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Tall Cardinals in Extender Models TALL CARDINALS IN EXTENDER MODELS GABRIEL FERNANDES* AND RALF SCHINDLERz Abstract. Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of 휆-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal 휅 is a tall cardinal if and only if it is either a strong cardinal or a measurable limit of strong cardinals. 1. Introduction Tall cardinals appeared in varying contexts as hypotheses in the work of Woodin and Gitik but they were only named as a distinct type of large cardinal by Hamkins in [Ham09]. Definition 1.1. Let 훼 be an ordinal and 휅 a cardinal. We say that 휅 is 훼-tall iff there is an elementary embedding j : V ! M such that the following holds: a) crit(j) = 휅, b) j(휅) > 훼, c) 휅M ⊆ M. We say that 휅 is a tall cardinal iff 휅 is 훼-tall for every ordinal 훼. One can compare this notion with that of strong cardinals. Definition 1.2. Let 훼 be an ordinal and 휅 a cardinal. We say that 휅 is 훼-strong iff there is an elementary embedding j : V ! M such that the following holds: a) crit(j) = 휅, b) j(휅) > 훼, c) V훼 ⊆ M. We say that 휅 is a strong cardinal iff 휅 is 훼-strong for every ordinal 훼. In this paper, working under the hypothesis that there is no inner model with a Woodin cardinal, we present a characterization of 휆-tall cardinals in `extender models' (see Definition 2.5) that are `self-iterable' (see Definition 3.10). Given a cardinal 휅, if 휅 is 훼-strong then 휅 is 훼-tall, and the existence of a strong cardinal is equiconsistent with the existence of a tall cardinal (see [Ham09]). We will prove that the following equivalence holds in extender models: Corollary A. Suppose that there is no inner model with a Woodin cardinal, V is an extender model of the form L[E] which is iterable. Then given a cardinal 휅 the following equivalence holds: 휅 is a tall cardinal iff 휅 is either a strong cardinal or a measurable limit of strong cardinals. 2010 Mathematics Subject Classification. Primary 03E55. Secondary 03E45. Key words and phrases. Tall cardinals, Strong cardinals, Extender models, Core model. *The author is funded by the European Research Council (grant agreement ERC-2018-StG 802756) as a postdoctoral fellow at Bar-Ilan University. zThe author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foun- dation) under Germany's Excellence Strategy EXC 2044 390685587, Mathematics M¨unster:Dy- namics - Geometry - Structure. 1 2 GABRIEL FERNANDES* AND RALF SCHINDLERz Remark 1.3. In contrast to Corollary A, Hamkins in [Ham09, Theorem 4.1] adapted Magidor's results in [Mag76] to prove that it is consistent (assuming the consistency of ZFC plus a strong cardinal) that there is a model of ZFC where there exists a cardinal 휅 such that 휅 is a tall cardinal which is neither a strong cardinal nor a limit of strong cardinals. Hence the equivalence from Corollary A does not hold in such model. An extender1 is a means of encoding elementary embeddings of models of (frag- ments of) ZFC in a set-size object. There are various ways to represent extenders. Extenders are a generalization of measures, in particular, notions such as a `critical point' which are used in the context of measures can also be used when talking about extenders. Extender models are a generalization of G¨odel'sconstructible universe that can accommodate large cardinals. In general, given a predicate E, which can be a set or a proper class, L[E] is the smallest inner model2 closed under the operation x 7! E \ x. Inner models of the form L[E] can be stratified using the J-hierarchy: E ∙ J; = ;, E E E ∙ J훼+1 := rudE(fJ훼 g [ J훼 ), E S E ∙ J훾 := 휉<훾 J휉 for 훾 a limit ordinal, S E ∙ L[E] = 휉2OR J휉 3 where rudE is the closure under rudimentary functions and the function x 7! E \ x. We are interested in the special case where E is such that E : OR ! V and for every ordinal 훼, either E훼 = ; or E훼 is a partial extender (see Definition 2.5). That is, E is a `sequence of extenders'. Convention. There are different ways of organizing sequences of extenders, we will use Jensen's 휆-indexing (see Remark 2.3). Definition 1.4. Suppose V is an extender model of the form L[E]. Given a cardinal 4 휅 we define o(휅) := otp(f훽 j crit(E훽) = 