 1 C1 holds. For the inductive step n ` 1, n ą 0, suppose we already have constructed a xp : q ă y ă sequence of pairs Ci`1,Ci`1 : i n such that for each i n, Y : Ď p :q 1. Ci`1 Ci`1 ji,i`1 Ci ;. | |“ : P 2. Ci`1 śi and Ci`1 i`1; 3. T ` “ C ` Ď S ` ; i 1 jăi`1 j 1 i 1 Ť : 1 1 : 4. C Ď C ,whereC “ta P j ` pC q| C ` Ă au, thus i`1 i`1 i`1 i,i 1 i i 1 r 1 ˚,i`1 @t P T ` @a P C @b P t rb Ă as; i 1 i`1 r ‘1 p : q P P D r “ 5. i`1 Ci`1 holds, i.e. for every r Ti`1 and every D i ,letkD ˚,i` r 1 p∅q r ă Ă Ă : rankji,i`1pDq ,thenkD and for every finite -increasing s Cn`1, ˚,i` r r 1 p q“ t r ´| |u (a) rankji,i`1pDq s max 0,kD s , | |ě r p ˚,i`1 p : q`p qq P p q (b) if s kD ,then r s, Ci`1 s ji,i`1 D ; ‘2 p : q P P E 6. i`1 Ci`1 holds, i.e. for any two r, s Ti`1 and every E i ,let

˚,i`1 ˚,i`1 kr,s “ rankr ,s p∅, ∅q, E ji,i`1pEq : then kr,s is finite and for any two finite Ă-increasing r1,s1 Ă C with |r1|“ E r i`1 |s1|, 1006 XIANGHUI SHI

˚,i`1 ˚,i`1 (a) rankr ,s pr1,s1q“maxt0,kr,s ´|r1|u, ji,i`1pEq E | 1|“| 1|ě r,s (b) if r s kE ,then pp ˚,i`1 1 p : q`p 1 Y 1qq p ˚,i`1 1 p : q`p 1 Y 1qqq P p q r r , Ci`1 r s , s s , Ci`1 r s ji,i`1 E . : : Now consider jn,n`1pCn q.Thisisamemberofn`1 as Cn P n. Using Lemma 4.4 : ` p ` p qq | ` |“ again, let Cn 1 be a subsetŤ of jn,n 1 CŤn n such that Cn 1 n. This ensures Ď | |ă P P p q that Tn`1 Sn`1.As Cn`1 n`1, Cn`1 n`1 n`1 .Then 1 : Ť C “ta P j ` pC q| C ` Ă auP ` . n`1 n,n 1 n n 1 r n 1 p ˚,n`1 1 q P Apply Lemma 3.18 in Rn`1 to conditions of the form r ,Cn`1 , r Tn`1,and D Ď 1 every member of jn,n`1“ n.Sincen`1 is n`1-complete, there is a set An`ś1 Cn`1 P ‘1 p q | |“ ă such that An`1 n`1 and n`1 An`1 holds. This is because Tn`1 iďni n`1 and |Dn|ăn`1. ˚,n`1 ˚,n`1 Next, apply Lemma 3.19 in Rn`1 ˆ Rn`1 to ppr ,An`1q, ps ,An`1qq for every pr, s qPTn`1 ˆ Tn`1, and every member of jn,n`11“En.Asn`1 is n`1- | ˆ |¨|E |ă : Ď complete and Tn`1 Tn`1 n n`1, one gets a Cn`1 An`1 such that : P ‘2 p : q : Ď ‘1 p : q Cn`1 n`1 and n`1 Cn`1 holds. Since Cn`1 An`1,wealsohave n`1 Cn`1 . The other conditions (1–4) in the inductive hypothesis are clear. This completes the construction of subtree T . Now we verify that T meets the requirements in the claim on page 36. Our tree T is designed to be ¯-splitting. (i) is due to the fact that T Ď S. (ii) follows from our arrangement of the applications of Lemma 3.18 to every member of jn,n`1“Dn at the pn ` 1q-st step and the fact that every dense open subset of R in M is of the form jn,pDq for some n ă and some D P Dn. Pr s : “p∅ :q : “pp æ q˚,n :q ą Fix an x T ,letp0 ,C0 and pn x n ,Cn for n 0. Then p :q“p ˚æ p :qq ă jn, pn x n, jn, Cn , for all n . : Ď 1 Ď p :q By our construction, Cn`1 Cn`1 jn,n`1 Cn ,sowehave : Ď p : qĎ p :q ă ď Cn`k jn`m,n`k Cn`m jn,n`k Cn , for m k . : Since Cn`1 Ď jn,n`1pCn q,wehaveform ă k ď ,

jn`m`1,n`k pxpn ` mqq P jn`m`1,n`k pCn`m`1q Ď p p : qq jn`m`1,n`k jn`m,n`m`1 Cn`m “ p : qĎ p :q jn`m,n`k Cn`m jn,n`k Cn .

By item 4 of the inductive hypothesis, for m ă k, j ` ` ` pxpn ` mqq Ă a for n m 1,n k r P : “p ær ` qq˚,n`k ă every a Cn`k.Letsn,k x n, n k . Then for n ,wehave Ă p :q : Ď p :q`p q ă sn,k jn,n`k Cn and Cn`k jn,n`k Cn sn,k , for k . ą ‘1 p :q Our construction ensures that for n 0, n Cn holds. Thus applying jn,n`k,we ˚,n 1 : x,n pxænq have ‘ pjn,n`k pCn qq:inparticular,fixaD P Dn´1,letk “ rank p∅q,then n D jn´1,npDq ě p x,nq“ x,n Ă p :q for k jn,n`k kD kD ,sincesn,k jn,n`k Cn , pp æp ` qq˚,n`k p p :qq`p qq P p q x n k , jn,n`k Cn sn,k jn´1,n`k D . AXIOM I0 AND HIGHER DEGREE THEORY 1007 : Ď p :qq`p q p q : P p q Since Cn`k jn,n`k Cn sn,k and jn´1,n`k D is open, pn`k jn´1,n`k D .By p : qP p q p : q ˚æp ` elementarity, jn`k, pn`k jn´1, D . The finite part of jn`k, pn`k is x n ˚ kq, thus the filter Gx˚ induced by x has a nonempty intersection with jn´1,pDq: : ˚ p qP ˚ X p q Pr s D jn`k, pn`k Gx jn´1, D . Therefore for every x T , x is jn,“ n-generic for all n ă , hence is fully M-generic for R. To see 4.6, suppose x, y PrT s. In our construction we arrange the applications of Lemma 3.19 for every member of En at the pn ` 1q-st step, argue similarly as we ˚ did for the D-genericity of x for x PrT s to show that every -liftoftheform ˚ ˚ 9 px ,y q, px, yqPrT sˆrT s,isE-generic. Moreover, as R “ P ‹ R{G and ˚ ˚ px q“py q“¯ by our construction, there are x1,y1 such that ˚ ˚ Mrx s“Mr¯srx1s and Mry s“Mr¯sry1s.

