THE MAXIMALITY of the CORE MODEL 1. Introduction This Paper
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Mice with Finitely Many Woodin Cardinals from Optimal Determinacy Hypotheses
MICE WITH FINITELY MANY WOODIN CARDINALS FROM OPTIMAL DETERMINACY HYPOTHESES SANDRA MULLER,¨ RALF SCHINDLER, AND W. HUGH WOODIN Abstract We prove the following result which is due to the third author. 1 1 Let n ≥ 1. If Πn determinacy and Πn+1 determinacy both hold true and 1 # there is no Σn+2-definable !1-sequence of pairwise distinct reals, then Mn 1 exists and is !1-iterable. The proof yields that Πn+1 determinacy implies # that Mn (x) exists and is !1-iterable for all reals x. A consequence is the Determinacy Transfer Theorem for arbitrary n ≥ 1, 1 (n) 2 1 namely the statement that Πn+1 determinacy implies a (< ! − Π1) de- terminacy. Contents Abstract 1 Preface 2 Overview 4 1. Introduction 6 1.1. Games and Determinacy 6 1.2. Inner Model Theory 7 1.3. Mice with Finitely Many Woodin Cardinals 9 1.4. Mice and Determinacy 14 2. A Model with Woodin Cardinals from Determinacy Hypotheses 15 2.1. Introduction 15 2.2. Preliminaries 16 2.3. OD-Determinacy for an Initial Segment of Mn−1 32 2.4. Applications 40 2.5. A Proper Class Inner Model with n Woodin Cardinals 44 3. Proving Iterability 49 First author formerly known as Sandra Uhlenbrock. The first author gratefully ac- knowledges her non-material and financial support from the \Deutsche Telekom Stiftung". The material in this paper is part of the first author's Ph.D. thesis written under the su- pervision of the second author. 1 2 SANDRA MULLER,¨ RALF SCHINDLER, AND W. HUGH WOODIN 3.1. -
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). -
Forcing Axioms and Projective Sets of Reals
FORCING AXIOMS AND PROJECTIVE SETS OF REALS RALF SCHINDLER Abstract. This paper is an introduction to forcing axioms and large cardinals. Specifically, we shall discuss the large cardinal strength of forcing axioms in the presence of regularity properties for projective sets of reals. The new result shown in this paper says that ZFC + the bounded proper forcing axiom (BPFA) + \every projective set of reals is Lebesgue measurable" is equiconsistent with ZFC + \there is a Σ1 reflecting cardinal above a remarkable cardinal." 1. Introduction. The current paper∗ is in the tradition of the following result. Theorem 1. ([HaSh85, Theorem B]) Equiconsistent are: (1) ZFC + there is a weakly compact cardinal, and (2) ZFC + Martin's axiom (MA) + every projective set of reals is Lebesgue measurable. This theorem links a large cardinal concept with a forcing axiom (MA) and a regularity property for the projective sets of reals. We shall be interested in considering forcing axioms which are somewhat stronger than MA. In particular, we shall produce the following new result.y Theorem 2. Equiconsistent are: (1) ZFC + there is a Σ1 reflecting cardinal above a remarkable car- dinal, and (2) ZFC + the bounded proper forcing axiom (BPFA) + every projec- tive set of reals is Lebesgue measurable. 1991 Mathematics Subject Classification. Primary 03E55, 03E15. Secondary 03E35, 03E60. Key words and phrases. set theory/forcing axioms/large cardinals/descriptive set theory. ∗The author would like to thank Ilijas Farah, Ernest Schimmerling, and Boban Velickovic for sharing with him mathematical and historical insights. yThis theorem was first presented in a talk at the 6e Atelier International de Th´eorie des Ensembles in Luminy, France, on Sept 20, 2000. -
Arxiv:1901.02074V1 [Math.LO] 4 Jan 2019 a Xoskona Lrecrias Ih Eal Ogv Entv a Definitive a Give T to of Able Basis Be the Might Cardinals” [17]
GENERIC LARGE CARDINALS AS AXIOMS MONROE ESKEW Abstract. We argue against Foreman’s proposal to settle the continuum hy- pothesis and other classical independent questions via the adoption of generic large cardinal axioms. Shortly after proving that the set of all real numbers has a strictly larger car- dinality than the set of integers, Cantor conjectured his Continuum Hypothesis (CH): that there is no set of a size strictly in between that of the integers and the real numbers [1]. A resolution of CH was the first problem on Hilbert’s famous list presented in 1900 [19]. G¨odel made a major advance by constructing a model of the Zermelo-Frankel (ZF) axioms for set theory in which the Axiom of Choice and CH both hold, starting from a model of ZF. This showed that the axiom system ZF, if consistent on its own, could not disprove Choice, and that ZF with Choice (ZFC), a system which suffices to formalize the methods of ordinary mathematics, could not disprove CH [16]. It remained unknown at the time whether models of ZFC could be found in which CH was false, but G¨odel began to suspect that this was possible, and hence that CH could not be settled on the basis of the normal methods of mathematics. G¨odel remained hopeful, however, that new mathemati- cal axioms known as “large cardinals” might be able to give a definitive answer on CH [17]. The independence of CH from ZFC was finally solved by Cohen’s invention of the method of forcing [2]. Cohen’s method showed that ZFC could not prove CH either, and in fact could not put any kind of bound on the possible number of cardinals between the sizes of the integers and the reals. -
