Hierarchies of Forcing Axioms, the Continuum Hypothesis and Square Principles
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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]). -
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. -
Martin's Maximum Revisited
Martin’s maximum revisited Matteo Viale Abstract We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with strong large cardinals, the forcing axiom ++ Π MM makes the 2-fragment of the theory of Hℵ2 invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a close to optimal generalization to Hℵ2 of Woodin’s absoluteness results for L(). In due course of proving this we shall give a new proof of some of Woodin’s results. 1 Introduction In this introduction we shall take a long detour to motivate the results we want to present and to show how they stem out of Woodin’s work on Ω-logic. We tried to make this introduction comprehensible to any person acquainted with the theory of forcing as presented for example in [7]. The reader may refer to subsection 1.1 for unexplained notions. Since its discovery in the early sixties by Paul Cohen [2], forcing has played a central role in the development of modern set theory. It was soon realized its fun- arXiv:1110.1181v2 [math.LO] 9 Feb 2012 damental role to establish the undecidability in ZFC of all the classical problems of set theory, among which Cantor’s continuum problem. Moreover, up to date, forcing (or class forcing) is the unique efficient method to obtain independence results over ZFC. This method has found applications in virtually all fields of pure mathematics: in the last forty years natural problems of group theory, func- tional analysis, operator algebras, general topology, and many other subjects were shown to be undecidable by means of forcing (see [4, 13] among others). -
Martin's Maximum and Weak Square
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000{000 S 0002-9939(XX)0000-0 MARTIN'S MAXIMUM AND WEAK SQUARE JAMES CUMMINGS AND MENACHEM MAGIDOR (Communicated by Julia Knight) Abstract. We analyse the influence of the forcing axiom Martin's Maximum on the existence of square sequences, with a focus on the weak square principle λ,µ. 1. Introduction It is well known that strong forcing axioms (for example PFA, MM) and strong large cardinal axioms (strongly compact, supercompact) exert rather similar kinds of influence on the combinatorics of infinite cardinals. This is perhaps not so sur- prising when we consider the widely held belief that essentially the only way to obtain a model of PFA or MM is to collapse a supercompact cardinal to become !2. We quote a few results which bring out the similarities: 1a) (Solovay [14]) If κ is supercompact then λ fails for all λ ≥ κ. 1b) (Todorˇcevi´c[15]) If PFA holds then λ fails for all λ ≥ !2. ∗ 2a) (Shelah [11]) If κ is supercompact then λ fails for all λ such that cf(λ) < κ < λ. ∗ 2b) (Magidor, see Theorem 1.2) If MM holds then λ fails for all λ such that cf(λ) = !. 3a) (Solovay [13]) If κ is supercompact then SCH holds at all singular cardinals greater than κ. 3b) (Foreman, Magidor and Shelah [7]) If MM holds then SCH holds. 3c) (Viale [16]) If PFA holds then SCH holds. In fact both the p-ideal di- chotomy and the mapping reflection principle (which are well known con- sequences of PFA) suffice to prove SCH. -
The Ultrapower Axiom and the Equivalence Between Strong Compactness and Supercompactness
The Ultrapower Axiom and the equivalence between strong compactness and supercompactness Gabriel Goldberg October 12, 2018 Abstract The relationship between the large cardinal notions of strong compactness and su- percompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equiv- alent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent. 1 Introduction How large is the least strongly compact cardinal? Keisler-Tarski [1] asked whether it must be larger than the least measurable cardinal, and Solovay later conjectured that it is much larger: in fact, he conjectured that every strongly compact cardinal is supercompact. His conjecture was refuted by Menas [2], who showed that the least strongly compact limit of strongly compact cardinals is not supercompact. But Tarski's question was left unresolved until in a remarkable pair of independence results, Magidor showed that the size of the least strongly compact cannot be determined using only the