A Class of Strong Diamond Principles
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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. -
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. -
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]. -
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. -
The Hierarchy of Ramsey-Like Cardinals
The hierarchy of Ramsey-like cardinals Victoria Gitman [email protected] http://victoriagitman.github.io Algebra and Logic Seminar University of Denver January 14, 2019 Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 1 / 30 Large cardinals Models of set theory A universe V of set theory satisfies the Zermelo-Fraenkel axioms ZFC. It is the union of the von Neamann hierarchy of the Vα's. V V0 = ; . Vα+1 = (Vα). Vλ P . S . Vλ = Vα for a limit ordinal λ. Vα+1 α<λ Vα S . V = Vα. α2Ord V !+1 V! Definition: A class M is an inner model of V if M is: M V transitive: if a 2 M and b 2 a, then b 2 M, Ord ⊆ M, M j= ZFC. G¨odel'sconstructible universe L is the smallest inner model. L L0 = ; V Lα+1 = \All definable subsets of Lα" S Lλ = α<λ Lα for a limit λ. S L = α2Ord Lα. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 2 / 30 Large cardinals A hierarchy of set-theoretic assertions There are many set theoretic assertions independent of ZFC. Some of these assertions are stronger than others. A theory T is stronger than a theory S if from a model of T we can construct a model of S, but not the other way around. Examples: ZFC + CH and ZFC + :CH are equally strong. Given a model of ZFC + CH, we can use forcing to build a model of ZFC + :CH and visa-versa. ZFC and ZFC + V = L are equally strong. -
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. -
Diamonds on Large Cardinals
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Helsingin yliopiston digitaalinen arkisto Diamonds on large cardinals Alex Hellsten Academic dissertation To be presented, with the permission of the Faculty of Science of the University of Helsinki, for public criticism in Auditorium III, Porthania, on December 13th, 2003, at 10 o’clock a.m. Department of Mathematics Faculty of Science University of Helsinki ISBN 952-91-6680-X (paperback) ISBN 952-10-1502-0 (PDF) Yliopistopaino Helsinki 2003 Acknowledgements I want to express my sincere gratitude to my supervisor Professor Jouko V¨a¨an¨anen for supporting me during all these years of getting acquainted with the intriguing field of set theory. I am also grateful to all other members of the Helsinki Logic Group for interesting discussions and guidance. Especially I wish to thank Do- cent Tapani Hyttinen who patiently has worked with all graduate students regardless of whether they study under his supervision or not. Professor Saharon Shelah deserve special thanks for all collaboration and sharing of his insight in the subject. The officially appointed readers Professor Boban Veliˇckovi´cand Docent Kerkko Luosto have done a careful and minute job in reading the thesis. In effect they have served as referees for the second paper and provided many valuable comments. Finally I direct my warmest gratitude and love to my family to whom I also wish to dedicate this work. My wife Carmela and my daughters Jolanda and Vendela have patiently endured the process and have always stood me by. -
Weak Prediction Principles
WEAK PREDICTION PRINCIPLES OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT Abstract. We prove the consistency of the failure of the weak diamond Φλ at strongly inaccessible cardinals. On the other hand we show that <λ λ the very weak diamond Ψλ is equivalent to the statement 2 < 2 and hence holds at every strongly inaccessible cardinal. 2010 Mathematics Subject Classification. 03E05. Key words and phrases. Weak diamond, very weak diamond, Radin forcing. 1 2 OMER BEN-NERIA, SHIMON GARTI, AND YAIR HAYUT 0. Introduction The prediction principle ♦λ (diamond on λ) was discovered by Jensen, [6], who proved that it holds over any regular cardinal λ in the constructible universe. This principle says that there exists a sequence hAα : α < λi of sets, Aα ⊆ α for every α < λ, such that for every A ⊆ λ the set fα < λ : A \ α = Aαg is stationary. Jensen introduced the diamond in 1972, and the main focus was the case of @0 @0 λ = @1. It is immediate that ♦@1 ) 2 = @1, but consistent that 2 = @1 along with :♦@1 . Motivated by algebraic constructions, Devlin and Shelah [2] introduced a weak form of the diamond principle which follows from the continuum hypothesis: Definition 0.1 (The Devlin-Shelah weak diamond). Let λ be a regular uncountable cardinal. The weak diamond on λ (denoted by Φλ) is the following principle: For every function c : <λ2 ! 2 there exists a function g 2 λ2 such that λ fα 2 λ : c(f α) = g(α)g is a stationary subset of λ whenever f 2 2. -
Parametrized ♢ Principles