Axiomatic Set Teory P.D.Welch
Axiomatic Set Teory P.D.Welch. August 16, 2020 Contents Page 1 Axioms and Formal Systems 1 1.1 Introduction 1 1.2 Preliminaries: axioms and formal systems. 3 1.2.1 The formal language of ZF set theory; terms 4 1.2.2 The Zermelo-Fraenkel Axioms 7 1.3 Transfinite Recursion 9 1.4 Relativisation of terms and formulae 11 2 Initial segments of the Universe 17 2.1 Singular ordinals: cofinality 17 2.1.1 Cofinality 17 2.1.2 Normal Functions and closed and unbounded classes 19 2.1.3 Stationary Sets 22 2.2 Some further cardinal arithmetic 24 2.3 Transitive Models 25 2.4 The H sets 27 2.4.1 H - the hereditarily finite sets 28 2.4.2 H - the hereditarily countable sets 29 2.5 The Montague-Levy Reflection theorem 30 2.5.1 Absoluteness 30 2.5.2 Reflection Theorems 32 2.6 Inaccessible Cardinals 34 2.6.1 Inaccessible cardinals 35 2.6.2 A menagerie of other large cardinals 36 3 Formalising semantics within ZF 39 3.1 Definite terms and formulae 39 3.1.1 The non-finite axiomatisability of ZF 44 3.2 Formalising syntax 45 3.3 Formalising the satisfaction relation 46 3.4 Formalising definability: the function Def. 47 3.5 More on correctness and consistency 48 ii iii 3.5.1 Incompleteness and Consistency Arguments 50 4 The Constructible Hierarchy 53 4.1 The L -hierarchy 53 4.2 The Axiom of Choice in L 56 4.3 The Axiom of Constructibility 57 4.4 The Generalised Continuum Hypothesis in L. -
O&ONSTRUCTIBLE UNIVERSE and MEASURABLE CARDINALS 0
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Annals of Pure and Applied Logic 30 (1986) 293-320 293 North-Holland o&ONSTRUCTIBLE UNIVERSE AND MEASURABLE CARDINALS Claude SURESON Dkpartement de Mathkmatiques, Universitk de Caen, 1403.2 Caen, France Communicated by A. Nerode Received 23 September 1984 In analogy with K. Godel’s model L, C. Chang [l] formulated the wr- constructible universe C”‘, using the infinitary language L,,,, instead of the language of Set Theory L,,. The cumulative hierarchy of sets obtained in this way has many similarities with the hierarchy of the constructible universe (except for a major point: the axiom of choice [l], [9]). C”’ can also be characterized as the least inner model closed under arbitrary countable sequences. This paper is inspired by results of R. Jensen and J. Silver concerning the existence of O# and the covering property for L. We consider here a stronger notion of indiscernibles for the model C”’ and we say that C”’ satisfies the ‘covering property’ if any set of ordinals X in the universe can be covered by a set in C”’ of cardinality ]X]‘O. The existence of ‘indiscernibles’ for C”’ is also linked to large cardinal assumptions, and our main result (in ZFC) can be summarized as shown in Diagram 1: Diagram 1. The first part is devoted to the study of indiscernibles for PI. We prove the implications (1) and (2). In the second section, we deal with the covering property and show (3). -
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. -
Force to Change Large Cardinal Strength
Force to change large cardinal strength by Erin Carmody A dissertation submitted to the Graduate Faculty in Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy, The City University of New York. 2015 arXiv:1506.03432v2 [math.LO] 12 Jun 2015 i ii ©2015 Erin Carmody All Rights Reserved iv Abstract This dissertation includes many theorems which show how to change large cardinal prop- erties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal κ, and ordinal α, if κ is α-inaccerssible, then there is a P forcing that κ which preserves that α-inaccessible but destorys that κ is (α+1)-inaccessible. I also consider Mahlo cardinals and degrees of Mahlo cardinals. I showed that for every cardinal κ, and ordinal α, there is a notion of forcing P such that κ is still α-Mahlo in the extension, but κ is no longer (α + 1)-Mahlo. I also show that a cardinal κ which is Mahlo in the ground model can have every possible inaccessible degree in the forcing extension, but no longer be Mahlo there. The thesis includes a collection of results which give forcing notions which change large cardinal strength from weakly compact to weakly measurable, including some earlier work by others that fit this theme. I consider in detail measurable cardinals and Mitchell rank. I show how to change a class of measurable cardinals by forcing to an extension where all measurable cardinals above some fixed ordinal α have Mitchell rank below α. -
