Large Cardinals with Forcing
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Set Theory, by Thomas Jech, Academic Press, New York, 1978, Xii + 621 Pp., '$53.00
BOOK REVIEWS 775 BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 1, July 1980 © 1980 American Mathematical Society 0002-9904/80/0000-0 319/$01.75 Set theory, by Thomas Jech, Academic Press, New York, 1978, xii + 621 pp., '$53.00. "General set theory is pretty trivial stuff really" (Halmos; see [H, p. vi]). At least, with the hindsight afforded by Cantor, Zermelo, and others, it is pretty trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of conventional mathematics, including the general theory of transfinite cardi nals and ordinals. This "trivial" part of set theory is well covered in standard texts, such as [E] or [H]. Jech's book is an introduction to the "nontrivial" part. Now, nontrivial set theory may be roughly divided into two general areas. The first area, classical set theory, is a direct outgrowth of Cantor's work. Cantor set down the basic properties of cardinal numbers. In particular, he showed that if K is a cardinal number, then 2", or exp(/c), is a cardinal strictly larger than K (if A is a set of size K, 2* is the cardinality of the family of all subsets of A). Now starting with a cardinal K, we may form larger cardinals exp(ic), exp2(ic) = exp(exp(fc)), exp3(ic) = exp(exp2(ic)), and in fact this may be continued through the transfinite to form expa(»c) for every ordinal number a. -
Forcing Axioms and Stationary Sets
ADVANCES IN MATHEMATICS 94, 256284 (1992) Forcing Axioms and Stationary Sets BOBAN VELICKOVI~ Department of Mathematics, University of California, Evans Hall, Berkeley, California 94720 INTRODUCTION Recall that a partial ordering 9 is called semi-proper iff for every suf- ficiently large regular cardinal 0 and every countable elementary submodel N of H,, where G9 is the canonical name for a Y-generic filter. The above q is called (9, N)-semi-generic. The Semi-Proper Forcing Axiom (SPFA) is the following statement: For every semi-proper 9 and every family 9 of Et, dense subsets of 9 there exists a filter G in 9 such that for every DE 9, GnD#@. SPFA+ states that if in addition a Y-name 3 for a stationary subset of wi is given, then a filter G can be found such that val,(S)= {c(<wi: 3p~Gp I~-EE$) is stationary. In general, if X is any class of posets, the forcing axioms XFA and XFA + are obtained by replacing “semi-proper LY” by “9 E X” in the above definitions. Sometimes we denote these axioms by FA(X) and FA+(.X). Thus, for example, FA(ccc) is the familiar Martin’s Axiom, MAN,. The notion of semi-proper, as an extension of a previous notion of proper, was defined and extensively studied by Shelah in [Shl J. These forcing notions generalize ccc and countably closed forcing and can be iterated without collapsing N, . SPFA is implicit in [Shl] as an obvious strengthening of the Proper Forcing Axiom (PFA), previously formulated and proved consistent by Baumgartner (see [Bal, Del]). -
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. -
Set-Theoretic Geology, the Ultimate Inner Model, and New Axioms
Set-theoretic Geology, the Ultimate Inner Model, and New Axioms Justin William Henry Cavitt (860) 949-5686 [email protected] Advisor: W. Hugh Woodin Harvard University March 20, 2017 Submitted in partial fulfillment of the requirements for the degree of Bachelor of Arts in Mathematics and Philosophy Contents 1 Introduction 2 1.1 Author’s Note . .4 1.2 Acknowledgements . .4 2 The Independence Problem 5 2.1 Gödelian Independence and Consistency Strength . .5 2.2 Forcing and Natural Independence . .7 2.2.1 Basics of Forcing . .8 2.2.2 Forcing Facts . 11 2.2.3 The Space of All Forcing Extensions: The Generic Multiverse 15 2.3 Recap . 16 3 Approaches to New Axioms 17 3.1 Large Cardinals . 17 3.2 Inner Model Theory . 25 3.2.1 Basic Facts . 26 3.2.2 The Constructible Universe . 30 3.2.3 Other Inner Models . 35 3.2.4 Relative Constructibility . 38 3.3 Recap . 39 4 Ultimate L 40 4.1 The Axiom V = Ultimate L ..................... 41 4.2 Central Features of Ultimate L .................... 42 4.3 Further Philosophical Considerations . 47 4.4 Recap . 51 1 5 Set-theoretic Geology 52 5.1 Preliminaries . 52 5.2 The Downward Directed Grounds Hypothesis . 54 5.2.1 Bukovský’s Theorem . 54 5.2.2 The Main Argument . 61 5.3 Main Results . 65 5.4 Recap . 74 6 Conclusion 74 7 Appendix 75 7.1 Notation . 75 7.2 The ZFC Axioms . 76 7.3 The Ordinals . 77 7.4 The Universe of Sets . 77 7.5 Transitive Models and Absoluteness . -