By the definition of members of E, px1,y1q is pR {¯ ˆ R{¯q-generic over Mr¯s. Therefore x1,y1 are mutually pMr¯s,R{¯q-generic. This implies that ˚ ˚ M rx sXMry s“Mr¯srx1sXMr¯sry1s“Mr¯s. This completes the proof of Theorem 4.1. %

§5. Posner–Robinson Theorems at . For sets X, Y Ă , X ďT Y if X is recursive in Y ,andX ”T Y if X ďT Y and Y ďT X .ForX, Y,Z Ă , X ďT pY, Z q if X is recursive in the pair pY, Z q, i.e. there is a recursive bijection  : Ñ ˆ such ´1 that X ďT  rY ˆ Zs. The classical Posner–Robinson theorem (see [32], [42]) asserts that for any non- recursive real A,thereisarealG such that A appears to be the Turing jump of 1 G modulo G, more precisely, G ”T pA, Gq. Shore and Slaman [40] generalize ă CK this to any α-REA operators, for any α 1 , the first admissible ordinal. More pαq precisely, for any A R Iα,thereisarealG such that G ”T pA, Gq,whereIα is the countable ideal consisting of reals which are not recursive in Hp q for any ă α,andX pαq denotes the α-th Turing jump of X , X Ă . Woodin later proved (unpublished) the Posner–Robinson Theorem for hyperarithmetic jump as well as for the sharp. The sharp version says: Assume @x Ă px7 existsq, then for any 7 nonconstructible real A,thereisarealG such that G ”T pA, Gq. The definition of sharps can be found in standard set theory textbooks such as Jech [15] and Kanamori [17]. The relevant point is that G 7, or the relevant jump of G,canbe coded as a subset of . The Posner–Robinson theorem and its generalizations play key roles in Shore–Slaman’s work on the definability of Turing jump (see [40]) and Slaman–Steel’s partial solution to Martin’s Conjecture (see [41]). Ă ď Suppose is an infinite cardinal. Define for subsets X, Y V, X T Y if X is both Σ1 and Π1 definable over pV, P,,Yq,where is a fixed wellordering of ” V of ordertype . The equivalence relation T is defined as usual. This gives a degree notion analogous to Turing degree. We mostly work with ’s that are strong limit singular cardinals, in which case subsets of V are often treated as subsets of . As a corollary of the -Perfect Set Theorem, we obtain a version of Posner– Robinson Theorem for sharps at V`1. In the introduction, to make our points, we stated the theorem for Zermelo degrees only, here we put it in a stronger form. 1008 XIANGHUI SHI

Theorem 5.1. Assume I0pq.ForeveryA P V`1 and for almost all (i.e. except at ě most many) B T A, the Posner–Robinson equation for B has a solution, i.e. there P p q” 7 exists a G V`1 such that B, G T G . Here G 7 is treated as a subset of . Our proof of this theorem works for most definability jump operators at . ProofofTheorem5.1. Again, we may assume that j is proper. Let ¯ be the associated critical sequence. For A P V`1,letϕpAq denote the following statement: D Ă p| |ď ^@ ě p R ÑD rp q” 7sqq XA V`1 XA B T A B XA G B, G T G p q” 1 Note that the relation  a, b “a is the sharp of b” can be expressed by Π1- formula. Here the superscript refers to second order quantifiers in the context of , 5 namely, quantifiers ranging over V`1. So @AϕpAq is a sentence over pV`1, Pq. Towards a contradiction, we fix an A0 P V`1 such that V`1 |ù ϕpA0q, i.e. for Ă ď R ě any X V`1 of size , there is a “bad” B X , namely such that B T A but @ rp qı 7s X G B, G T G .Let 0 be the set of bad B’s, i.e. X “t P | ě D p q” 7u 0 B V`1 B T A0 and G, B, G T G .

ϕpA0q implies that |X0|ą. By the “projective” -Perfect Set Theorem, X0 contains a -perfect subset of V`1. We shall derive a contradiction. As remarked on page 31, members of V`1 can be coded by cofinal branches of ¯ Y¯ (see page 31) via the -sequence and a fixed nice well order of V, and further by cofinal branches of T¯ via a simple injective function that embeds Y¯ into T¯. P p q For X V`1,letc X denote the cofinal branch of T¯ that codes X . Note that this coding function is continuous thus can be treated as a member of V`1,and ď ¯ moreover, it is bijective and is T -reducible to . By Proposition 3.21, there is a ¯ “ p q r sĎ X Ďr s uniform -splitting tree T ran  such that T c“ 0 T¯ ,where is the uniform mapping witnessing this, it can be coded by a subset of V. T is a “copy” of the standard ¯-splitting tree, so one can pick a path B through 7 T to code W ,whereW “pA0,q. The coding is defined as follows: B is a cofinal branch through T such that for each n ă , Bæn “ pcpW 7qæmqæn, for sufficiently large m ă . W 7 “computes” W , thus it “computes” .  “computes” ¯, and hence c. Therefore 7 7 ě p q W “computes” B, and hence W T B, W . On the other hand, since  is an injection, for each m ă , cpW 7qæm “ ´1pBænqæm for sufficiently large n ă , therefore B and  together can “compute” cpW 7q.  can “compute” c´1,sowe p q” 7 Pr sĎX % have B, W T W .ButB T 0. This is a contradiction! Remark 5.2. Note that the definition of sharp is not essential in the proof. The same conclusion holds if the sharp operator is replaced by any reasonable jump

