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# Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic

History and Philosophy of Logic 23 (1):1-30 (2002)

 Fraenkel-Carnap properties.G. Au George Weaver - 2005 - Mathematical Logic Quarterly 51 (3):285.In the 1920's Fraenkel and Carnap raised the question of whether or not every finitely axiomatizable semantically complete theory formulated in the theory of types is categorical. Partial answers to this and a related question are presented for theories formulated in second-order logic. Download     Export citation     Bookmark   4 citations Fraenkel–Carnap Questions for Equivalence Relations.George Weaver & Irena Penev - 2011 - Australasian Journal of Logic 10:52-66.An equivalence is a binary relational system A = (A,ϱA) where ϱA is an equivalence relation on A. A simple expansion of an equivalence is a system of the form (Aa1…an) were A is an equivalence and a1,…,an are members of A. It is shown that the Fraenkel-Carnap question when restricted to the class of equivalences or to the class of simple expansions of equivalences has a positive answer: that the complete second-order theory of such a system is categorical, if (...) Download     Export citation     Bookmark   2 citations A General Setting for Dedekind's Axiomatization of the Positive Integers.George Weaver - 2011 - History and Philosophy of Logic 32 (4):375-398.A Dedekind algebra is an ordered pair (B, h), where B is a non-empty set and h is a similarity transformation on B. Among the Dedekind algebras is the sequence of the positive integers. From a contemporary perspective, Dedekind established that the second-order theory of the sequence of the positive integers is categorical and finitely axiomatizable. The purpose here is to show that this seemingly isolated result is a consequence of more general results in the model theory of second-order languages. (...) Download     Export citation     Bookmark Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.We try to answer the question which is the “right” foundation of mathematics, second order logic or set theory. Since the former is usually thought of as a formal language and the latter as a first order theory, we have to rephrase the question. We formulate what we call the second order view and a competing set theory view, and then discuss the merits of both views. On the surface these two views seem to be in manifest conflict with each (...) Download     Export citation     Bookmark   17 citations Tracing Internal Categoricity.Jouko Väänänen - 2021 - Theoria 87 (4):986-1000.Theoria, Volume 87, Issue 4, Page 986-1000, August 2021. Download     Export citation     Bookmark   2 citations Concept Formation and Scientific Objectivity: Weyl’s Turn against Husserl.Iulian D. Toader - 2013 - Hopos: The Journal of the International Society for the History of Philosophy of Science 3 (2):281-305.This paper argues that Weyl's view that scientific objectivity requires that concepts be freely created, i.e., introduced via Hilbert-style axiomatizations, led him to abandon the phenomenological view of objectivity. Download     Export citation     Bookmark   4 citations Book reviews. [REVIEW]John Symons - 2008 - Studia Logica 89 (2):285-289. Download     Export citation     Bookmark Completeness and Categoricity, Part II: Twentieth-Century Metalogic to Twenty-first-Century Semantics.Steve Awodey & Erich H. Reck - 2002 - History and Philosophy of Logic 23 (2):77-94.This paper is the second in a two-part series in which we discuss several notions of completeness for systems of mathematical axioms, with special focus on their interrelations and historical origins in the development of the axiomatic method. We argue that, both from historical and logical points of view, higher-order logic is an appropriate framework for considering such notions, and we consider some open questions in higher-order axiomatics. In addition, we indicate how one can fruitfully extend the usual set-theoretic semantics (...) Download     Export citation     Bookmark   16 citations Logic in the 1930s: Type Theory and Model Theory.Georg Schiemer & Erich H. Reck - 2013 - Bulletin of Symbolic Logic 19 (4):433-472.In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style ofPrincipia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of (...) Download     Export citation     Bookmark   5 citations Carnap’s Early Semantics.Georg Schiemer - 2013 - Erkenntnis 78 (3):487-522.This paper concerns Carnap’s early contributions to formal semantics in his work on general axiomatics between 1928 and 1936. Its main focus is on whether he held a variable domain conception of models. I argue that interpreting Carnap’s account in terms of a fixed domain approach fails to describe his premodern understanding of formal models. By drawing attention to the second part of Carnap’s unpublished manuscript Untersuchungen zur allgemeinen Axiomatik, an alternative interpretation of the notions ‘model’, ‘model extension’ and ‘submodel’ (...) Download     Export citation     Bookmark   12 citations Carnap’s early metatheory: scope and limits.Georg Schiemer, Richard Zach & Erich Reck - 2017 - Synthese 194 (1):33-65.In Untersuchungen zur allgemeinen Axiomatik and Abriss der Logistik, Carnap attempted to formulate the metatheory of axiomatic theories within a single, fully interpreted type-theoretic framework and to investigate a number of meta-logical notions in it, such as those of model, consequence, consistency, completeness, and decidability. These attempts were largely unsuccessful, also in his own considered judgment. A detailed assessment of Carnap’s attempt shows, nevertheless, that his approach is much less confused and hopeless than