Forcing in Proof Theory∗
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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. -
Arxiv:1804.02439V1
DATHEMATICS: A META-ISOMORPHIC VERSION OF ‘STANDARD’ MATHEMATICS BASED ON PROPER CLASSES DANNY ARLEN DE JESUS´ GOMEZ-RAM´ ´IREZ ABSTRACT. We show that the (typical) quantitative considerations about proper (as too big) and small classes are just tangential facts regarding the consistency of Zermelo-Fraenkel Set Theory with Choice. Effectively, we will construct a first-order logic theory D-ZFC (Dual theory of ZFC) strictly based on (a particular sub-collection of) proper classes with a corresponding spe- cial membership relation, such that ZFC and D-ZFC are meta-isomorphic frameworks (together with a more general dualization theorem). More specifically, for any standard formal definition, axiom and theorem that can be described and deduced in ZFC, there exists a corresponding ‘dual’ ver- sion in D-ZFC and vice versa. Finally, we prove the meta-fact that (classic) mathematics (i.e. theories grounded on ZFC) and dathematics (i.e. dual theories grounded on D-ZFC) are meta-isomorphic. This shows that proper classes are as suitable (primitive notions) as sets for building a foundational framework for mathematics. Mathematical Subject Classification (2010): 03B10, 03E99 Keywords: proper classes, NBG Set Theory, equiconsistency, meta-isomorphism. INTRODUCTION At the beginning of the twentieth century there was a particular interest among mathematicians and logicians in finding a general, coherent and con- sistent formal framework for mathematics. One of the main reasons for this was the discovery of paradoxes in Cantor’s Naive Set Theory and related sys- tems, e.g., Russell’s, Cantor’s, Burati-Forti’s, Richard’s, Berry’s and Grelling’s paradoxes [12], [4], [14], [3], [6] and [11]. -
Coll041-Endmatter.Pdf
http://dx.doi.org/10.1090/coll/041 AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS VOLUME 41 A FORMALIZATION OF SET THEORY WITHOUT VARIABLES BY ALFRED TARSKI and STEVEN GIVANT AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1985 Mathematics Subject Classification. Primar y 03B; Secondary 03B30 , 03C05, 03E30, 03G15. Library o f Congres s Cataloging-in-Publicatio n Dat a Tarski, Alfred . A formalization o f se t theor y withou t variables . (Colloquium publications , ISS N 0065-9258; v. 41) Bibliography: p. Includes indexes. 1. Se t theory. 2 . Logic , Symboli c an d mathematical . I . Givant, Steve n R . II. Title. III. Series: Colloquium publications (American Mathematical Society) ; v. 41. QA248.T37 198 7 511.3'2 2 86-2216 8 ISBN 0-8218-1041-3 (alk . paper ) Copyright © 198 7 b y th e America n Mathematica l Societ y Reprinted wit h correction s 198 8 All rights reserve d excep t thos e grante d t o th e Unite d State s Governmen t This boo k ma y no t b e reproduce d i n an y for m withou t th e permissio n o f th e publishe r The pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . @ Contents Section interdependenc e diagram s vii Preface x i Chapter 1 . Th e Formalis m £ o f Predicate Logi c 1 1.1. Preliminarie s 1 1.2. -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
A General Setting for the Pointwise Investigation of Determinacy
A General Setting for the Pointwise Investigation of Determinacy Yurii Khomskii? Institute of Logic, Language and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands Abstract. It is well-known that if we assume a large class of sets of reals to be determined then we may conclude that all sets in this class have certain regularity properties: we say that determinacy implies regularity properties classwise. In [L¨o05]the pointwise relation between determi- nacy and certain regularity properties (namely the Marczewski-Burstin algebra of arboreal forcing notions and a corresponding weak version) was examined. An open question was how this result extends to topological forcing no- tions whose natural measurability algebra is the class of sets having the Baire property. We study the relationship between the two cases, and using a definition which adequately generalizes both the Marczewski- Burstin algebra of measurability and the Baire property, prove results similar to [L¨o05]. We also show how this can be further generalized for the purpose of comparing algebras of measurability of various forcing notions. 1 Introduction The classical theorems due to Mycielski-Swierczkowski, Banach-Mazur and Mor- ton Davis respectively state that under the Axiom of Determinacy all sets of reals are Lebesgue measurable, have the Baire property and the perfect set property (see, e.g., [Ka94, pp 373{377]). In fact, these proofs give classwise implications, i.e., if Γ is a boldface pointclass (closed under continuous preimages and inter- sections with basic open sets) such that all sets in Γ are determined, then all sets in Γ have the corresponding regularity property. -