5The key to the two layers of quantifiers is the wellfoundedness condition. Normally, if a is (or codes) a countable theory, the statement “a satisfies the wellfoundedness condition” (hence “a is the sharp of 1 Ă b”) is Π2 in the context of (cf. Theorem 14.11 in [17]). But for an uncountable theory, say a , is uncountable, to say that “a satisfies the wellfoundedness condition”, it is equivalent to say that “every countable subset of a (or the theory coded by a) satisfies the wellfoundedness condition”. The latter Ă 1 1 is saying “for every countable c a,aΠ2 statement (in context) holds for c”. This is only a Π1 1 statement in the context of , since Π2 formula in the context contributes no second order quantifier in the context of . AXIOM I0 AND HIGHER DEGREE THEORY 1009 operator defined on members of V`1, since the complexity of the coding method used in our proof is below Δ1. For instance, in analogue to the situation at , define the n-th jump of X to be the complete Σn set over pV, P,,Xq.Thenwehavea Posner–Robinson Theorem at V`1 for the n-th jump operator.

§6. Degree Determinacy at . Let pD, ďT q denote the Turing degree poset at . A cone is a subset of the form td P D | a ďT du for some a P D. The classical Turing Determinacy (TD) states that every subset of pD, ďT q either contains a cone or is disjoint from a cone. TD follows from the Axiom of Determinacy (AD), and many consequences of AD actually can be proved from TD. For instance, Sami [35] shows that in LpRq, Turing Determinacy implies the Perfect Set Theorem. Let be a cardinal, ďΓ a (definability) degree notion at .Forb Ď ,acone above b is the set of Ppq of the form tx | b ďΓ xu.Whenwesay“ϕpuq holds for a cone of u P Ppq”, it means that the set tu Ď | ϕpuqu contains a cone. A set X Ď Ppq is Γ-degree invariant if x P X and x ”Γ y implies y P X . We often omit Γ when it is clear from the context.

Definition 6.1. Degree Determinacy for Γ-degrees at ,DetpDΓq, is the follow- ing statement: Every Γ-degree invariant subset of Ppq either contains a cone or is disjoint from a cone.

The classical Turing Determinacy axiom is the case that “ and the partial order is given by Turing reduction, i.e. Γ “ Δ1. Despite that the obvious generalization of AD, determinacy for -step games with plays from , is immediately false in ZFC, and the fact that AD and TD are equivalent over LpRq, assuming ZF ` DC (this was shown by Woodin in 80’s), it is not obvious whether Degree Determinacy for the analogous degree notion at is incompatible with I0. Under I0, Woodin has obtained many structural properties over LpV`1q parallel to the situations over LpRq under AD assumption. However, in this section, we show that this analogy can not be extended to degree determinacy. In this section, we show that assuming I0 holds at plus an indestructibility requirement on critpjq, degree determinacy for any reasonable degree notions at is false. As a consequence, I0 does not imply degree determinacy at . Notice that, as a contrast, degree determinacy at follows from Axiom of Determinacy. 6.1. The failure of Degree Determinacy. Here we state the result only for ZFC- degrees, to reduce notational cumbersomeness, but our argument works for degree ď ZFC notions (as in Definition 2.1) as weak as T (see page 40). In the case of -degree, the jump operator is the sharp function. More precisely, we show that:

Theorem 6.2 (ZFC). Assume j is an I0pq-embedding. Let κ “ critpjq and suppose V |ù “the supercompactness of κ is indestructible by κ-directed closed posets”. Then LpPpqq |ù DetpDZFCq. We follow Laver [21], a poset P is κ-directed closed if whenever D Ď P is directed and |D|ăκ, then there is a p P P with p ďP q for all q P D. Notice that LpV`1q contains a wellordering of reals. Our strategy is to show that assuming the hypothesis in Theorem 6.2, if degree determinacy for ZFC-degrees at is true in LpV`1q,thenevery1-Suslin set of reals is determined, which implies 1010 XIANGHUI SHI

that there is no 1-sequence of distinct reals. This will lead us to a contradiction. Now we assume that degree determinacy for ZFC-degrees at holds in LpV`1q. Let rκs 1 denote the set of all ordertype 1 subsets of κ. We identify members of rκs1 with their increasing enumerations. A measure  on rκs1 is coherent if for any pattern P Pr1s 1 ,foranyX Ďrκs 1 , pX q“1 ñ pX |Pq“1, where 1 zæP “def tzpiq|i P Pu for z Prκs and X |P “def txæP | x P X u.Belowisa corollary to the proofs of results in Kechris–Kleinberg–Moschovakis–Woodin [18] (Theorem 2.2) and Woodin [46] (Theorem 2.1). Corollary 6.3 (ZF ` DC). If there is a countably additive coherent measure on rκs 1 , for some cardinal κ ą 1,thenevery1-Suslin set of reals is determined. ` Therefore, our goal is to produce a countably additive coherent measure on r s1 . In his proof of the partition relation 1 Ñp1q from AD (in fact the proof uses TD), Martin uses a lemma of Jensen to transfer the cone measure on R given by the determinacy assumption to a measure on r1s (see Theorem 2.6 [19]). Jensen’s lemma generalizes Sacks’ result that every countable admissible ordinal is the first admissible ordinal relativized to some real.

Lemma 6.4 (Jensen [16]). Suppose A “tαi : i ă u is a set of a-admissible ordinals, a Ă , and otppAq“. Then there exists b ěT a s.t. A “ first many b-admissible ordinals.

