Gödel on Finitism, Constructivity and Hilbert's Program
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Finitism, Divisibility, and the Beginning of the Universe: Replies to Loke and Dumsday
This is a preprint of an article whose final and definitive form will be published in the Australasian Journal of Philosophy. The Australasian Journal of Philosophy is available online at: http://www.tandf.co.uk/journals/. FINITISM, DIVISIBILITY, AND THE BEGINNING OF THE UNIVERSE: REPLIES TO LOKE AND DUMSDAY Stephen Puryear North Carolina State University Some philosophers contend that the past must be finite in duration, because otherwise reaching the present would have involved the sequential occurrence of an actual infinity of events, which they regard as impossible. I recently developed a new objection to this finitist argument, to which Andrew Ter Ern Loke and Travis Dumsday have replied. Here I respond to the three main points raised in their replies. Keywords: Space, time, continuity, infinity, creation 1. Introduction Some philosophers argue that the universe must have had a beginning, because otherwise reaching the present would have required something impossible, namely, the occurrence or ‘traversal’ of an actually infinite sequence of events. But this argument faces the objection that actual infinities are traversed all the time: for example, whenever a finite period of time elapses [Morriston 2002: 162; Sinnott-Armstrong 2004: 42–3]. Let us call these the finitist argument and the Zeno objection. In response to the Zeno objection, the most ardent proponent of the finitist argument in our day, William Lane Craig [Craig and Sinclair 2009: 112–13, 119; cf. Craig and Smith 1993: 27–30], has taken the position that ‘time is logically prior to the divisions we make within it’. On this view, time divides into parts only in so far as we divide it, and thus divides into only a finite number of parts. -
Aristotelian Finitism
Synthese DOI 10.1007/s11229-015-0827-9 S.I. : INFINITY Aristotelian finitism Tamer Nawar1 Received: 12 January 2014 / Accepted: 25 June 2015 © Springer Science+Business Media Dordrecht 2015 Abstract It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments con- sidered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logi- cal’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed. Keywords Aristotle · Aristotelian commentators · Infinity · Mathematics · Metaphysics 1 Introduction It is widely known that Aristotle embraced some sort of finitism and denied the exis- tence of so-called ‘actual infinities’ while allowing for the existence of ‘potential infinities’. It is difficult to overestimate the influence of Aristotle’s views on this score and the denial of the (actual) existence of infinities became a commonplace among philosophers for over two thousand years. However, the precise grounds for Aristo- tle’s finitism have not been discussed in much detail and, insofar as they have received attention, his reasons for ruling out the existence of (actual) infinities have often been B Tamer Nawar [email protected] 1 University of Oxford, 21 Millway Close, Oxford OX2 8BJ, UK 123 Synthese deemed obscure or ad hoc (e.g. -
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 . -
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. -
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. -
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. -
Project Abstract the Summer Program for Diversity in Logic for Undergraduates Builds Upon the PIKSI Summer Program Model, Focusi
Project Abstract The Summer Program for Diversity in Logic for Undergraduates builds upon the PIKSI Summer Program Model, focusing on Logic, an area in philosophy needing to increase diversity. We request seed funding for the pilot run of the program in May of 2016, when we will offer 12 students the opportunity to explore an exciting research theme in Logic –Paradoxes—receive small-group tutoring in formal techniques, receive mentoring and support for professionalization, experience validation, understanding and advice regarding diversity issues they have encountered –sexism, racism, ableism (and which the students may worry about encountering them in the profession), and develop a sense of community with students and faculty with whom they can identify and, in turn, come to strengthen their own identities. Project Purpose Our goal is to empower students to conceive of themselves as aspiring logicians, philosophers of logic and formal philosophers who belong in our profession. As Audrey Yap has noted, a majority of philosophy majors encounter some logic as a part of their undergraduate curriculum. Increasing diversity in logic is not simply a problem of exposure, but concretely addressing underlying pressures women and minority students experience, particularly stereotype threat and pernicious ideas about “natural aptitude.” (1) These pressures are especially strong in the subfield of logic, where almost no women and minorities have contributed to the research literature until very recently, and women and minorities remain underrepresented to a higher degree than in philosophy more broadly. Women and minorities learning logic can be vulnerable to feeling that a field like logic, that tends to be male and white dominated on the whole, is not welcoming to them. -