standard axioms of set theory (ZFC). More precisely, it is consistent with ZFC that the least strongly compact cardinal is the least measurable cardinal, but it is also consistent that the least strongly compact cardinal is the least supercompact cardinal. One of the most prominent open questions in set theory asks whether a weak variant of Solovay's conjecture might still be true: is the existence of a strongly compact cardinal arXiv:1810.05058v1 [math.LO] 9 Oct 2018 equiconsistent with the existence of a supercompact cardinal? Since inner model theory is essentially the only known way of proving nontrivial consistency strength lower bounds, answering this question probably requires generalizing inner model theory to the level of strongly compact and supercompact cardinals. -
UC Irvine UC Irvine Electronic Theses and Dissertations
UC Irvine UC Irvine Electronic Theses and Dissertations Title Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms Permalink https://escholarship.org/uc/item/9875g511 Author Schatz, Jeffrey Robert Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA, IRVINE Axiom Selection by Maximization: V = Ultimate L vs Forcing Axioms DISSERTATION submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in Philosophy by Jeffrey Robert Schatz Dissertation Committee: Distinguished Professor Penelope Maddy, Chair Professor Toby Meadows Dean’s Professor Kai Wehmeier Professor Jeremy Heis 2019 All materials c 2019 Jeffrey Robert Schatz DEDICATION To Mike and Lori Schatz for always encouraging me to pursue my goals AND In memory of Jackie, a true and honest friend. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iv CURRICULUM VITAE vi ABSTRACT OF THE DISSERTATION vii CHAPTER 1: Two Contemporary Candidates for A Strong Theory of Sets 1 1 The Axiomatization of Set Theory and The Question of Axiom Selection 2 1.1 The Inner Model Program 14 1.2 The Forcing Axiom Program 27 1.3 How to Settle This Dispute? 36 1.4 CHAPTER 2: Axiom Selection and the Maxim of ‘Maximize’ 38 2 ‘Maximize’ as a Methodological Maxim 39 2.1 Two Notions of Maximize 43 2.2 Re-characterizing M-Max for Extensions of ZFC 50 2.3 CHAPTER 3: Applying ‘Maximize’ to Contemporary Axiom Candidates 53 3 Towards a Theory for Ultimate L 54 3.1 S-Max: -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title The Combinatorics and Absoluteness of Definable Sets of Real Numbers Permalink https://escholarship.org/uc/item/0v60v6m4 Author Norwood, Zach Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles ¿e Combinatorics and Absoluteness of Denable Sets of Real Numbers A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Zach Norwood 2018 © Copyright by Zach Norwood 2018 ABSTRACT OF THE DISSERTATION ¿e Combinatorics and Absoluteness of Denable Sets of Real Numbers by Zach Norwood Doctor of Philosophy in Mathematics University of California, Los Angeles, 2018 Professor Itay Neeman, Chair ¿is thesis divides naturally into two parts, each concerned with the extent to which the theory of LR can be changed by forcing. ¿e rst part focuses primarily on applying generic-absoluteness principles to show that denable sets of reals enjoy regularity properties. ¿e work in Part I is joint with Itay Neeman and is adapted from our forthcoming paper [33]. ¿is project was motivated by questions about mad families, maximal families of innite subsets of ω any two of which have only nitely many members in common. We begin, in the spirit of Mathias [30], by establishing (¿eorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem [48] that there are no innite mad families in the Solovay model. In Chapter3 we stray from the main line of inquiry to briey study a game-theoretic characteri- zation of lters with the Baire Property. -
Bounded Forcing Axioms and the Continuum