Here an ordinal α is x-admissible if Lαrxs is admissible, namely a model of KP. Ana¨ıve attempt is to prove an analogue of Jensen’s lemma for ZFC-degrees at . ` To get a coherent measure on r s1 , it is necessary to get a version of the lemma ` that works for A’s,assubsetsof ,ofordertype1. However, the singularity of presents an obstacle for a direct generalization. cfpqă1 seems to prevent us from getting up to A’s of ordertype 1. Even though we manage to prove analogues of this lemma for A of any countable ordertype, we are unable to give a uniform argument. To overcome this, our strategy is to work in a forcing extension: First get a coherent measure at some ą with uncountable cofinality, and then collapse to . The indestructibility requirement is a price we pay to get this to work. We only get a weak version of Jensen’s lemma. Fortunately, it is enough to derive a coherent ` measure on r s1 . Recall that for x Ă ,wecallanordinalα ą a ZFC-ordinal for x if α is the height of a ZFC-model containing x.Forx Ă ,letZx denote the set of the first 1 many ordinals α ą that are ZFC-ordinals for x.As1 is smaller than and hence less than the least ZFC-ordinal-for-x limit of ZFC-ordinals for x,setsofthe form Zx are all scattered sets. A set X Ď Ord is scattered if α ą suppX X αq for every α P X .

Lemma 6.5. Assume as in the hypothesis of Theorem 6.2. Suppose P Ď 1 is a set of ordertype 1, and u Ă . Then there exists a, b Ă such that u ďZFC a, u ďZFC b, and Zb “ Za æP “def tZa piq|i P Pu.

We will prove this lemma in §6.2. Assuming DetpDZFCq, one can transfer the ` ` cone measure on Ppq to a measure  on r s1 as follows: For any A Ďr s1 ,

pAq“1 ðñ Za P A for a cone of a P Ppq. (˚) AXIOM I0 AND HIGHER DEGREE THEORY 1011

It is easy to see that every countable set of degrees has an upper bound, therefore the measure  defined by (˚) is countably additive. Using Lemma 6.5, we show that  can mimic the partition measure used in showing that 1-Suslin sets are determined, in other words, that  is coherent. Lemma 6.6. Assume as in the hypothesis of Theorem 6.2, and in addition assume ` DetpDZFCq. Then the measure on r s 1 defined by p˚q is coherent, i.e. for any P Ď 1 ` of ordertype 1, and any A Ăr s 1 , pAq“1 ùñ pA|Pq“1.

Proof. Towards a contradiction, suppose P Ď 1 is a “bad” set of ordertype 1 and x Ă is such that A contains the sets associated to the cone above x, i.e. A Ě Cx “def tZa | x ďZFC au,butpA|Pq“0. Then there is a y Ă such that pA|PqXCy “ ∅. By Lemma 6.5, letting u “px, yq,wehave

Da, b pu ďZFC a ^ u ďZFC b ^ Zb “ Za æPq.

Therefore pCpx,yq|PqXCpx,yq ‰ ∅. By the assumption, pCx|PqXCy “ ∅,but Cpx,yq Ă Cx X Cy ,sopCpx,yq|PqXCpx,yq “ ∅. Contradiction! % Now we summarize the argument.

Proof of Theorem 6.2. Assume j is an I0pq-embedding. Then κ “ critpjq is supercompact in V and, by assumption, the supercompactness of κ is indestructible by κ-directed posets in V. By Lemma 6.6, if one assumes DetpDZFCq, then there is ` a countably additive coherent measure on r s1 . By Corollary 6.3, this implies that every 1-Suslin set of reals is determined. Hence, there is no sequence of distinct reals of length 1. But in our situation, V is well-orderable, there is a well ordering of all the reals. Contradiction! % 6.2. Proof of Lemma 6.5. Now we prove Lemma 6.5. But in this subsection we only give an outline of the argument, and the technical lemmas will be proved in the subsequent subsections. Assume toward a contradiction that for some pattern P Ă 1,thereisanu Ă such that there do not exist a, b Ă such that u ďZFC a, b and Zb is Za thinned by the pattern P, i.e. Zb “ Za æP. Let pM,j0, q be the -th iterate of pLpV`1q,jq. κ is supercompact in V, so “ j0,pκq is supercompact in M and remains supercompact in MrGs by indestructibility, where G is M-generic for any -directed poset in M. Let be the least measurable cardinal of M above and for each ă , “ be the -th strongly inaccessible cardinal of M above .Let sup ă .Fixa z Ď which codes an enumeration of M| “def M X V of length .Forx Ă , ą ZFC r s|ùZFC ˚ call an ordinal α a -ordinal for x if Lα z, x .LetZx be the set of first 1 many ZFC-ordinals for x. Let P be the full product of the partial orders P , ă , where each P adds a Cohen generic subset of : conditions in P are functions p with domppq“ and p qPP ă P P ď p qď p q P p for each ,andforp, q , p P q iff p P q for all . This P is clearly -directed closed, so by Laver (see [21]), forcing with P over M preserves the supercompactness of (this is the only instance of indestructibility that we need). In particular, remains to be pă q-supercompact in the extension, which is witnessed by a tower of measures on Pp q, ă . 1012 XIANGHUI SHI

For ą α ą , letting  ,α be two supercompact measures on Pp q and Ppαq, respectively, we say  projects to α , denoted as α “  |α,ifX P α if and only if there is a Z P  such that X “tz X α | z P Zu.LetI “t | ă u.Let P  PpM q and p0 P P be such that p0 forces that  is a tower (indexed by I )of measures x9α : α P I y such that

1. 9α “ 9 |α,forα, P I and α ă . 2. x9α : α P I y witnesses the pă q-supercompactness of .

