Finitary and Infinitary Mathematics, the Possibility of Possibilities and the Definition of Probabilities
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Philosophy 405: Knowledge, Truth and Mathematics Hamilton College Spring 2008 Russell Marcus M, W: 1-2:15Pm [email protected]
Philosophy 405: Knowledge, Truth and Mathematics Hamilton College Spring 2008 Russell Marcus M, W: 1-2:15pm [email protected] Class 15: Hilbert and Gödel I. Hilbert’s programme We have seen four different Hilberts: the term formalist (mathematical terms refer to inscriptions), the game formalist (ideal terms are meaningless), the deductivist (mathematics consists of deductions within consistent systems) the finitist (mathematics must proceed on finitary, but not foolishly so, basis). Gödel’s Theorems apply specifically to the deductivist Hilbert. But, Gödel would not have pursued them without the term formalist’s emphasis on terms, which lead to the deductivist’s pursuit of meta-mathematics. We have not talked much about the finitist Hilbert, which is the most accurate label for Hilbert’s programme. Hilbert makes a clear distinction between finite statements, and infinitary statements, which include reference to ideal elements. Ideal elements allow generality in mathematical formulas, and require acknowledgment of infinitary statements. See Hilbert 195-6. When Hilbert mentions ‘a+b=b+a’, he is referring to a universally quantified formula: (x)(y)(x+y=y+x) x and y range over all numbers, and so are not finitary. See Shapiro 159 on bounded and unbounded quantifiers. Hilbert’s blocky notation refers to bounded quantifiers. Note that a universal statement, taken as finitary, is incapable of negation, since it becomes an infinite statement. (x)Px is a perfectly finitary statement -(x)Px is equivalent to (x)-Px, which is infinitary - uh-oh! The admission of ideal elements begs questions of the meanings of terms in ideal statements. To what do the a and the b in ‘a+b=b+a’ refer? See Hilbert 194. -
Arxiv:1906.04871V1 [Math.CO] 12 Jun 2019
Nearly Finitary Matroids By Patrick C. Tam DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Eric Babson Jes´usDe Loera arXiv:1906.04871v1 [math.CO] 12 Jun 2019 Matthias K¨oppe Committee in Charge 2018 -i- c Patrick C. Tam, 2018. All rights reserved. To... -ii- Contents Abstract iv Acknowledgments v Chapter 1. Introduction 1 Chapter 2. Summary of Main Results 17 Chapter 3. Finitarization Spectrum 20 3.1. Definitions and Motivation 20 3.2. Ladders 22 3.3. More on Spectrum 24 3.4. Nearly Finitary Matroids 27 3.5. Unionable Matroids 33 Chapter 4. Near Finitarization 35 4.1. Near Finitarization 35 4.2. Independence System Example 36 Chapter 5. Ψ-Matroids and Related Constructions 39 5.1. Ψ-Matroids 39 5.2. P (Ψ)-Matroids 42 5.3. Matroids and Axiom Systems 44 Chapter 6. Thin Sum Matroids 47 6.1. Nearly-Thin Families 47 6.2. Topological Matroids 48 Bibliography 51 -iii- Patrick C. Tam March 2018 Mathematics Nearly Finitary Matroids Abstract In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system for this thesis. In Chapter 2, we summarize our main results after reviewing historical background and motivation. In Chapter 3, we define a notion of spectrum for matroids. Moreover, we show that the spectrum of a nearly finitary matroid can be larger than any fixed finite size. -
Formalism II Gödel’S Incompleteness Theorems Hilbert's Programme and Its Collapse Curry
Outline Hilbert’s programme Formalism II Gödel’s incompleteness theorems Hilbert's Programme and its collapse Curry 13 April 2005 Joost Cassee Gustavo Lacerda Martijn Pennings 1 2 Previously on PhilMath… Foundations of Mathematics Axiomatic methods Logicism: Principia Mathematica Logicism Foundational crisis Formalism Russell’s paradox Term Formalism Cantor’s ‘inconsistent multitudes’ Game Formalism Law of excluded middle: P or not-P Intuitionism Deductivism = Game Formalism + preservation of truth Consistency paramount Quiet period for Hilbert: 1905 ~ 1920 No intuition needed Rejection of logicism 3 4 1 Birth of Finitism Finitary arithmetic Hermann Weyl: “The new foundational crisis in Basic arithmetic: 2+3=5, 7+7≠10, … mathematics” (1921) Intuitionistic restrictions – crippled mathematics Bounded quantifiers Bounded: 100 < p < 200 and p is prime Hilbert’s defense: finitism Unbounded: p > 100 and p and p+2 are prime ‘Basing the if of if-then-ism’ Core: finitary arithmetic All sentences effectively decidable Construction of all mathematics from core Finite algorithm Epistemologically satisfying 5 6 Finitary arithmetic – ontology Ideal mathematics – the rest Meaningful, independent of logic – Kantian Game-Formalistic rules Correspondance to finitary arithmetic Concrete symbols themselves: |, ||, |||, … … but not too concrete – not physical Restriction: consistency with finitary arithmetic Essential to human thought – not reducible Formal systems described in finitary arithmetic Finitary proof of -