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 109 (2001) 179–203 www.elsevier.com/locate/apal Bounded forcing axioms and the continuum David AsperÃo, Joan Bagaria ∗ Departament de Logica, Historia i Filosoÿa de la Ciencia, Universitat de Barcelona, Baldiri Reixac, s/n, 08028 Barcelona, Catalonia, Spain Received 31 March 1999; received in revised form 3 November 1999; accepted 19 June 2000 Communicated by T. Jech Abstract We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2;!2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L atthelevel !2, and therefore with the existence of an !2-Suslin tree. We also ℵ1 show that the axiom we call BMMℵ3 implies ℵ2 =ℵ2, as well as a stationary re7ection principle which has many of the consequences of Martin’s Maximum for objects of size ℵ2. Finally, we ℵ0 give an example of a so-called boldface bounded forcing axiom implying 2 = ℵ2. c 2001 Elsevier Science B.V. All rights reserved. MSC: 03E35; 03E50; 03E05; 03E65 Keywords: Bounded forcing axioms; Gaps; Open coloring axiom; The continuum; Boldface bounded forcing axioms 1. Introduction Strong forcing axioms, like the Proper Forcing Axiom, the Semiproper Forcing Ax- iom, or Martin’s Maximum, have natural bounded forms, the so-called Bounded Forcing Axioms [3, 13, 16]. They are the natural generalizations of Martin’s Axiom, a bounded forcing axiom itself. -
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. -
The Virtual Large Cardinal Hierarchy
The Virtual Large Cardinal Hierarchy Stamatis Dimopoulos, Victoria Gitman and Dan Saattrup Nielsen September 10, 2020 Abstract. We continue the study of the virtual large cardinal hierarchy, initi- ated in [GS18], by analysing virtual versions of superstrong, Woodin, Vopěnka, and Berkeley cardinals. Gitman and Schindler showed that virtualizations of strong and supercompact cardinals yield the same large cardinal notion [GS18]. We show the same result for a (weak) virtualization of Woodin and a virtual- ization of Vopěnka cardinals. We also show that there is a virtually Berkeley cardinal if and only if the virtual Vopěnka principle holds, but On is not Mahlo. Contents 1 Introduction 1 2 Preliminaries 3 3 Virtually supercompact cardinals 4 4 Virtual Woodin and Vopěnka cardinals 12 5 Virtual Berkeley cardinals 18 6 Questions 24 A Chart of virtual large cardinals 25 1 Introduction The study of generic large cardinals, being cardinals that are critical points of elementary embeddings existing in generic extensions, goes back to the 1970’s. At that time, the primary interest was the existence of precipitous and saturated ideals on small cardinals like !1 and !2. Research in this area later moved to the study of more general generic embeddings, both defined on V , but also on rank-initial segments of V — these were investigated by e.g. [DL89] and [FG10]. 1 The move to virtual large cardinals happened when [Sch00] introduced the remarkable cardinals, which it turned out later were precisely a virtualization of supercompactness. Various other virtual large cardinals were first investi- gated in [GS18]. The key difference between virtual large cardinals and generic versions of large cardinals studied earlier is that in the virtual case we require the embedding to be between sets with the target model being a subset of the ground model. -
Large Cardinals and the Iterative Conception of Set
Large cardinals and the iterative conception of set Neil Barton∗ 4 December 2017y Abstract The independence phenomenon in set theory, while pervasive, can be par- tially addressed through the use of large cardinal axioms. One idea sometimes alluded to is that maximality considerations speak in favour of large cardinal ax- ioms consistent with ZFC, since it appears to be ‘possible’ (in some sense) to continue the hierarchy far enough to generate the relevant transfinite number. In this paper, we argue against this idea based on a priority of subset formation under the iterative conception. In particular, we argue that there are several con- ceptions of maximality that justify the consistency but falsity of large cardinal axioms. We argue that the arguments we provide are illuminating for the debate concerning the justification of new axioms in iteratively-founded set theory. Introduction Large cardinal axioms (discussed in detail below) are widely viewed as some of the best candidates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical consequences for indepen- dence results (both as a tool for generating new models and for consequences within the models in which they reside), and often appear natural to the working set theo- rist, providing fine-grained information about different properties of transfinite sets. They are considered mathematically interesting and central for the study of set the- ory and its philosophy. In this paper, we do not deny any of the above views. We will, however, argue that the status of large cardinal axioms as maximality principles is questionable.