Let 9 be an pM, Pq-name of their projection(s) to Pp q.SinceP is more than ` ` |Pp q| -closed (as |Pp q| ă minpI q), we may assume that p0 decides the value of 9 . Let a0 Ă be a set in M which codes p0 and z (the code for M| ). As we will show in Lemma 6.7, one can choose two conditions p, q P P below p and two 9 0 P-names a,9 b forsubsetsof such that • p , “a9 “paˇ , G9 q^Z˚ “ Z˚ ”, 0 a9 aˇ0 (saving all the ZFC-ordinals for a0) 9 9 ˚ ˚ • q , “b “paˇ0, Gq^Z9 “ Z æPˇ ”. b aˇ0 (thinning the ZFC-ordinals for a0 according to the prescribed pattern P)

Now choose two pM , Pq-generic filters, Gp and Gq , such that

1. p P Gp and q P Gq, 2. MrGps“MrGqs. Item 2 can be obtained by the homogeneity of P. By Proposition 3.9, P is -good in M,sosuchGp, Gq can be found in V . Interpreting  by Gp and Gq gives two towers of measures that project to the same measure on Pp q,asp0 P Gp X Gq and p0 decides the value of 9 . Next we use a mixed Prikry tower forcing Q defined over the normal measure at (it is the same measure in all generic extensions MrGs with G Ă P such that p0 P G) and the interpretation of  (see §6.4 for the definition of Q). A Q-generic xp q ă y x ă y can be viewed as a countable sequence i ,ui : i ,where i : i is a Prikry sequence for the measure  and xui : i ă y is a diagonal Prikry sequence for the sequence of measures xi : i ă y, where each i is the fine normal measure P p q § Q on i given by the interpretation of .In 6.4, we will show that collapses to and makes cfpq“. Choose Q-generics Hp over M rGps and Hq over M rGqs in the same manner with respect to the interpretations of  via Gp and Gq respectively. By Proposi- tion 6.11, Q is -good in MrGps as well as in MrGqs, and hence by Proposi- 9 tion 3.9-2, P ‹ Q is -good in M.SoHp and Hq can be found in V . Notice that is collapsed to in both MrGpsrHps and MrGqsrHqs, a0 can be viewed as a subset M rGps of in these two models. Let a “ a9 , the interpretation of a9 in MrGps,and 9 M rGq s similarly b “ b . Thus a “pa0,Gpq and b “pa0,Gq q.Let ˚ • a be the subset of given by pa, Hpq, ˚ • b be the subset of given by pb, Hq q.

Since Q collapses to and preserves ZFC-ordinals above ,asifα P Za˚ then Hp ˚ ˚ Q9 Gp r s ˚ “ ˚ “ is -generic over Lα a, Gp ,wehaveZa Za . Similarly, Zb Zb . Therefore Zb˚ is obtained by thinning Za˚ according to the pattern P. AXIOM I0 AND HIGHER DEGREE THEORY 1013

Notice that Hp and Hq project to the same -supercompact Prikry generic 9 Gp 9 Gq on Pp q,asQ and Q use the same -supercompact measure on Pp q in M rGps and MrGqs, respectively. Call this generic H . Similar to Proposition 4.2, ˚ in both MrGpsrHps and MrGqsrHqs, V`1 X MrH s has size .Botha and ˚ b can compute an enumeration of V`1 X MrH s of ordertype , therefore both of them have ZFC-degrees above all members of V`1 X MrH s.Thisisakey point. In MrH s,cfpq“, so by Generic Absoluteness, V`1 X MrH s ă V`1.Let ϕpuq be the formula

Da, b pu ďZFC a ^ u ďZFC b ^ Zb “ Za |Pq.

6 Clearly, ϕpuq is a formula over pV`1, Pq . Note that for every P Ă 1 of ordertype ˚ ˚ 1,asa and b compute every set in V`1 X MrH s,wehaveforeveryu P ˚ ˚ V`1 X MrH s, u ďZFC a ^ u ďZFC b and Zb˚ “ Za˚ |P, and thus V`1 |ù ϕpuq. By elementarity, ϕpuq is also true in V`1 X MrH s.SoV`1 X MrH s|ù@uϕpuq. Using elementarity again, we have V`1 |ù @ uϕpuq. This contradicts our initial assumption. To complete the proof, we have two more technical lemmas to prove. 6.3. Saving or killing ZFC-ordinals.

Lemma 6.7 (ZFC). Let , , , P be as in §6.2 and V “ M . Suppose a Ă , P Ď 1 is a set of ordertype 1 and p P P. Then there is a q P P such that q ď p and , ˚ “ ˚æ “p q ˚ ZFC q Zb Za P,whereb a, G and Zx is the set of first 1 many -ordinals for x above . Proof. | ˚ “t | ă u Fix an enumeration z of M of length .LetZa αi i 1 enumerate the first 1 ZFC-ordinals above in increasing order. For i ă 1,letNi r s Ď denote Lαi z, a . Fix a prescription set P 1. We shall construct an 1 sequence of decreasing conditions xqi : i ă 1y inductively so that any M-generic filter G containing the coordinate-wise infimum of this sequence realizes the equation ˚ “ ˚æ P Zpa,Gq Za P. Keep in mind that the definition of is absolute between Ni and Ni Ni M , thus P “ P X Ni ,fori ă 1;andifD is a dense open subset of P in Ni , ˚ ˚ then there is a dense subset D Ă P in M such that D “ D X Ni —simply take ˚ D “ D YpPzNi q. Assume we have obtained this sequence up to i, xqj : j ă iy. Assume that for ă P , ˚ æ “ ˚æp X q each j i, qj Nj and qj PNj Zpa,Gq j Za P j . This sequence is in Ni , ą ˚ 1 as αi supjăi αj (asŤZa is scattered). Let qi be the coordinate-wise limit of this 1 1 p q“ p q ă PNi sequence, i.e. qi jăi qj for .Thenqi is in . Let G Ă P be a M-generic filter such that qi P G.ThenG gives a -sequence “x ă y ă 1 g¯ g : such that each g is a Cohen subset of , .Soqi forces that