Parametrized ♦ principles Justin Tatch Moore, Michael Hruˇs´ak, Mirna Dˇzamonja ∗† September 2, 2003 Abstract We will present a collection of guessing principles which have a similar relationship to ♦ as cardinal invariants of the continuum have to CH. The purpose is to provide a means for systematically analyzing ♦ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of CH and ♦ in models such as those of Laver, Miller, and Sacks. 1 Introduction Very early on in the course of modern set theory, Jensen isolated the following combinatorial principle known as ♦: ♦ There is a sequence Aα (α < ω1) such that for all α < ω1, Aα ⊆ α and if X is a subset of ω1 then set {α < ω1 : X ∩ α = Aα} is stationary. ∗The first and third authors received support from EPSRC grant GR/M71121 for the research of this paper. The research of the second author was supported in part by the Netherlands Organization for Scientific Research (NWO) – Grant 613.007.039, and in part by the Grant Agency of the Czech Republic – Grant GACRˇ 201/00/1466. †2000 Mathematics Subject Classification. Primary 03E17, 03E65. Key words: Dia- mond, weak diamond, cardinal invariant, guessing principle. 1 Jensen used this principle to construct a Suslin tree [17] and later many other constructions were carried out using ♦ as an assumption — see [9]. The purpose of this paper is to provide a broad framework for analyzing the consequences of Jensen’s ♦ principle. Our intent is to present an array of ♦-principles which have the same relation to ♦ as the cardinal invariants of the continuum (see e.g. -
Arxiv:2009.07164V1 [Math.LO] 15 Sep 2020 Ai—U Nta Eeyb Etoigfaue Htuiul Char Uniquely That Isomorphism
CATEGORICAL LARGE CARDINALS AND THE TENSION BETWEEN CATEGORICITY AND SET-THEORETIC REFLECTION JOEL DAVID HAMKINS AND HANS ROBIN SOLBERG Abstract. Inspired by Zermelo’s quasi-categoricity result characterizing the models of second-order Zermelo-Fraenkel set theory ZFC2, we investigate when those models are fully categorical, characterized by the addition to ZFC2 ei- ther of a first-order sentence, a first-order theory, a second-order sentence or a second-order theory. The heights of these models, we define, are the categor- ical large cardinals. We subsequently consider various philosophical aspects of categoricity for structuralism and realism, including the tension between categoricity and set-theoretic reflection, and we present (and criticize) a cat- egorical characterization of the set-theoretic universe hV, ∈i in second-order logic. Categorical accounts of various mathematical structures lie at the very core of structuralist mathematical practice, enabling mathematicians to refer to specific mathematical structures, not by having carefully to prepare and point at specially constructed instances—preserved like the one-meter iron bar locked in a case in Paris—but instead merely by mentioning features that uniquely characterize the structure up to isomorphism. The natural numbers hN, 0,Si, for example, are uniquely characterized by the Dedekind axioms, which assert that 0 is not a successor, that the successor func- tion S is one-to-one, and that every set containing 0 and closed under successor contains every number [Ded88, Ded01]. We know what we mean by the natural numbers—they have a definiteness—because we can describe features that com- pletely determine the natural number structure. -
Ordinal Notation
Dmytro Taranovsky Last Update: December 31, 2018 Ordinal Notation Abstract: We introduce a framework for ordinal notation systems, present a family of strong yet simple systems, and give many examples of ordinals in these systems. While much of the material is conjectural, we include systems with conjectured strength beyond second order arithmetic (and plausibly beyond ZFC), and prove well-foundedness for some weakened versions. Contents: 1 Introduction 1.1 Introduction 1.2 Prior Work 1.3 Goals of Ordinal Analysis 2 A Framework for Ordinal Notations — defines the general structure for the notations. 2.1 Definition of the General Notation 2.2 Basic Properties 2.3 Bachmann-Howard Ordinal 2.4 One Variable C 2.5 Reflection Configurations 3 Degrees of Recursive Inaccessibility — presents a notation system for ATR0 plus "for every ordinal a, there is recursively a-inaccessible ordinal". 3.1 Definition of the Notation 3.2 Comparison Relation 3.3 Examples 4 Degrees of Reflection — presents a notation system for KP + Πn reflection and somewhat stronger systems, with the canonical assignment for nonrecursive ordinals. 4.1 Definition of Degrees of Reflection 4.2 Examples and Additional Properties 4.3 Assignment of Degrees 4.4 A Step towards Second Order Arithmetic 5 Main Ordinal Notation System — presents the main notation system, which plausibly reaches second order arithmetic (Z2) or higher. 5.1 Definition and Basic Properties 5.2 Strength of the Main System 5.3 Old Analysis and Examples 6 Beyond Second Order Arithmetic — gives insights for going beyond second order arithmetic and ZFC 6.1 Old Analysis 6.2 New Analysis 7 Iteration of n-built from below — proposes a strengthening of the main system.