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Even Ordinals and the Kunen Inconsistency∗
Even ordinals and the Kunen inconsistency∗ Gabriel Goldberg Evans Hall University Drive Berkeley, CA 94720 July 23, 2021 Abstract This paper contributes to the theory of large cardinals beyond the Kunen inconsistency, or choiceless large cardinal axioms, in the context where the Axiom of Choice is not assumed. The first part of the paper investigates a periodicity phenomenon: assuming choiceless large cardinal axioms, the properties of the cumulative hierarchy turn out to alternate between even and odd ranks. The second part of the paper explores the structure of ultrafilters under choiceless large cardinal axioms, exploiting the fact that these axioms imply a weak form of the author's Ultrapower Axiom [1]. The third and final part of the paper examines the consistency strength of choiceless large cardinals, including a proof that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. embedding j : Vλ+3 ! Vλ+3 implies that every subset of Vλ+1 has a sharp. We show that the existence of an elementary embedding from Vλ+2 to Vλ+2 is equiconsistent with the existence of an elementary embedding from L(Vλ+2) to L(Vλ+2) with critical point below λ. We show that assuming DC, the existence of an elementary embedding j : Vλ+3 ! Vλ+3 implies the consistency of ZFC + I0. By a recent result of Schlutzenberg [2], an elementary embedding from Vλ+2 to Vλ+2 does not suffice. 1 Introduction Assuming the Axiom of Choice, the large cardinal hierarchy comes to an abrupt halt in the vicinity of an !-huge cardinal. -
Tall, Strong, and Strongly Compact Cardinals ∗†
Tall, Strong, and Strongly Compact Cardinals ∗y Arthur W. Apter z Department of Mathematics Baruch College of CUNY New York, New York 10010 USA and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, New York 10016 USA http://faculty.baruch.cuny.edu/aapter [email protected] August 12, 2017 Abstract We construct three models in which there are different relationships among the classes of strongly compact, strong, and non-strong tall cardinals. In the first two of these models, the strongly compact and strong cardinals coincide precisely, and every strongly compact/strong cardinal is a limit of non-strong tall cardinals. In the remaining model, the strongly compact cardinals are precisely characterized as the measurable limits of strong cardinals, and every strongly compact cardinal is a limit of non-strong tall cardinals. These results extend and generalize those of of [3] and [1]. 1 Introduction and Preliminaries We begin with some definitions. Suppose κ is a cardinal and λ ≥ κ is an arbitrary ordinal. κ is λ tall if there is an elementary embedding j : V ! M with critical point κ such that j(κ) > λ and M κ ⊆ M. κ is tall if κ is λ tall for every ordinal λ. Hamkins made a systematic study of tall cardinals in [10]. In particular, among many other results, he showed that every cardinal which is ∗2010 Mathematics Subject Classifications: 03E35, 03E55. yKeywords: Supercompact cardinal, strongly compact cardinal, strong cardinal, tall cardinal, non-reflecting stationary set of ordinals, indestructibility. zThe author wishes to thank Stamatis Dimopoulos for helpful email correspondence on the subject matter of this paper. -
I0 and Rank-Into-Rank Axioms
I0 and rank-into-rank axioms Vincenzo Dimonte∗ July 11, 2017 Abstract Just a survey on I0. Keywords: Large cardinals; Axiom I0; rank-into-rank axioms; elemen- tary embeddings; relative constructibility; embedding lifting; singular car- dinals combinatorics. 2010 Mathematics Subject Classifications: 03E55 (03E05 03E35 03E45) 1 Informal Introduction to the Introduction Ok, that’s it. People who know me also know that it is years that I am ranting about a book about I0, how it is important to write it and publish it, etc. I realized that this particular moment is still right for me to write it, for two reasons: time is scarce, and probably there are still not enough results (or anyway not enough different lines of research). This has the potential of being a very nasty vicious circle: there are not enough results about I0 to write a book, but if nobody writes a book the diffusion of I0 is too limited, and so the number of results does not increase much. To avoid this deadlock, I decided to divulge a first draft of such a hypothetical book, hoping to start a “conversation” with that. It is literally a first draft, so it is full of errors and in a liquid state. For example, I still haven’t decided whether to call it I0 or I0, both notations are used in literature. I feel like it still lacks a vision of the future, a map on where the research could and should going about I0. Many proofs are old but never published, and therefore reconstructed by me, so maybe they are wrong. -