휅g) and O(휅) := supf훽 j crit(E훽) = 휅g. Our main result, Theorem A, is a level-by-level version of Corollary A. The statement of Theorem A uses the notion of 휇-stable premouse which is introduced in Definition 3.11. Theorem A. Suppose that there is no inner model with a Woodin cardinal and that the universe V is an iterable extender model L[E]. Let 휅 < 휇 be regular cardinals. Suppose further that L[E]j휇 is 휇-stable above 휅. Then 휅 is 휇-tall iff o(휅) > 휇 or (︁ )︁ o(휅) > 휅+ ^ supf휈 < 휅 j o(휈) > 휇g = 휅 We prove Theorem A in Section 4. The rest of this introduction gives a technical overview of our proof of Theorem A. 1For an introduction to the theory of extender we recommend [Kan09]. 2An inner model is a transitive proper class that models ZF . 3 See [SZ10] for the definition of Erud . 4Note that our definitions of O(휅) and o(휅) are not standard, because we do not only consider extenders which are total, but also consider partial extenders. For this reason, our definitions differ from those in other references such as[Zem02] and [GM96]. TALL CARDINALS IN EXTENDER MODELS 3 In order to prove Theorem A we will need some results from core model theory. Specifically, we shall need the core model K below a Woodin cardinal. This model is an extender model5 that generalizes the covering, absoluteness, and definability properties of L. The following result due to Jensen and Steel guarantees that such a model exists. Theorem 1.5. ([JS13]) There are Σ2 formulae K(v) and Σ(v) such that, if there is no inner model with a Woodin cardinal, then (i) K = fv j K(v)g is an inner model satisfying ZFC, V V [g] V V [g] (ii) K = K , and Σ = Σ \ V , whenever g is V -generic over a poset of set size, (iii) For every singular strong limit cardinal 휅, 휅+ = (휅+)K, (iv) fv j Σ(v)g is an iteration strategy for K for set-sized iteration trees, and moreover the unique such strategy, (v) Kj!1 is Σ1 definable over J!1 (R). We now describe our strategy for proving Theorem A. One direction is due to Hamkins (see Theorem 4.26 and Theorem 4.27), so we start from the assumption that 휅 and 휇 are cardinals such that 휇 > 휅, 휅 is 휇-tall and j witnesses that 휅 is 휇-tall. Note that this implies that 휅 is measurable and that 휇 < j(휅). We now consider two cases. Either 휅 is a limit of cardinals 훽 such that o(훽) > 휇, in which we case we get the second alternative of the direction we are proving using Lemma 4.23. So, suppose that 휅 is not a limit of cardinals 훽 such that o(훽) > 휇, and then we work towards proving that o(휅) > 휇. As a first step, we combine Lemma 4.12 and Theorem 4.15 to obtain that j is an T iteration map coming from an iteration tree T on L[E] such that j = 휋0;1. This is Lemma 4.17. We will spend the main part of the proof of Theorem A analyzing the iteration tree T . We shall prove that o(휅) > 휇 by contradiction. That is, we shall start with the assumption that o(휅) ≤ 휇. Then, we will find Θ and 훽* such that 훽* < 휅 ≤ Θ ≤ 휇 and 휏 2 (훽*; Θ] implies o(휏) < Θ (see Lemma 3.24). Using the upper bounds that T we obtain in Section3 we shall finally prove that j(휅) = 휋0;1(휅) ≤ Θ ≤ 휇, which T will contradict the fact that 휇 < j(휅) = 휋0;1(휅). The following is essentially a reformulation of Theorem A which we also prove in Section 4. Theorem B. Suppose there is no inner model with a Woodin cardinal and L[E] is an extender model that is self-iterable. Let 휅, 휇 be ordinals such that 휅 < 휇 and 휇 is a regular cardinal. If L[E]j휇 is 휇-stable above 휅, then (휅 is 휇-tall)L[E] iff (o(휅)) > 휇)L[E] (1) or (o(휅) > 0 ^ 휅 = supf휈 < 휅 j o(휈) > 휇g)L[E] It follows that if L[E] is weakly iterable and L[E]j휇 is 휇-stable above 휅, then (휅 is 휇-tall)L[E] iff (1) holds. 2. Preliminaries In this section we summarize the notation that will be used in this paper. We follow closely the notation used in [Zem02]. 5See [SW16] for an example where K is not an extender model. 4 GABRIEL FERNANDES* AND RALF SCHINDLERz E Definition 2.1. We say that M := hJ훼 ; 2;E 훼; E훼i is an acceptable J-structure iff M is a transitive amenable structure and for every 휉 < 훼 and 휏 < 훼! if (P(휏) \ E E E E J휉+1) n J휉 6= ;, then there is f : 휏 ! J휉 surjective such that f 2 J휉+1.
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