6 To be precise, the first two components are Π1-formulae (“b is in every ZFC-model containing a”) and the third is Δ2 over pV`1, Pq (The key point is that: Let ϕpx, M, Y,αq denote the formula saying that “M is a ZFC-model containing x, Y Ă M has otppY q“α and consists of ZFC-ordinals for x, i.e. N ordinals of the form Ord for some N P M containing x”. Then ϕpx, M, Y,1q is Δ1 with parameter px, M, Y q over pV`1, Pq.The“Zb “ Za æP”, P Ď 1, statement says that minimal pairs pMi ,Yi q, i Pta, bu, with properties ϕpa, Ma ,Ya ,1q, ϕpb, Mb ,Yb ,1q respectively satisfy that Yb “ Ya æP.So it is both Σ2 and Π2 over pV`1, Pq.). 1014 XIANGHUI SHI r sr s ZFC P X ZFC P z Lαj z, a g¯ is a model of for j P i, and not a model of for j i P. ď 1 We need to find a qi PNi qi to take care of i.Therearetwocases. P ď 1 Case 1. i P. This means that we need to find a qi qi that saves the i-th ZFC , ZFC -ordinal for a above , i.e. qi PNi “αi remains to be a -ordinal for b”. To achieve this, it suffices to ensure that G X Ni is Ni -generic, for any M-generic G containing qi . |ù | |“ “ Work in Ni`1. Note that Ni`1 “ Ni ”and sup ă .Thereisa Ď x i ă y -increasing sequence Γ : in Ni`1 such that ă i P ă 1. for eachŤ ,Γ Ni and has size . i “ i P X 2. Γ ă Γ enumerates all the dense open subsets of Ni in Ni . Fix some notations. At any ă , P can be factored as Pă ˆ Pě ,where ś ś ă ě P “ tP | ă u and P “ tP | ď ă u.

Let D Ď P be any dense open set. For ă ,letDă , Dě denote their projections to Pă and Pě respectively, i.e.

Dă “tqær0, q|q P Du and Dě “tqær , q|q P Du. An useful fact is that ě 1 ě ă Fact 6.8. For any r P P ,thereisar ďPě r in D such that for any s P P , 1 1 1 there is an s ďPă s such that s r P D. ă ě ă The key is that |P |ă and P is -closed. Enumerate P as ttk | k ă u, ˚ ˚ 1 “ ď ď ă 1 ă let r0 r. For each k , first pick a rk such that rk P rk for all k k,and ě 1 then pick a rk P D such that for every k ă k,thereisansk1 ďPă tk1 such that   ˚  1 1 ď 1 1 P “ sk rk P tk rk and sk rk D. At the end, let r r .Thisprovesthefact. We build by induction a sequence of conditions xr : ă y in Ni`1 such that

1. r P P X Ni for ă , ď 1 2. their diagonal supremum (see below) qi P qi ,and i 3. for any M-generic G such that qi P G, G X Ni X D ‰ ∅ for every D P Γ . “ 1 Let r´1 qi .Atstage , apply the aforementionedŞ fact within Ni successively to each ě i P tp qNi | P i u ď ě member of Γ to find a condition r D D Γ such that r P r ă ă ă P i PpP qNi “ P for , and such that for every D Γ and for every s ,thereis 1 1 ď ă P i Yt i uĂ | i |ă an s P s such that s r D. Such r exists in Ni ,asΓ Γ Ni , Γ ě and P X Ni is -closed in Ni . So for each ă , we have defined a r P Ni . At the end, set qi to be the diagonal supremum of xr : ă y, which is defined as: 1 p q“ p q ă ď ær qď ě ă qi r for .Soqi P qi and qi , P r for every .Since x i ă yP P Γ : Ni`1,wehaveqi Ni`1. Let G be a M-generic filter over P such that qi P G.Sinceqi ďP qj for j ă i, ˚ æ “ ˚æp X q P i ă ˚ Ă P we have Zpa,Gq i Za P i . Suppose D Γ for some .LetD be ˚ ˚ ˚ dense open and such that D “ D X Ni .SoG X D ‰ ∅.Letq P G X D .Thisq might not be in Ni . Note that M and Ni agree on V . By the property of r ,theset ă  ă Ni ă  ts P P | s r P Du is dense in pP q “ P .soE “tp P P | pær0, q r P Du 1  is dense in P, so we may assume that q P E as well. Let q “ qær0, q r ,then 1 ` ` ` q P D.Sinceqi and q are in G,thereisanq P G such that q ďP q and q ďP qi . ` (Or simply assume that the q we chose is such that q ďP qi and then let q “ q.) AXIOM I0 AND HIGHER DEGREE THEORY 1015

1 ` 1 1 Note that qi ær , qďPě r , by the definition of q , q ďP q , thus q P G. Therefore pG X Ni qXD “ G X D ‰ ∅. R ď 1 Case 2. i P. This means that we need to find a qi P qi that kills the i-th ZFC-ordinal for a above , i.e. qi ,P “αi is not a ZFC-ordinal for b”. To r s achieve this, we code a surjection from αi to in G so that Lαi z, a, G is no longer a model of ZFC. Note that such surjection exists in Ni`1.Letu be a binary sequence of length in Ni`1 that codes this surjection. We break it into blocks, t | ă u “ ∅ “ æ ă ă z , as follows, z0 and z u sup ă for 0 . Define p q“ 1p q ă qi as qi qi z (as concatenation of two binary sequences), for . ą ă ă p q P Notice that sup ă for 0 ,soeachqi is a condition in . Thus qi P P X Ni`1. Now let G Ă P be a M-generic filter such that qi P G.Let¯g be the corresponding generic -sequence xg : ă y of Cohen sets. Since qi ďP qj for j ă i,wehave ˚ æ “ ˚æp X q 1 P x ă y Zpa,Gq i Za P i . Note that qi Ni .Using¯g and : , one can x ă y r s“ r sr s r sr s compute z : and hence u in Ni g¯ Lαi z, a g¯ . Thus Lαi z, a g¯ is no longer a model of ZFC. This takes care of the case i R P. x ă y ď P Now, we have this 1 sequence qi : i 1 of PŤ-decreasing conditions in . Let q be the coordinate-wise limit of q ’s, i.e. qp q“ q p q for ă .Then i iă1 i q P P, q ďP p and q clearly works as required in the lemma. %