An Introduction to Itreated Ultrapowers John Steel
An Introduction to Itreated Ultrapowers John Steel Contents Introduction iii Lecture 1. Measures and Embeddings 1 Lecture 2. Iterated Ultrapowers 5 Lecture 3. Canonical Inner Models and Comparison 11 Lecture 4. Extenders 17 Lecture 5. Linear Iteration via Extenders 23 Lecture 6. Iteration Trees of Length ! 25 Lecture 7. Iteration Trees of Transfinite Length 33 Bibliography 37 i Introduction In these notes, we develop the basic theory of iterated ultrapowers of models of set theory. The notes are intended for a student who has taken one or two semesters of graduate-level set theory, but may have little or no prior exposure to ultrapowers and iteration. We shall develop the pure theory, which centers on the question of well-foundedness for the models produced in various iteration processes. In addition, we consider two sorts of application: (1) Large cardinal hypotheses yield regularity properties for definable sets of reals. Large cardinal hypotheses yield that logical simple sentences are absolute between V and its generic extensions. (2) Large cardinal hypotheses admit canonical inner models having wellorders of R which are simply definable Roughly, applications of type (1) involves using the large cardinal hypotheses to construct complicated iterations. Applications of type (2) involves bounding the complexity of the iterations one can produce under a given large cardinal hypothesis. The notes are organized as follows. In lecture 1, we develop the basic theory of ultrapower Ult(M; U), where M is a transitive model of ZFC and U is an ultrafilter over M. In lecture 2, we develop the pure theory of iterations of such ultrapowers, and present some applications of type (1). -
Are Large Cardinal Axioms Restrictive?
Are Large Cardinal Axioms Restrictive? Neil Barton∗ 24 June 2020y Abstract The independence phenomenon in set theory, while perva- sive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality prin- ciples depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardi- nals are still important axioms of set theory and can play many of their usual foundational roles. Introduction Large cardinal axioms are widely viewed as some of the best candi- dates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical con- sequences for questions independent from ZFC (such as consistency statements and Projective Determinacy1), and appear natural to the ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@uni- konstanz.de. yI would like to thank David Aspero,´ David Fernandez-Bret´ on,´ Monroe Eskew, Sy-David Friedman, Victoria Gitman, Luca Incurvati, Michael Potter, Chris Scam- bler, Giorgio Venturi, Matteo Viale, Kameryn Williams and audiences in Cambridge, New York, Konstanz, and Sao˜ Paulo for helpful discussion. Two anonymous ref- erees also provided helpful comments, and I am grateful for their input. I am also very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foundations of Mathematics. -
Stably Measurable Cardinals
Stably measurable cardinals P.D. Welch School of Mathematics, University of Bristol, Bristol, BS ÕTW, England Dec. óß’th óþÕ Abstract We dene a weak iterability notion that is sucient for a number of arguments concerning Σ-denability at uncountable regular cardinals. In particular we give its exact consistency strength rstly in terms of the second uniform indiscernible for bounded subsets of : u( ), and secondly to give the consistency strength of a property of Lucke’s.¨ eorem: e following are equiconsistent: (i) ere exists which is stably measurable ; (ii) for some cardinal , u( ) = ( ); (iii) e Σ-club property holds at a cardinal . M H + H Here ( ) is the height of the smallest ăΣ ( ) containing + and all of ( ). Let R R Φ( ) be the assertion: @X Ď @r P [X is Σ( , r)-denable ←→ X P Σ(r)]. eorem: Assume is stably measurable. en Φ(). And a form of converse: eorem: Suppose there is no sharp for an inner model with a strong cardinal. en in the core model K we have: “DΦ()” is (set)-generically absolute ←→ ere are arbitrarily large stably measurable cardinals. When u( ) < ( ) we give some results on inner model reection. Õ Õ Introduction ere are a number of properties in the literature that fall in the region of being weaker than measur- ability, but stronger than j, and thus inconsistent with the universe being that of the constructible sets. Actual cardinals of this nature have been well known and are usually of ancient pedigree: Ramsey cardinals, Rowbottom cardinals, Erdos˝ cardinals, and the like (cf. for example, [ß]). -
Arxiv:2101.07455V2 [Math.LO] 13 Feb 2021 Ilas Ics Ti Hssection