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. -
Singular Cardinals: from Hausdorff's Gaps to Shelah's Pcf Theory
SINGULAR CARDINALS: FROM HAUSDORFF’S GAPS TO SHELAH’S PCF THEORY Menachem Kojman 1 PREFACE The mathematical subject of singular cardinals is young and many of the math- ematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, Istv´an Juh´asz, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to Istv´an Juh´asz, for his patient reading for me from the Russian text of Alexandrov and Urysohn’s Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff’s writing, and to Stefan Geschke, who helped me with the German texts I needed to read and sometimes translate. I am also indebted to the Hausdorff project in Bonn, for publishing a beautiful annotated volume of Hausdorff’s monumental Grundz¨uge der Mengenlehre and for Springer Verlag, for rushing to me a free copy of this book; many important details about the early history of the subject were drawn from this volume. The wonderful library and archive of the Institute Mittag-Leffler are a treasure for anyone interested in mathematics at the turn of the 20th century; a particularly pleasant duty for me is to thank the institute for hosting me during my visit in September of 2009, which allowed me to verify various details in the early research literature, as well as providing me the company of many set theorists and model theorists who are interested in the subject. -
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. -
Ineffability Within the Limits of Abstraction Alone
Ineffability within the Limits of Abstraction Alone Stewart Shapiro and Gabriel Uzquiano 1 Abstraction and iteration The purpose of this article is to assess the prospects for a Scottish neo-logicist foundation for a set theory. The gold standard would be a theory as rich and useful as the dominant one, ZFC, Zermelo-Fraenkel set theory with choice, perhaps aug- mented with large cardinal principles. Although the present paper is self-contained, we draw upon recent work in [32] in which we explore the power of a reasonably pure principle of reflection. To establish terminology, the Scottish neo-logicist program is to develop branches of mathematics using abstraction principles in the form: xα = xβ $ α ∼ β where the variables α and β range over items of a certain sort, x is an operator taking items of this sort to objects, and ∼ is an equivalence relation on this sort of item. The standard exemplar, of course, is Hume’s principle: #F = #G ≡ F ≈ G where, as usual, F ≈ G is an abbreviation of the second-order statement that there is a one-one correspondence from F onto G. Hume’s principle, is the main support for what is regarded as a success-story for the Scottish neo-logicist program, at least by its advocates. The search is on to develop more powerful mathematical theories on a similar basis. As witnessed by this volume and a significant portion of the contemporary liter- ature in the philosophy of mathematics, Scottish neo-logicism remains controver- sial, even for arithmetic. There are, for example, issues of Caesar and Bad Com- pany to deal with. -
Notes on Set Theory, Part 2
§11 Regular cardinals In what follows, κ , λ , µ , ν , ρ always denote cardinals. A cardinal κ is said to be regular if κ is infinite, and the union of fewer than κ sets, each of whose cardinality is less than κ , is of cardinality less than κ . In symbols: κ is regular if κ is infinite, and κ 〈 〉 ¡ κ for any set I with I ¡ < and any family Ai i∈I of sets such that Ai < ¤¦¥ £¢ κ for all i ∈ I , we have A ¡ < . i∈I i (Among finite cardinals, only 0 , 1 and 2 satisfy the displayed condition; it is not worth including these among the cardinals that we want to call "regular".) ℵ To see two examples, we know that 0 is regular: this is just to say that the union of finitely ℵ many finite sets is finite. We have also seen that 1 is regular: the meaning of this is that the union of countably many countable sets is countable. For future use, let us note the simple fact that the ordinal-least-upper-bound of any set of cardinals is a cardinal; for any set I , and cardinals λ for i∈I , lub λ is a cardinal. i i∈I i α Indeed, if α=lub λ , and β<α , then for some i∈I , β<λ . If we had β § , then by i∈I i i β λ ≤α β § λ λ < i , and Cantor-Bernstein, we would have i , contradicting the facts that i is β λ β α β ¨ α α a cardinal and < i .