Introduction Concrete Mathematical Incompleteness
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Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae Available Online
Donald Knuth Fletcher Jones Professor of Computer Science, Emeritus Curriculum Vitae available Online Bio BIO Donald Ervin Knuth is an American computer scientist, mathematician, and Professor Emeritus at Stanford University. He is the author of the multi-volume work The Art of Computer Programming and has been called the "father" of the analysis of algorithms. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar,[4] Knuth created the WEB and CWEB computer programming systems designed to encourage and facilitate literate programming, and designed the MIX/MMIX instruction set architectures. As a member of the academic and scientific community, Knuth is strongly opposed to the policy of granting software patents. He has expressed his disagreement directly to the patent offices of the United States and Europe. (via Wikipedia) ACADEMIC APPOINTMENTS • Professor Emeritus, Computer Science HONORS AND AWARDS • Grace Murray Hopper Award, ACM (1971) • Member, American Academy of Arts and Sciences (1973) • Turing Award, ACM (1974) • Lester R Ford Award, Mathematical Association of America (1975) • Member, National Academy of Sciences (1975) 5 OF 44 PROFESSIONAL EDUCATION • PhD, California Institute of Technology , Mathematics (1963) PATENTS • Donald Knuth, Stephen N Schiller. "United States Patent 5,305,118 Methods of controlling dot size in digital half toning with multi-cell threshold arrays", Adobe Systems, Apr 19, 1994 • Donald Knuth, LeRoy R Guck, Lawrence G Hanson. -
Arxiv:2011.02915V1 [Math.LO] 3 Nov 2020 the Otniy Ucin Pnst Tctr)Ue Nti Ae R T Are Paper (C 1.1
REVERSE MATHEMATICS OF THE UNCOUNTABILITY OF R: BAIRE CLASSES, METRIC SPACES, AND UNORDERED SUMS SAM SANDERS Abstract. Dag Normann and the author have recently initiated the study of the logical and computational properties of the uncountability of R formalised as the statement NIN (resp. NBI) that there is no injection (resp. bijection) from [0, 1] to N. On one hand, these principles are hard to prove relative to the usual scale based on comprehension and discontinuous functionals. On the other hand, these principles are among the weakest principles on a new com- plimentary scale based on (classically valid) continuity axioms from Brouwer’s intuitionistic mathematics. We continue the study of NIN and NBI relative to the latter scale, connecting these principles with theorems about Baire classes, metric spaces, and unordered sums. The importance of the first two topics re- quires no explanation, while the final topic’s main theorem, i.e. that when they exist, unordered sums are (countable) series, has the rather unique property of implying NIN formulated with the Cauchy criterion, and (only) NBI when formulated with limits. This study is undertaken within Ulrich Kohlenbach’s framework of higher-order Reverse Mathematics. 1. Introduction The uncountability of R deals with arbitrary mappings with domain R, and is therefore best studied in a language that has such objects as first-class citizens. Obviousness, much more than beauty, is however in the eye of the beholder. Lest we be misunderstood, we formulate a blanket caveat: all notions (computation, continuity, function, open set, et cetera) used in this paper are to be interpreted via their higher-order definitions, also listed below, unless explicitly stated otherwise. -
On the Reverse Mathematics of General Topology Phd Dissertation, August 2005
The Pennsylvania State University The Graduate School Department of Mathematics ON THE REVERSE MATHEMATICS OF GENERAL TOPOLOGY A Thesis in Mathematics by Carl Mummert Copyright 2005 Carl Mummert Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2005 The thesis of Carl Mummert was reviewed and approved* by the following: Stephen G. Simpson Professor of Mathematics Thesis Advisor Chair of Committee Dmitri Burago Professor of Mathematics John D. Clemens Assistant Professor of Mathematics Martin F¨urer Associate Professor of Computer Science Alexander Nabutovsky Professor of Mathematics Nigel Higson Professor of Mathematics Chair of the Mathematics Department *Signatures are on file in the Graduate School. Abstract This thesis presents a formalization of general topology in second-order arithmetic. Topological spaces are represented as spaces of filters on par- tially ordered sets. If P is a poset, let MF(P ) be the set of maximal fil- ters on P . Let UF(P ) be the set of unbounded filters on P . If X is MF(P ) or UF(P ), the topology on X has a basis {Np | p ∈ P }, where Np = {F ∈ X | p ∈ F }. Spaces of the form MF(P ) are called MF spaces; spaces of the form UF(P ) are called UF spaces. A poset space is either an MF space or a UF space; a poset space formed from a countable poset is said to be countably based. The class of countably based poset spaces in- cludes all complete separable metric spaces and many nonmetrizable spaces including the Gandy–Harrington space. All poset spaces have the strong Choquet property. -