HOW STRONG IS A REINHARDT SET OVER EXTENSIONS OF CZF? HANUL JEON Abstract. We investigate the lower bound of the consistency strength of CZF with Full Separation Sep and a Reinhardt set, a constructive analogue of Reinhardt cardinals. We show that CZF + Sep with a Reinhardt set interprets ZF− with a cofinal elementary embedding j : V ≺ V . We also see that CZF + Sep with a − Reinhardt set interprets ZF with a model of ZF + WA0, the Wholeness axiom for bounded formulas. 1. Introduction Large cardinals are one of the important topics of set theory: the linear hierarchy of large cardinals provides a scale to fathom the consistency strength of a given set-theoretic statement. Reinhardt introduced a quite strong notion of large cardinal, now known as Reinhardt cardinal. Unfortunately, Kunen [15] showed that Reinhardt cardinals do not exist in ZFC. However, Kunen’s proof relied on the Axiom of Choice, so it remains the hope that Reinhardt cardinals are consistent if we dispose of the Axiom of Choice. Recently, there are attempts to study Reinhardt cardinals over ZF and find intrinsic evidence for choiceless large cardinals, cardinals that are incompatible with the Axiom of Choice. We may ask choiceless large cardinals are actually consistent with subtheories of ZFC, and there are some positive answers for this question. For example, Schultzenberg [21] showed that ZF with an elementary embedding j : Vλ+2 ≺ Vλ+2 is consistent if ZFC with I0 is. Matthews [17] proved that Reinhardt cardinal is − compatible with ZFC if we assume the consistency of ZFC with I1. We may also ask the lower bound of the consistency strength of large cardinals over subtheories of ZFC, which is quite non-trivial if we remove axioms other than choice. -
A Premouse Inheriting Strong Cardinals from $ V$
A premouse inheriting strong cardinals from V Farmer Schlutzenberg1 Abstract We identify a premouse inner model L[E], such that for any coarsely iterable background universe R modelling ZFC, L[E]R is a proper class premouse of R inheriting all strong and Woodin cardinals from R. Moreover, for each α ∈ OR, L[E]R|α is (ω, α)-iterable, via iteration trees which lift to coarse iteration trees on R. We prove that (k + 1)-condensation follows from (k + 1)-solidity together with (k,ω1 + 1)-iterability (that is, roughly, iterability with respect to normal trees). We also prove that a slight weakening of (k + 1)-condensation follows from (k,ω1 + 1)-iterability (without the (k + 1)-solidity hypothesis). The results depend on the theory of generalizations of bicephali, which we also develop. Keywords: bicephalus, condensation, normal iterability, inner model, strong cardinal 2010 MSC: 03E45, 03E55 1. Introduction M Consider fully iterable, sound premice M,N with M|ρ = N|ρ and ρω = ρ = N ρω . Under what circumstances can we deduce that either M E N or N E M? This conclusion follows if ρ is a cutpoint of both models. By [2, Lemma 3.1],1 the conclusion also follows if ρ is a regular uncountable cardinal in V and there is no premouse with a superstrong extender. We will show that if M||ρ+M = N||ρ+N and M,N have a certain joint iterability property, then M = N. The joint iterability property and the proof that M = N, is motivated by arXiv:1506.04116v3 [math.LO] 27 Apr 2020 the bicephalus argument of [3, §9]. -
Arxiv:1608.00592V3 [Math.LO] 3 Mar 2018 Ria Eso.Mlie Ute Rvdi 1]Ta H Olcino S of Furthe Collection Space
THE HALPERN-LAUCHLI¨ THEOREM AT A MEASURABLE CARDINAL NATASHA DOBRINEN AND DAN HATHAWAY Abstract. Several variants of the Halpern-L¨auchli Theorem for trees of un- countable height are investigated. For κ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite d ≥ 2, we prove the consistency of the Halpern-L¨auchli Theorem on d many normal κ-trees at a measurable cardinal κ, given the consistency of a κ + d-strong cardinal. This follows from a more general consistency result at measurable κ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD. 1. Introduction Halpern and L¨auchli proved their celebrated theorem regarding Ramsey theory on products of trees in [12] as a necessary step for their construction in [13] of a model of ZF in which the Boolean Prime Ideal Theorem holds but the Axiom of Choice fails. Since then, many variations have been established and applied. One of the earliest of these is due to Milliken, who extended the Halpern-L¨auchli Theorem to colorings of finite products of strong trees in [16]. As the two versions are equivalent, this version is often used synonymously with Halpern and L¨auchli’s orginal version. Milliken further proved in [17] that the collection of strong trees forms, in modern terminology, a topological Ramsey space. Further variations and applications include ω many perfect trees in [15]; partitions of products in [2]; the density version in [5]; the dual version in [21]; canonical equivalence relations on finite strong trees in [22]; and applications to colorings of subsets of the rationals in [1] and [23], and to finite Ramsey degrees of the Rado graph in [18] which in turn was applied to show the Rado graph has the rainbow Ramsey property in [4], to name just a few. -