V •. •, </>O a <#>I a • • • A<I>„ A
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 79, Number 2, June 1980 THE HANF NUMBER OF L„iUi JOUKO VAÄNÄNEN1 Abstract. A model of set theory is constructed in which the Hanf number of £„,„, is below the first weakly compact cardinal. This answers a question of J. Silver. 1. Introduction. J. Silver proved in [9] that if the smallest Erdös cardinal k(cS) exists, then hu u, the Hanf number of La a, exceeds k(w), and therefore also the first weakly compact cardinal. He left it as an open problem whether h could, in the absence of k(<o), be below the first weakly compact cardinal. The purpose of this paper is to give an affirmative answer to this problem. K. Kunen proved in [5] that hu u can be even larger than a measurable cardinal (indeed this always takes place in the inner model LM), but for trivial reasons hu a is below the first strongly compact cardinal. M. Magidor showed in [7] that the first measurable cardinal could be strongly compact, whence ha u could be below the first measurable cardinal. We prove this result without assuming the consistency of a strongly compact cardinal. Likewise our method yields a model in which ha u is below the first Ramsey cardinal. 2. Preliminaries. The infinitary language Lau extends the usual first order logic by allowing countable disjunctions and conjunctions *oV *i V • ■. V*„ V •. •, </>oA <#>i A • • • A<i>„A • •., and quantification over countably many variables 3x0.. .x„... <p(x0, . , x„, . ), Vx0...x„... </>(*„,. ., xn, . ). For a rigorous definition of Lu a see [1]. -
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]. -
SUBTLE CARDINALS and LINEAR ORDERINGS by Harvey M
1 SUBTLE CARDINALS AND LINEAR ORDERINGS by Harvey M. Friedman Department of Mathematics Ohio State University [email protected] www.math.ohio-state.edu/~friedman/ April 9, 1998 INTRODUCTION The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen, and a number of basic facts were proved there. These concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in [Ba75], where a highly developed theory is presented. This important level of the large cardinal hierarchy was discussed in some detail in the survey article [KM78], section 20. However, discussion was omitted in the subsequent [Ka94]. This level is associated with the discrete/finite combinatorial independence results in [Fr98] and [Fr¥]. In section 1 we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established in [Ba75]. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three properly intertwined hierarchies with the same limit, lying strictly above Òtotal indescribabilityÓ and strictly below Òarrowing w.Ó The innovation here is presented in section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy 2 by means of a striking universal second order property of linear orderings. As is usual in set theory, we treat cardinals and ordinals as von Neumann ordinals. -
Generalized Solovay Measures, the HOD Analysis, and the Core Model Induction
Generalized Solovay Measures, the HOD Analysis, and the Core Model Induction by Nam Duc Trang A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor John Steel, Chair Professor W. Hugh Woodin Professor Sherrilyn Roush Fall 2013 Generalized Solovay Measures, the HOD Analysis, and the Core Model Induction Copyright 2013 by Nam Duc Trang 1 Abstract Generalized Solovay Measures, the HOD Analysis, and the Core Model Induction by Nam Duc Trang Doctor of Philosophy in Mathematics University of California, Berkeley Professor John Steel, Chair This thesis belongs to the field of descriptive inner model theory. Chapter 1 provides a proper context for this thesis and gives a brief introduction to the theory of AD+, the theory of hod mice, and a definition of KJ (R). In Chapter 2, we explore the theory of generalized Solovay measures. We prove structure theorems concerning canonical models of the theory \AD+ + there is a generalized Solovay measure" and compute the exact consistency strength of this theory. We also give some applications relating generalized Solovay measures to the determinacy of a class of long games. In Chapter 3, we give a HOD analysis of AD+ + V = L(P(R)) models below \ADR + Θ is regular." This is an application of the theory of hod mice developed in [23]. We also analyze HOD of AD+-models of the form V = L(R; µ) where µ is a generalized Solovay measure. In Chapter 4, we develop techniques for the core model induction.