Equivalents to the Axiom of Choice and Their Uses A
EQUIVALENTS TO THE AXIOM OF CHOICE AND THEIR USES A Thesis Presented to The Faculty of the Department of Mathematics California State University, Los Angeles In Partial Fulfillment of the Requirements for the Degree Master of Science in Mathematics By James Szufu Yang c 2015 James Szufu Yang ALL RIGHTS RESERVED ii The thesis of James Szufu Yang is approved. Mike Krebs, Ph.D. Kristin Webster, Ph.D. Michael Hoffman, Ph.D., Committee Chair Grant Fraser, Ph.D., Department Chair California State University, Los Angeles June 2015 iii ABSTRACT Equivalents to the Axiom of Choice and Their Uses By James Szufu Yang In set theory, the Axiom of Choice (AC) was formulated in 1904 by Ernst Zermelo. It is an addition to the older Zermelo-Fraenkel (ZF) set theory. We call it Zermelo-Fraenkel set theory with the Axiom of Choice and abbreviate it as ZFC. This paper starts with an introduction to the foundations of ZFC set the- ory, which includes the Zermelo-Fraenkel axioms, partially ordered sets (posets), the Cartesian product, the Axiom of Choice, and their related proofs. It then intro- duces several equivalent forms of the Axiom of Choice and proves that they are all equivalent. In the end, equivalents to the Axiom of Choice are used to prove a few fundamental theorems in set theory, linear analysis, and abstract algebra. This paper is concluded by a brief review of the work in it, followed by a few points of interest for further study in mathematics and/or set theory. iv ACKNOWLEDGMENTS Between the two department requirements to complete a master's degree in mathematics − the comprehensive exams and a thesis, I really wanted to experience doing a research and writing a serious academic paper. -
The Strength of Mac Lane Set Theory
The Strength of Mac Lane Set Theory A. R. D. MATHIAS D´epartement de Math´ematiques et Informatique Universit´e de la R´eunion To Saunders Mac Lane on his ninetieth birthday Abstract AUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and S Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, TCo, of Transitive Containment, we shall refer as MAC. His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasizes, one that is adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke{Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory Z, and obtain an apparently new proof that Z is not finitely axiomatisable; we study Friedman's strengthening KPP + AC of KP + MAC, and the Forster{Kaye subsystem KF of MAC, and use forcing over ill-founded models and forcing to establish independence results concerning MAC and KPP ; we show, again using ill-founded models, that KPP + V = L proves the consistency of KPP ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret -
Self-Similarity in the Foundations
Self-similarity in the Foundations Paul K. Gorbow Thesis submitted for the degree of Ph.D. in Logic, defended on June 14, 2018. Supervisors: Ali Enayat (primary) Peter LeFanu Lumsdaine (secondary) Zachiri McKenzie (secondary) University of Gothenburg Department of Philosophy, Linguistics, and Theory of Science Box 200, 405 30 GOTEBORG,¨ Sweden arXiv:1806.11310v1 [math.LO] 29 Jun 2018 2 Contents 1 Introduction 5 1.1 Introductiontoageneralaudience . ..... 5 1.2 Introduction for logicians . .. 7 2 Tour of the theories considered 11 2.1 PowerKripke-Plateksettheory . .... 11 2.2 Stratifiedsettheory ................................ .. 13 2.3 Categorical semantics and algebraic set theory . ....... 17 3 Motivation 19 3.1 Motivation behind research on embeddings between models of set theory. 19 3.2 Motivation behind stratified algebraic set theory . ...... 20 4 Logic, set theory and non-standard models 23 4.1 Basiclogicandmodeltheory ............................ 23 4.2 Ordertheoryandcategorytheory. ...... 26 4.3 PowerKripke-Plateksettheory . .... 28 4.4 First-order logic and partial satisfaction relations internal to KPP ........ 32 4.5 Zermelo-Fraenkel set theory and G¨odel-Bernays class theory............ 36 4.6 Non-standardmodelsofsettheory . ..... 38 5 Embeddings between models of set theory 47 5.1 Iterated ultrapowers with special self-embeddings . ......... 47 5.2 Embeddingsbetweenmodelsofsettheory . ..... 57 5.3 Characterizations.................................. .. 66 6 Stratified set theory and categorical semantics 73 6.1 Stratifiedsettheoryandclasstheory . ...... 73 6.2 Categoricalsemantics ............................... .. 77 7 Stratified algebraic set theory 85 7.1 Stratifiedcategoriesofclasses . ..... 85 7.2 Interpretation of the Set-theories in the Cat-theories ................ 90 7.3 ThesubtoposofstronglyCantorianobjects . ....... 99 8 Where to go from here? 103 8.1 Category theoretic approach to embeddings between models of settheory . -
Tuple Routing Strategies for Distributed Eddies