6.4. A mixed Prikry tower forcing. In this subsection, we prove some facts about the mixed Prikry tower forcing Q, which is the second step forcing used in the proof of Lemma 6.5 (see page 45). Let  be a normal measure on .Letx : ă y be the sequence such that each is the -th strongly inaccessible cardinal above ,andforeach ă ,let be a fine normal -supercompact measure on Pp q. Let X “tp ,uq| ă ^ u P Pp qu.Letď be the following prewellordering on X:forp ,uq, p  ,vqPX, p ,uq ď p  ,vqô ď  .Letă be the strict part of ď.LetrXsă denote the set of linearly ă-ordered finite subsets of X. ˚ With  and  , ă , one can define a product measure  on subsets of X Ťas follows: A subset X Ď X is called a measure-one box if it is the form X “ pt uˆ q P P P ˚ PA X ,whereA  and X  for each A.  is the filter generated by measure-one boxes, i.e. for X Ď X, X P ˚ iff X contains a measure one box. Clearly ˚ is an ultrafilter on X. Since the measures involved in the definition of ˚ are all at least -complete, ˚ is -complete. Our forcing poset Q is a tree Prikry forcing associated to ˚ (see §1.2 of [12] for the definition). Briefly speaking, conditions in Q are pairs of the form pt, T q,where T is a subtree of prXsă, q with trunk t and such that every node in T -above t has measure-one-box many immediate successors. Here s0  s1 if s0 is an initial segment of s1 in the increasing ă-order. Two conditions pt0,T0qďQ pt1,T1q iff T0 ˚ is a subtree of T1 with trunk t1 such that t0  t1.Andwritept0,T0qďQ pt1,T1q if in addition t0 “ t1.ThisQ is an instance of Q discussed in §3.4.2. A generic filter G Ă Q is often viewed as a subset of X of ă-ordertype , xp q ă y i ,ui : i , which is the union of the trunks of the trees in G. Though not relevant to this paper, it is worth pointing out that such x i : i ă y is a generic sequence for the standard Prikry forcing P,andxui : i ă y is generic for the P “x ă y diagonal supercompact Prikry forcing ¯ ,where¯  i : i . 1016 XIANGHUI SHI

Below are some standard facts about this forcing. 1. Q is `-cc, therefore it preserves cardinals ě `. ˚ ˚ 2. Since  is -complete, pQ, ďQq is -closed. As Q satisfies Prikry condition (see Lemma 3.16, page 28), Q adds no bounded subsets of , therefore it preserves cardinals ď . As for cardinals in p, s,wehave Lemma . Q “ p q“ 6.9 collapses sup ă to and makes cf .

Proof. Let G Ă Q be a generic filter and WG be the corresponding -sequence. Identify W with its range. We claim that G Ť Claim 6.10. “ tu |pα, uqPWG ,forsomeαu. This follows from genericity. Suppose P .Let Ť D “tpt, T qPQ | P tu |pα, uqPt,forsomeαuu.

We show that D is dense in Q.Letp “pt, T q be an arbitrary condition in Q. 1 “p 1qď We prove somethingŤ stronger. We show that there is a T such that q t, T p , ˇ P t |p qP 9 u and q u α, u ŤWG ,forsomeα . We may assume that R tu |pα, uqPt,forsomeαu.Theset X “tpα, uqPX | t Ytpα, uqu P T ^ α ą u Ť pt uˆ q P P is a measure-one box. Write X as PA X ,whereA  and each X  , P “t P | P uP AŤ.Aseach is a fine measure, the set X u X u  .Let 1 “ pt uˆ q 1 1 P 1 X PA X .LetT be the part of T restricted to X ,namely,s T iff 1 1 for some x P X , s  t Ytxu or t Ytxu  s.Clearly,q “pt, T qPQ, q ďQ p and it worksasdesired. For every pα, uqPWG , u has cardinality ă , therefore in the generic extension, cfp q“ and | |“, and hence cfpq“. % Furthermore, as a corollary of Proposition 3.17, the -completeness of ˚ gives us that Proposition 6.11. Q is -good.

6.5. I0 and the indestructibility condition. In this subsection, we show that the indestructibility assumption in Theorem 6.2 is consistent relative to ZFC ` I0.The main result of this subsection is a corollary of a theorem of Dimonte–Friedman. Theorem 6.12 (see Theorem 3.5 and Corollary 3.8 of [5]). Suppose j is an I0pq-embedding and P is a directed closed, -bounded, j-coherent, backward Easton iterated forcing. Then for any P-generic filter G, j lifts to an elementary embedding j¯ : LpV`1qrGsÑLpV`1qrGs.IfP in addition is above then LpV`1qrGs“ LpV rGs`1q, therefore in this case j¯ witnesses I0 in V rGs. Below are relevant definitions.

Definition 6.13. Let P be a forcing iteration of length ,where is either a strong limit ordinal or equals to Ord. For each ă ,letP denote the -th iterate and Q denote the -th iterand. AXIOM I0 AND HIGHER DEGREE THEORY 1017

• P is directed closed if every Q , ă ,isă -directed closed, i.e. for any D Ă Q of size ă such that for any two d0,d1 P D there is an e ď d0 ^ e ď d1,then there is a p P Q such that p ď d for all d P D. • P is -bounded if every Q , ă ,hassizeă. • Suppose j is an elementary embedding such that j“ Ď and P Ď dompjq. We say P is j-coherent if for any ă , jpP q“Pjp q. • P is above if P`1 is a trivial poset. Now we state the main result of this subsection.

Proposition 6.14 (ZFC). Assume j is an I0pq-embedding, then there is a generic extension V rGs in which V rGs |ù “the supercompactness of κ is indestructible by κ-directed closed posets” and j lifts to an I0pq-embedding in V rGs. Proof. We show that the conclusion holds in extensions obtained by a modified version of Laver’s preparation posets for making the supercompactness of one supercompact κ indestructible by κ-directed posets (see [21]). Letκ ¯ “xκn : n ă y be the critical sequence associated to j. Thus κ “ κ0.Notice ă ă Ñ that I0 implies that κn is -supercompact, for every n .Letf0 : κ0 Vκ0 be the Laver function for κ0.Forn ă ,letfn`1 “ jpfnq. By elementarity, each fn æ “ is the Laver function for κŤn. Moreover, it is not difficult to see that fn`1 κn fn ă “ for every n .Letf năfn. We modify Laver’s preparation forcing for κ-directed posets (see [21]) as follows: 9 Let P be the iteration of xPα, Qα : α ă y with backward Easton support, where