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]. -
Leading the Way in Scientific Computing. Addison-Wesley. I When Looking for the Best in Scientific Computing, You've Come to Rely on Addison-Wesley
Leading the way in scientific computing. Addison-Wesley. I When looking for the best in scientific computing, you've come to rely on Addison-Wesley. I Take a moment to see how we've made our list even stronger. The LATEX Companion sional programming languages. This book is the definitive user's Michael Goossens, Frank Mittelbach, and Alexander Samarin guide and reference manual for the CWEB system. This book is packed with information needed to use LATEX even 1994 (0-201 -57569-8) approx. 240 pp. Softcover more productively. It is a true companion to Leslie Lamport's users guide as well as a valuable complement to any LATEX introduction. Concrete Mathematics. Second Edition Describes the new LATEX standard. Ronald L. Graham, Donald E. Knuth, and Oren Patashnick 1994 (0-201 -54 199-8) 400 pp. Softcover With improvements to almost every page, the second edition of this classic text and reference introduces the mathematics that LATEX: A Document Preparation System, Second Edition supports advanced computer programming. Leslie Lamport 1994 (0-201 -55802-5) 672 pp. Hardcover The authoritative user's guide and reference manual has been revised to document features now available in the new standard Applied Mathematics@: Getting Started, Getting It Done software release-LATEX~E.The new edition features additional styles William T. Shaw and Jason Tigg and functions, improved font handling, and much more. This book shows how Mathematics@ can be used to solve problems in 1994 (0-201-52983-1) 256 pp. Softcover the applied sciences. Provides a wealth of practical tips and techniques. 1 994 (0-201 -542 1 7-X) 320 pp. -
Reverse Mathematics Carl Mummert Communicated by Daniel J
B OOKREVIEW Reverse Mathematics Carl Mummert Communicated by Daniel J. Velleman the theorem at hand? This question is a central motiva- Reverse Mathematics: Proofs from the Inside Out tion of the field of reverse mathematics in mathematical By John Stillwell logic. Princeton University Press Mathematicians have long investigated the problem Hardcover, 200 pages ISBN: 978-1-4008-8903-7 of the Parallel Postulate in geometry: which theorems require it, and which can be proved without it? Analo- gous questions arose about the Axiom of Choice: which There are several monographs on aspects of reverse math- theorems genuinely require the Axiom of Choice for their ematics, but none can be described as a “general audi- proofs? ence” text. Simpson’s Subsystems of Second Order Arith- In each of these cases, it is easy to see the importance metic [3], rightly regarded as a classic, makes substantial of the background theory. After all, what use is it to prove assumptions about the reader’s background in mathemat- a theorem “without the Axiom of Choice” if the proof ical logic. Hirschfeldt’s Slicing the Truth [2] is more ac- uses some other axiom that already implies the Axiom cessible but also makes assumptions beyond an upper- of Choice? To address the question of necessity, we must level undergraduate background and focuses more specif- begin by specifying a precise set of background axioms— ically on combinatorics. The field has been due for a gen- our base theory. This allows us to answer the question of eral treatment accessible to undergraduates and to math- whether an additional axiom is necessary for a particular ematicians in other areas looking for an easily compre- hensible introduction to the field. -
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. -
Forcing in Proof Theory∗
Forcing in proof theory¤ Jeremy Avigad November 3, 2004 Abstract Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound e®ects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation re- sults for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments. 1 Introduction In 1963, Paul Cohen introduced the method of forcing to prove the indepen- dence of both the axiom of choice and the continuum hypothesis from Zermelo- Fraenkel set theory. It was not long before Saul Kripke noted a connection be- tween forcing and his semantics for modal and intuitionistic logic, which had, in turn, appeared in a series of papers between 1959 and 1965. By 1965, Scott and Solovay had rephrased Cohen's forcing construction in terms of Boolean-valued models, foreshadowing deeper algebraic connections between forcing, Kripke se- mantics, and Grothendieck's notion of a topos of sheaves. In particular, Lawvere and Tierney were soon able to recast Cohen's original independence proofs as sheaf constructions.1 It is safe to say that these developments have had a profound impact on most branches of mathematical logic. -
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 . -
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 . -
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.