Tuple Routing Strategies for Distributed Eddies Feng Tian David J. DeWitt Department of Computer Sciences University of Wisconsin, Madison Madison, WI, 53706 {ftian, dewitt}@cs.wisc.edu Abstract data stream management systems has begun to receive the attention of the database community. Many applications that consist of streams of data Many of the fundamental assumptions that are the are inherently distributed. Since input stream basis of standard database systems no longer hold for data rates and other system parameters such as the stream management systems [8]. A typical stream query is amount of available computing resources can long running -- it listens on several continuous streams fluctuate significantly, a stream query plan must and produces a continuous stream as its result. The notion be able to adapt to these changes. Routing tuples of running time, which is used as an optimization goal by between operators of a distributed stream query a classic database optimizer, cannot be directly applied to plan is used in several data stream management a stream management system. A data stream management systems as an adaptive query optimization system must use other performance metrics. In addition, technique. The routing policy used can have a since the input stream rates and the available computing significant impact on system performance. In this resources will usually fluctuate over time, an execution paper, we use a queuing network to model a plan that works well at query installation time might be distributed stream query plan and define very inefficient just a short time later. Furthermore, the performance metrics for response time and “optimize-then-execute” paradigm of traditional database system throughput. -
EQUICONSISTENCIES at SUBCOMPACT CARDINALS We
EQUICONSISTENCIES AT SUBCOMPACT CARDINALS ITAY NEEMAN AND JOHN STEEL Abstract. We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ , a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both 2(δ) and 2δ fail, then δ is subcompact in a class inner 2 + 2 model. If in addition (δ ) fails, we prove that δ is Π1 subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing 2 Axiom implies the existence of a class inner model with a Π1 subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+(n) supercompact for all n < !. We state some results at this level, and indicate how they are proved. MSC 2010: 03E45, 03E55. Keywords: subcompact cardinals, inner models, long extenders, coherent sequences, square. We dedicate this paper to Rich Laver, a brilliant mathematician and a kind and generous colleague. x1. Introduction. We present equiconsistency results at the level of sub- compact cardinals. The methods we use extend further, to levels which are interlaced with the axioms κ is κ+(n) supercompact, for n < !. The extensions will be carried out in a sequel to this paper, Neeman-Steel [7], but we indicate in this paper some of the main ideas involved. Our reversals assume iterability for countable substructures of V . -
Introduction: the 1930S Revolution
PROPERTY OF MIT PRESS: FOR PROOFREADING AND INDEXING PURPOSES ONLY Introduction: The 1930s Revolution The theory of computability was launched in the 1930s by a group of young math- ematicians and logicians who proposed new, exact, characterizations of the idea of algorithmic computability. The most prominent of these young iconoclasts were Kurt Gödel, Alonzo Church, and Alan Turing. Others also contributed to the new field, most notably Jacques Herbrand, Emil Post, Stephen Kleene, and J. Barkley Rosser. This seminal research not only established the theoretical basis for computability: these key thinkers revolutionized and reshaped the mathematical world—a revolu- tion that culminated in the Information Age. Their motive, however, was not to pioneer the discipline that we now know as theoretical computer science, although with hindsight this is indeed what they did. Nor was their motive to design electronic digital computers, although Turing did go on to do so (in fact producing the first complete paper design that the world had seen for an electronic stored-program universal computer). Their work was rather the continuation of decades of intensive investigation into that most abstract of subjects, the foundations of mathematics—investigations carried out by such great thinkers as Leopold Kronecker, Richard Dedekind, Gottlob Frege, Bertrand Russell, David Hilbert, L. E. J. Brouwer, Paul Bernays, and John von Neumann. The concept of an algorithm, or an effective or computable procedure, was central during these decades of foundational study, although for a long time no attempt was made to characterize the intuitive concept formally. This changed when Hilbert’s foundation- alist program, and especially the issue of decidability, made it imperative to provide an exact characterization of the idea of a computable function—or algorithmically calculable function, or effectively calculable function, or decidable predicate. -
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.