Hilbert's Program Then And
Hilbert’s Program Then and Now Richard Zach Abstract Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic sys- tems. Although G¨odel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial suc- cesses, and generated important advances in logical theory and meta- theory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical un- derpinnings and consequences, and its subsequent development and influ- ences since the 1930s. Keywords: Hilbert’s program, philosophy of mathematics, formal- ism, finitism, proof theory, meta-mathematics, foundations of mathemat- ics, consistency proofs, axiomatic method, instrumentalism. 1 Introduction Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of G¨ottingen and elsewhere in the 1920s and 1930s. Briefly, Hilbert’s proposal called for a new foundation of mathematics based on two pillars: the axiomatic method, and finitary proof theory. Hilbert thought that by formalizing mathematics in axiomatic systems, and subsequently proving by finitary methods that these systems are consistent (i.e., do not prove contradictions), he could provide a philosophically satisfactory grounding of classical, infinitary mathematics (analysis and set theory). -
Formal Semantics and Logic.Pdf
Formal Semantics and Logic Bas C. van Fraassen Copyright c 1971, Bas C. van Fraassen Originally published by The Macmillan Company, New York This eBook was published by Nousoul Digital Publishers. Its formatting is optimized for eReaders and other electronic reading devices. For information, the publisher may be con- tacted by email at: [email protected] 2 To my parents 3 Preface to the .PDF Edition With a view to the increasing academic importance of digital media this electronic edition was created by Nousoul Digital Publishers. Thanks to the diligent work and expertise of Brandon P. Hopkins this edition has fea- tures that no book could have in the year of its original publication: searchable text and hyperlinked notes. The text itself remains essentially unchanged, but in addition to typographical corrections there are also some substantive corrections. Apart from the change in the solution to exercise 5.4 of Chapter 3, none comprise more than a few words or symbols. However, as different as digital media are from print media, so too is digital for- matting different from print formatting. Thus there are significant formatting differences from the earlier edition. The font and page dimensions differ, as well as the page numbering, which is made to accord with the pagina- tion automatically assigned to multi-paged documents by most standard document-readers. Bas van Fraassen 2016 4 Contents Preface (1971)9 Introduction: Aim and Structure of Logical Theory 12 1 Mathematical Preliminaries 19 1.1 Intuitive Logic and Set Theory....... 19 1.2 Mathematical Structures.......... 23 1.3 Partial Order and Trees......... -
The Structure of Groups of Finitary Transformations
JOURNAL OF ALGEBRA 119, 4OC448 (1988) The Structure of Groups of Finitary Transformations RICHARD E. PHILLIPS Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 Communicated byPeter M. Neumann Received May 27, 1987 1. INTRODUCTION Let V be a vector space over a field X. A linear transformation g on I’ is called linitary if is a finite dimensional subspace of V, i.e., if the linear transformation g - 1 has finite dimensional range. The set of all such g which are also invertible is easily seen to be a group (a subgroup of GL( V, .X)) and will be called the group of$nitary transformations of V. We denote this group by FGL( V, X). It is not difficult to see that FGL(V, X) equals the set of all g E GL( V, X) such that C,(g)= {YE Vlug=o} has finite codimension in V. A X-linitary representation .of a group G is a homomorphism r from G into FGL(V, .X) for some X-space V. As usual, such a representation is faithful if ker(z) = { 1 }. Our aim is to determine which groups G (in some reasonable class) have faithful X-linitary representations (for various choices of X). Such a group G will be called a X-finitary linear group (or simply a tinitary linear group if the nature of the field X is irrelevant). Certainly, finite dimen- sional X-linear groups are X-tinitary linear (rather than use the customary term “linear” for groups with faithful representations of finite degree, we here use the term “finite dimensional linear”). -
Beyond First Order Logic: from Number of Structures to Structure of Numbers Part I