THE LOOICAL ATOMISM J. D. MACKENZIE Date Submitted Thesis
THE LOOICAL ATOMISM OF F. P. RAMSEY J. D. MACKENZIE Date submitted Thesis submitted for the Degree of Master of Arts in the School of Philosophy University of New South Wales (i) SYNOPSIS The first Chapter sets Ramsey in histor:iealperspective as a Logical Atomist. Chapter Two is concerned with the impasse in which Russell found himself ,d.th general propositions, Wittgenstein's putative solution in terms of his Doctrine of Showing, and Ramsey's "Wittgensteinian" solution, which is not satisfactory. An attempt is then ma.de to describe a Ramseian solution on the basis of what he says about the Axiom of Infi- nity, and to criticize this solution. In Chapter Three Ramsay's objections to the Pl4 definition of identity are considered, and consequences of his rejection of that definition for the Theory of Classes and the Axiom of Choice are drawn. In Chapter Four, Ramsey•s modifications to Russell's Theory of Types are discussed. His division of the Paradoxes into two groups is defended, but his redefinition of 'predicative' is rejected. Chapter Five deals with Ra.msey's analysis of propositional attitudes and negative propositions, and Chapter Six considers the dispute between Russell and Ramsey over the nature and status of universals. In Chapter Seven, the conclusions are summarized, and Ramsay's contribution to Logical Atom.ism are assessed. His main fail ing is found to be his lack of understanding of impossibility, especially with regard to the concept of infinity. (ii) PREFACE The thesis is divided into chapters, which are in turn divided into sections. -
Lorenzen's Proof of Consistency for Elementary Number Theory [With An
Lorenzen’s proof of consistency for elementary number theory Thierry Coquand Computer science and engineering department, University of Gothenburg, Sweden, [email protected]. Stefan Neuwirth Laboratoire de mathématiques de Besançon, Université Bourgogne Franche-Comté, France, [email protected]. Abstract We present a manuscript of Paul Lorenzen that provides a proof of con- sistency for elementary number theory as an application of the construc- tion of the free countably complete pseudocomplemented semilattice over a preordered set. This manuscript rests in the Oskar-Becker-Nachlass at the Philosophisches Archiv of Universität Konstanz, file OB 5-3b-5. It has probably been written between March and May 1944. We also com- pare this proof to Gentzen’s and Novikov’s, and provide a translation of the manuscript. arXiv:2006.08996v1 [math.HO] 16 Jun 2020 Keywords: Paul Lorenzen, consistency of elementary number the- ory, free countably complete pseudocomplemented semilattice, inductive definition, ω-rule. We present a manuscript of Paul Lorenzen that arguably dates back to 1944 and provide an edition and a translation, with the kind permission of Lorenzen’s daughter, Jutta Reinhardt. It provides a constructive proof of consistency for elementary number theory by showing that it is a part of a trivially consistent cut-free calculus. The proof resorts only to the inductive definition of formulas and theorems. More precisely, Lorenzen proves the admissibility of cut by double induction, on the complexity of the cut formula and of the derivations, without using any ordinal assignment, contrary to the presentation of cut elimination in most standard texts on proof theory. -
Hunting the Story of Moses Schönfinkel
Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel Stephen Wolfram* Combinators were a key idea in the development of mathematical logic and the emergence of the concept of universal computation. They were introduced on December 7, 1920, by Moses Schönfinkel. This is an exploration of the personal story and intellectual context of Moses Schönfinkel, including extensive new research based on primary sources. December 7, 1920 On Tuesday, December 7, 1920, the Göttingen Mathematics Society held its regular weekly meeting—at which a 32-year-old local mathematician named Moses Schönfinkel with no known previous mathematical publications gave a talk entitled “Elemente der Logik” (“Elements of Logic”). This piece is included in S. Wolfram (2021), Combinators: A Centennial View, Wolfram Media. (wolframmedia.com/products/combinators-a-centennial-view.html) and accompanies arXiv:2103.12811 and arXiv:2102.09658. Originally published December 7, 2020 *Email: [email protected] 2 | Stephen Wolfram A hundred years later what was presented in that talk still seems in many ways alien and futuristic—and for most people almost irreducibly abstract. But we now realize that that talk gave the first complete formalism for what is probably the single most important idea of this past century: the idea of universal computation. Sixteen years later would come Turing machines (and lambda calculus). But in 1920 Moses Schönfinkel presented what he called “building blocks of logic”—or what we now call “combinators”—and then proceeded to show that by appropriately combining them one could effectively define any function, or, in modern terms, that they could be used to do universal computation. -
Common Sense for Concurrency and Strong Paraconsistency Using Unstratified Inference and Reflection
Published in ArXiv http://arxiv.org/abs/0812.4852 http://commonsense.carlhewitt.info Common sense for concurrency and strong paraconsistency using unstratified inference and reflection Carl Hewitt http://carlhewitt.info This paper is dedicated to John McCarthy. Abstract Unstratified Reflection is the Norm....................................... 11 Abstraction and Reification .............................................. 11 This paper develops a strongly paraconsistent formalism (called Direct Logic™) that incorporates the mathematics of Diagonal Argument .......................................................... 12 Computer Science and allows unstratified inference and Logical Fixed Point Theorem ........................................... 12 reflection using mathematical induction for almost all of Disadvantages of stratified metatheories ........................... 12 classical logic to be used. Direct Logic allows mutual Reification Reflection ....................................................... 13 reflection among the mutually chock full of inconsistencies Incompleteness Theorem for Theories of Direct Logic ..... 14 code, documentation, and use cases of large software systems Inconsistency Theorem for Theories of Direct Logic ........ 15 thereby overcoming the limitations of the traditional Tarskian Consequences of Logically Necessary Inconsistency ........ 16 framework of stratified metatheories. Concurrency is the Norm ...................................................... 16 Gödel first formalized and proved that it is not possible -
On Proofs of the Consistency of Arithmetic
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 5 (18) 2002 Roman Murawski Adam Mickiewicz University, Pozna´n ON PROOFS OF THE CONSISTENCY OF ARITHMETIC 1. The main aim and purpose of Hilbert’s programme was to defend the integrity of classical mathematics (refering to the actual infinity) by showing that it is safe and free of any inconsistencies. This problem was formulated by him for the first time in his lecture at the Second International Congress of Mathematicians held in Paris in August 1900 (cf. Hilbert, 1901). Among twenty three problems Hilbert mentioned under number 2 the problem of proving the consistency of axioms of arithmetic (under the name “arithmetic” Hilbert meant number theory and analysis). Hilbert returned to the problem of justification of mathematics in lectures and papers, especially in the twentieth 1, where he tried to describe and to explain the problem more precisely (in particular the methods allowed to be used) and simultaneously presented the partial solutions obtained by his students. Hilbert distinguished between the unproblematic, finitistic part of mathematics and the infinitistic part that needed justification. Finitistic mathematics deals with so called real sentences, which are completely meaningful because they refer only to given concrete objects. Infinitistic mathematics on the other hand deals with so called ideal sentences that contain reference to infinite totalities. It should be justified by finitistic methods – only they can give it security (Sicherheit). Hilbert proposed to base mathematics on finitistic mathematics via proof theory (Beweistheorie). It should be shown that proofs which use ideal elements in order to prove results in the real part of mathematics always yield correct results, more exactly, that (1) finitistic mathematics is conservative over finitistic mathematics with respect to real sentences and (2) the infinitistic 1 More information on this can be found for example in (Mancosu, 1998).