1. at limit stage ă , P is the set of those sequences in the inverse limit of xP : ă y whose supports are Easton set of ordinals; 9 2. at the successor stage, for every α ă ,letQα “ α when α ą is strongly inaccessible and fnpαq“xα ,αy with α being a Pα-name for a α-directed Pα poset in the V of V and α’s being ordinals such that ă α Ñ ă α,and 9 let Qα “ ∅ otherwise. The technical detail for the choice of α is irrelevant here, we refer the readers to [21] for further details regarding the choice of α’s and the rest of the proof. As a corollary of Laver’s proof (see also the last paragraph of [21]), this poset makes the ă-supercompactnesses of κ0 indestructible by κ0-directed closed forcing posets in V. It is easy to see that P is directed closed, -bounded, j-coherent, and above .LetG be a P-generic filter. By Dimonte–Friedman’s theorem, j lifts to an I0pq-embedding in V rGs. %

This shows that it is consistent relative to I0 that the ă-supercompactness of κ is indestructible by κ-directed closed posets in V. Hamkins–Shelah [13] shows that after small forcing (forcing of size ăκ), any ăκ-closed forcing will destroy the super- compactness of κ. Therefore it is consistent relative to I0 that the supercompactness of critpjq is destructible by posets in V. In short,

Corollary 6.15 (ZFC). The statement that V |ù “the supercompactness of κ is indestructible by κ-directed closed posets” is independent of I0pq.

6.6. A conjecture. We just showed that DetpDZFCq does not follow from ZFC ` I0pq. As we remarked at the beginning of §6, our argument works for any reasonable 1018 XIANGHUI SHI

degree notions, including Zermelo degree discussed in §2. The analyses in §2 seem to suggest that degree determinacy for Zermelo degrees at countable cofinality strong limit cardinals are always false in canonical inner models. But unlike the proof of Theorem 6.2 which exploits the richness of the degree structure at , our analyses (in §2) of degree structures in fine structure models give the failure of degree determinacy via the simplicity of the degree structure. Together with evidences to be discussed later, we propose the following conjecture. Conjecture 6.16 (ZFC). Suppose is an uncountable cardinal. Then

LpPpqq |ù DetpDZq. We have already analyzed structures of Zermelo degrees at countable cofinality strong limit singular cardinals in §2. The case that is a strong limit singular cardinal of uncountable cofinality follows from the following result of Shelah [38]. Theorem 6.17 (Shelah [38]). Assume ZFC. Then for every strong limit singular cardinal of uncountable cofinality, LpPpqq |ù Axiom of Choice. Using the choice, one can select in LpPpqq two disjoint unbounded sets of degrees, which witness the failure of degree determinacy for Zermelo degrees at . As for regular cardinals, we first look at the case that is regular and satisfies the weak power condition, i.e. 2ă “ . Jensen’s lemma on page 43 can be generalized to such —Jensen’s proof in the context of (see [16]) can be literally adapted for such . More precisely, we have the following lemma. Lemma 6.18 (Jensen). Assume ZF ` `-DC. Suppose ą is a regular cardinal and 2ă “ . Suppose a Ă and A Ăp, `q is a scattered set such that every α P A is a Zermelo ordinal for a. Suppose otppAqď. Suppose B Ď A and otppBq“otppAq. Then there is a b Ă such that b ěZ a and B is the set of the α-th Zermelo ordinal for b, α ă otppAq.

If DetpDZq were true in LpPpqq, then applying this lemma to A Ă that is scattered and otppAq“1, one would get a countably additive coherent measure on ` r s1 . As we argued before §6.2, this implies the determinacy for sets of reals that are 1-Suslin, hence there is no sequence of distinct reals of length 1, contradicting to the assumption that V is wellordered. So in ZFC models, DetpDZq fails in ´ LpPpqq for regular cardinal such that 2ă “ .LetZFC  be a fragment of ZFC sufficient for proving this. Now consider the case that is regular but 2ă ą . If degree determinacy for Zermelo degrees at were true in LpPpqq, then there would be a (in fact a cone ` of) u Ă and an u ă such that r s|ùZFC´ ` pPp qq |ù pD q L u u “L Det Z ”. “ Such u and u can be obtained as follows: start with N0 L 0 ,whereŤ 0 is least Ă |ù ZFC´ “ such that L 0 and L 0 ; for limit ordinal α,letNα ăαN ;and “ r s p q Ă ą let Nα`1 L α`1 uα ,where uα, α`1 is such that uα , α`1 α and is least such that ´ Nα`1 |ù ZFC ` “DetpDZq holds for degree invariant sets in Nα”. |ù ZFC´ ă ` Let be the least limit ordinal such that N .As , the sequence xuα : α ă y can be coded by a single u Ă .Let u “ limαă α. By induction, ` ` α ă for every α ă , thus u ă . AXIOM I0 AND HIGHER DEGREE THEORY 1019 r s ă “ Ă ă However L u u “thinks” 2 . This is because if x , then there are α, ă such that x P Lαru X s; then it follows that |Pp q| ď .There- r s|ù ă “ r s fore L u u “2 ”. Clearly is regular in L α u . But then according to r s|ù pPp qq |ù pD q the previous discussion, it must be that L u u “L Det Z ”. Contradiction! This concludes the case that is regular. So at least, we know for sure that Theorem 6.19. If ą is a regular cardinal or a strong limit singular cardinal of uncountable cofinality, then LpPpqq |ù DetpDZq. At this point, very little is known about singular cardinals that are not strong limit.

§7. Acknowledgments. The author has discussed contents in this paper exten- sively with W.Hugh Woodin, and would like to thank him for his insightful advices on this project. The author would like to thank Gitik, Magidor, Mitchell, Schindler, and Wu for many inspiring discussions. The Posner–Robinson result was first submitted in 2010, but was retrieved later due to new developments. A new proof is included in this submission, together with several new discover- ies in the investigation of the connection between large cardinals and generalized degree structures. The author is grateful to the referees of both submissions for their patience and the long lists of errors, comments, and suggestions, which lead to the significant improvement over the early drafts. During this project, the author visited National University of Singapore, York University, and Fields Institute, he wishes to express his gratitude to C.T.Chong and I. Farah for their support and hospitality.

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