Bulletin of the Iranian Mathematical Society Vol. XX No. X (201X), pp XX-XX. BEYOND FIRST ORDER LOGIC: FROM NUMBER OF STRUCTURES TO STRUCTURE OF NUMBERS PART I JOHN BALDWIN, TAPANI HYTTINEN AND MEERI KESÄLÄ Communicated by Abstract. The paper studies the history and recent developments in non-elementary model theory focusing in the framework of ab- stract elementary classes. We discuss the role of syntax and seman- tics and the motivation to generalize first order model theory to non-elementary frameworks and illuminate the study with concrete examples of classes of models. This first part introduces the main conceps and philosophies and discusses two research questions, namely categoricity transfer and the stability classification. 1. Introduction Model theory studies classes of structures. These classes are usually a collection of structures that satisfy an (often complete) set of sentences of first order logic. Such sentences are created by closing a family of basic relations under finite conjunction, negation and quantification over individuals. Non-elementary logic enlarges the collection of sentences by allowing longer conjunctions and some additional kinds of quantification. In this paper we first describe for the general mathematician the history, key questions, and motivations for the study of non-elementary logics and distinguish it from first order model theory. We give more detailed examples accessible to model theorists of all sorts. We conclude with MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. Keywords: mathematical logic, model theory. Received: 30 April 2009, Accepted: 21 June 2010. ∗Corresponding author c 2011 Iranian Mathematical Society. 1 2 Baldwin, Hyttinen and Kesälä questions about countable models which require only a basic background in logic. -
A Concise Introduction to Mathematical Logic
Wolfgang Rautenberg A Concise Introduction to Mathematical Logic Textbook Third Edition Typeset and layout: The author Version from June 2009 corrections included Foreword by Lev Beklemishev, Moscow The field of mathematical logic—evolving around the notions of logical validity, provability, and computation—was created in the first half of the previous century by a cohort of brilliant mathematicians and philosophers such as Frege, Hilbert, Gödel, Turing, Tarski, Malcev, Gentzen, and some others. The development of this discipline is arguably among the highest achievements of science in the twentieth century: it expanded mathe- matics into a novel area of applications, subjected logical reasoning and computability to rigorous analysis, and eventually led to the creation of computers. The textbook by Professor Wolfgang Rautenberg is a well-written in- troduction to this beautiful and coherent subject. It contains classical material such as logical calculi, beginnings of model theory, and Gödel’s incompleteness theorems, as well as some topics motivated by applica- tions, such as a chapter on logic programming. The author has taken great care to make the exposition readable and concise; each section is accompanied by a good selection of exercises. A special word of praise is due for the author’s presentation of Gödel’s second incompleteness theorem, in which the author has succeeded in giving an accurate and simple proof of the derivability conditions and the provable Σ1-completeness, a technically difficult point that is usually omitted in textbooks of comparable level. This work can be recommended to all students who want to learn the foundations of mathematical logic. v Preface The third edition differs from the second mainly in that parts of the text have been elaborated upon in more detail. -
Arxiv:Cs/0202031V1
Nonmonotonic inference operations ∗ Michael Freund D´epartement de Math´ematiques, Universit´ed’Orl´eans, 45067 Orl´eans, C´edex 2 (France) Daniel Lehmann Department of Computer Science, Hebrew University, Jerusalem 91904 (Israel) Abstract A. Tarski [21] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the mo- notonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both hu- mans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting fami- lies of inference operations. This study of infinitary inference operations is inspired by the results of [11] on finitary nonmonotonic operations, but this paper is self-contained. 1 Introduction Since Tarski [21] and Gentzen [8], logicians have had to choose between two possible frameworks for the study of logics. A set L of formulas being given, one may model a logic by a mapping C :2 L −→ 2 L and for every subset X arXiv:cs/0202031v1 [cs.AI] 20 Feb 2002 of L, C(X) is understood to be the set of all the consequences of the set X of assumptions. But one may think that only finite sets of assumptions have well-defined consequences, in which case one generally represents a logic by a relation ⊢ between a finite set of formulas on the left and a single formula on the right. For a finite set A of formulas and a formula b, the relation A ⊢ b holds when b is a consequence of the finite set A of assumptions. -
