Forcing Axioms and Projective Sets of Reals
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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]). -
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). -
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. -
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. -
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]. -
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: -
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. -
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. -
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. -
Ineffable Limits of Weakly Compact Cardinals and Similar Results
DOI: https://doi.org/10.15446/recolma.v54n2.93846 Revista Colombiana de Matem´aticas Volumen 54(2020)2, p´aginas181-186 Ineffable limits of weakly compact cardinals and similar results L´ımitesinefables de cardinales d´ebilmente compactos Franqui Cardenas´ Universidad Nacional de Colombia, Bogot´a,Colombia Abstract. It is proved that if an uncountable cardinal κ has an ineffable subset of weakly compact cardinals, then κ is a weakly compact cardinal, and if κ has an ineffable subset of Ramsey (Rowbottom, J´onsson,ineffable or subtle) cardinals, then κ is a Ramsey (Rowbottom, J´onsson,ineffable or subtle) cardinal. Key words and phrases. Weakly compact cardinal, subtle cardinal, ineffable car- dinal, ineffable set, J´onssoncardinal, Rowbottom cardinal, Ramsey cardinal. 2020 Mathematics Subject Classification. 03E55, 03E05. Resumen. Se prueba que si un cardinal no contable κ tiene un subconjunto casi inefable de cardinales d´ebilmente compactos entonces κ es un cardinal d´ebilmente compacto. Y si κ tiene un conjunto inefable de cardinales de Ram- sey (Rowbottom, J´onsson,inefables o sutiles) entonces κ es cardinal de Ram- sey (Rowbottom, J´onsson,inefable o sutil). Palabras y frases clave. Cardinal d´ebilmente compacto, cardinal sutil, cardinal inefable, conjunto inefable, cardinal J´onsson,cardinal Rowbottom, cardinal Ramsey. Large cardinals imply the existence of stationary subsets of smaller large cardi- nals. For instance weakly compact cardinals have a stationary subset of Mahlo cardinals, measurable cardinals imply the set of Ramsey cardinals below the measurable cardinal κ has measure 1 and Ramsey cardinals imply the set of weakly compact cardinals below the Ramsey cardinal κ is a stationary subset of κ.