A Class of Strong Diamond Principles
A class of strong diamond principles Joel David Hamkins Georgia State University & The City University of New York∗ Abstract. In the context of large cardinals, the classical diamond principle 3κ is easily strengthened in natural ways. When κ is a measurable cardinal, for example, one might ask that a 3κ sequence anticipate every subset of κ not merely on a stationary set, but on a set of normal measure one. This . is equivalent to the existence of a function ` . κ → Vκ such that for any A ∈ H(κ+) there is an embedding j : V → M having critical point κ with j(`)(κ) = A. This and the similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ram- sey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardi- nals for these weaker large cardinal notions. The resulting Laver diamond ? principles \ κ can hold or fail in a variety of interesting ways. The classical diamond principle 3κ, for an infinite cardinal κ, is a gem of modern infinite combinatorics; its reflections have illuminated the path of in- numerable combinatorial constructions and unify an infinite amount of com- binatorial information into a single, transparent statement. When κ exhibits any of a variety of large cardinal properties, however, the principle can be easily strengthened in natural ways, and in this article I introduce and survey the resulting class of strengthened principles, which I call the Laver diamond principles. MSC Subject Codes: 03E55, 03E35, 03E05. Keywords: Large cardinals, Laver dia- mond, Laver function, fast function, diamond sequence. -
Generic Large Cardinals and Systems of Filters
Generic Large Cardinals and Systems of Filters Giorgio Audrito, Silvia Steila Abstract We introduce the notion of C-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties. Contents 1 Introduction 2 1.1 Notation.................................. 3 1.2 Generalized stationarity . 4 1.3 Standardextendersandtowers . 5 2 Systems of filters 6 2.1 Standard extenders and towers as C-systemsoffilters. 9 2.2 Systems of filters in V and generic systems of ultrafilters . 11 2.3 Embedding derived from a system of ultrafilters . 12 2.4 System of ultrafilters derived from an embedding . 16 3 Generic large cardinals 20 3.1 Deriving large cardinal properties from generic systems of filters . 22 3.2 Consistency of small generic large cardinals . ..... 23 3.3 Combinatorial equivalents of ideally large cardinal properties . 27 3.3.1 Critical point and tallness . 27 3.3.2 Measurability........................... 28 arXiv:1511.05326v2 [math.LO] 5 Apr 2017 3.3.3 Strongness ............................ 30 3.3.4 Closure .............................. 31 3.4 Distinction between generic large cardinal properties ......... 33 4 Conclusions and open problems 34 1 1 Introduction Large cardinals have been among the most important axioms extending ZFC since the very beginning of modern set theory. On the one hand they provide a fine scale to measure the consistency strength of a rich variety of combinatorial principles, on the other hand they also solve important questions within set theory. -
Condensation for Mouse Pairs John Steel∗ Nam
CONDENSATION FOR MOUSE PAIRS JOHN STEEL∗ NAM TRANGy Abstract This is the first of two papers on the fine structure of HOD in models of the Axiom of + Determinacy (AD). Let M AD + V = L(}(R)). [10] shows that under a natural hypothesis on the existence of iteration strategies, the basic fine structure theory for pure extender models goes over to HODM . In this paper, we prove a fine condensation theorem, quite similar to Theorem 9.3.2 of Zeman's book [14], except that condensation for iteration strategies has been M added to the mix. In the second paper, we shall use this theorem to show that in HOD , κ holds iff κ is not subcompact. 1. INTRODUCTION One goal of descriptive inner model theory is to elucidate the structure of HOD (the universe of hereditarily ordinal definable sets) in models M of the Axiom of Determinacy. HODM is close to M + 1 in various ways; for example, if M AD + V = L(}(R)) , then M can be realized as a symmetric forcing extension of HODM , so that the first order theory of M is part of the first order theory of its HOD. 2 For this and many other reasons, the study of HOD in models of AD has a long history. We refer the reader to [11] for a survey of this history. The study of HOD involves ideas from descriptive set theory (for example, games and definable scales) and ideas from inner model theory (mice, comparison, fine structure). One early result showing that inner model theory is relevant is due to the first author, who showed in 1994 ([9]) that if there are ! Woodin cardinals with a measurable above them all, then in L(R), HOD up to θ is a pure extender mouse.