BARWISE: INFINITARY LOGIC and ADMISSIBLE SETS Contents 0
The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 BARWISE: INFINITARY LOGIC AND ADMISSIBLE SETS H. JEROME KEISLER AND JULIA F. KNIGHT Contents 0. Introduction 1 1. Background on infinitary logic 2 1.1. Expressive power of L!1! 2 1.2. The back-and-forth construction 3 1.3. The Scott isomorphism theorem 4 1.4. !-logic 7 1.5. Familiar theorems 8 1.6. Failure of compactness 9 2. Background on admissible sets 10 2.1. ∆0 formulas and Σ-formulas in set theory 10 2.2. Axioms of KP 11 2.3. Examples of admissible sets 12 CK 2.4. The admissible set L(!1 ) 12 3. Admissible fragments 14 3.1. Completeness and compactness 14 3.2. Computable structures via Barwise compactness 15 3.3. Other applications of Barwise compactness 16 4. Admissible sets over 19 4.1. KP with urelementsM 19 4.2. Truncation lemma 20 4.3. Admissible sets above 21 4.4. Inductive definitionsM 21 5. Saturation properties 23 5.1. Computable saturation 23 5.2. ΣA-saturation 26 6. Conclusion 27 0. Introduction. In [15], Barwise described his graduate study at Stanford. x He told of his interactions with Kreisel and Scott, and said how he chose Feferman as his advisor. He began working on admissible fragments of infinitary logic after reading and giving seminar talks on two Ph.D. theses which had recently been c 0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00 1 2 H. JEROME KEISLER AND JULIA F. KNIGHT completed: that of Lopez-Escobar, at Berkeley, on infinitary logic [45] (see also the papers [46], [47]), and that of Platek [57], at Stanford, on admissible sets. -
Finitary Inductively Presented Logics by Solomon Feferman1 Departments of Mathematics and Philosophy Stanford University
Finitary inductively presented logics by Solomon Feferman1 Departments of Mathematics and Philosophy Stanford University Abstract Anotionoffinitary inductively presented (f.i.p.) logic is proposed here, which in- cludes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach may be used under (iii)for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way. What is a logic? The question here is not “What is logic?”, which (tendentiously) seeks to canonize some one distinguished system of reasoning as being the only true one. But also, here, we are not after any logic—only those that are syntactically described, or formal, as distinguished from those that are semantically described. For the latter, a reasonable general basic notion has evolved, that of model-theoretic logic; cf. e.g. Barwise-Feferman [1985]. Curiously (for a subject so devoted to foundational matters), there is no corresponding generally accepted basic notion for the formal logics. Such should cover as special cases propositional and predicate calculi of various kinds (classical, intuitionistic, many-valued, modal, temporal, deontic, relevance, etc.)and styles (Hilbert, Gentzen-natural deduction, Gentzen- sequential, linear, etc.), as well as equational calculi, lambda calculi, combinatory calculi (typed and untyped), and various applied logics (theories of arithmetic, algebraic systems, analysis, types, sets, etc.)and logics of programs. -
The Philosophies of Mathematics
Swarthmore College Works Philosophy Faculty Works Philosophy 2-1-2017 The Philosophies Of Mathematics Alan Richard Baker Swarthmore College, [email protected] Follow this and additional works at: https://works.swarthmore.edu/fac-philosophy Part of the Philosophy Commons Let us know how access to these works benefits ouy Recommended Citation Alan Richard Baker. (2017). "The Philosophies Of Mathematics". American Mathematical Monthly. Volume 124, Issue 2. 188-192. https://works.swarthmore.edu/fac-philosophy/459 This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Philosophy Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. The American Mathematical Monthly ISSN: 0002-9890 (Print) 1930-0972 (Online) Journal homepage: http://www.tandfonline.com/loi/uamm20 Reviews Jason Rosenhouse & Alan Baker To cite this article: Jason Rosenhouse & Alan Baker (2017) Reviews, The American Mathematical Monthly, 124:2, 188-192 To link to this article: https://doi.org/10.4169/amer.math.monthly.124.2.188 Published online: 13 Dec 2017. Submit your article to this journal Article views: 1 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=uamm20 REVIEWS Edited by Jason Rosenhouse Department of Mathematics and Statistics, James Madison University, Harrisonburg, VA 22807 The Philosophies of Mathematics Alan Baker What does it mean to assert a mathematical claim, for example that there is a prime between 5 and 10? If the claim is true, then what makes it true? And how do we come to know it in the first place? It is apparently basic questions such as these that drive the field of philosophy of mathematics.