it has often been made out to (...) Download     Export citation     Bookmark   4 citations Carnap on extremal axioms, "completeness of the models," and categoricity.Georg Schiemer - 2012 - Review of Symbolic Logic 5 (4):613-641.This paper provides a historically sensitive discussion of Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnapcompleteness of the modelss different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time. Download     Export citation     Bookmark   9 citations Axioms in Mathematical Practice.Dirk Schlimm - 2013 - Philosophia Mathematica 21 (1):37-92.On the basis of a wide range of historical examples various features of axioms are discussed in relation to their use in mathematical practice. A very general framework for this discussion is provided, and it is argued that axioms can play many roles in mathematics and that viewing them as self-evident truths does not do justice to the ways in which mathematicians employ axioms. Possible origins of axioms and criteria for choosing axioms are also examined. The distinctions introduced aim at (...) Download     Export citation     Bookmark   18 citations On the Axiom of Canonicity.Jerzy Pogonowski - forthcoming - Logic and Logical Philosophy:1-29.The axiom of canonicity was introduced by the famous Polish logician Roman Suszko in 1951 as an explication of Skolem's Paradox and a precise representation of the axiom of restriction in set theory proposed much earlier by Abraham Fraenkel. We discuss the main features of Suszko's contribution and hint at its possible further applications. Download     Export citation     Bookmark   1 citation A Note on Intended and Standard Models.Jerzy Pogonowski - 2020 - Studia Humana 9 (3-4):131-139.This note discusses some problems concerning intended, standard, and nonstandard models of mathematical theories. We pay attention to the role of extremal axioms in attempts at a unique characterization of the intended models. We recall also Jan Woleński’s views on these issues. Download     Export citation     Bookmark   1 citation Russell's completeness proof.Peter Milne - 2008 - History and Philosophy of Logic 29 (1):31-62.Bertrand Russell’s 1906 article ‘The Theory of Implication’ contains an algebraic weak completeness proof for classical propositional logic. Russell did not present it as such. We give an exposition of the proof and investigate Russell’s view of what he was about, whether he could have appreciated the proof for what it is, and why there is no parallel of the proof in Principia Mathematica. Download     Export citation     Bookmark American Postulate Theorists and Alfred Tarski.Michael Scanlan - 2003 - History and Philosophy of Logic 24 (4):307-325.This article outlines the work of a group of US mathematicians called the American Postulate Theorists and their influence on Tarski's work in the 1930s that was to be foundational for model theory. The American Postulate Theorists were influenced by the European foundational work of the period around 1900, such as that of Peano and Hilbert. In the period roughly from 1900???1940, they developed an indigenous American approach to foundational investigations. This made use of interpretations of precisely formulated axiomatic theories (...) Download     Export citation     Bookmark   10 citations The development of mathematical logic from Russell to Tarski, 1900-1935.Paolo Mancosu, Richard Zach & Calixto Badesa - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press.The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) Download     Export citation     Bookmark   25 citations Submodels in Carnap’s Early Axiomatics Revisited.Iris Loeb - 2014 - Erkenntnis 79 (2):405-429.G. Schiemer has recently ascribed to Carnap the so-called domains-as-fields conception of models, which he subsequently used to defend Carnap’s treatment of extremal axioms against J. Hintikka’s criticism that the number of tuples in a relation, and not the domain of discourse, is optimised in Carnap’s treatment. We will argue by a careful textual analysis, however, that this domains-as-fields conception cannot be applied to Carnap’s early semantics, because it includes a notion of submodel and subrelation that is not only absent (...) Download     Export citation     Bookmark   3 citations What is Tarski's common concept of consequence?Ignacio Jané - 2006 - Bulletin of Symbolic Logic 12 (1):1-42.In 1936 Tarski sketched a rigorous definition of the concept of logical consequence which, he claimed, agreed quite well with common usage-or, as he also said, with the common concept of consequence. Commentators of Tarski's paper have usually been elusive as to what this common concept is. However, being clear on this issue is important to decide whether Tarski's definition failed (as Etchemendy has contended) or succeeded (as most commentators maintain). I argue that the common concept of consequence that Tarski (...) Download     Export citation     Bookmark   15 citations Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...) Download     Export citation     Bookmark   7 citations Zermelo: definiteness and the universe of definable sets.Heinz-Dieter Ebbinghaus - 2003 - History and Philosophy of Logic 24 (3):197-219.Using hitherto unpublished manuscripts from the Zermelo Nachlass, I describe the development of the notion of definiteness and the discussion about it, giving a conclusive picture of Zermelo's thoughts up to the late thirties. As it turns out, Zermelo's considerations about definiteness are intimately related to his concept of a Cantorian universe of categorically definable sets that may be considered an inner model of set theory in an ideationally given universe of classes. Download     Export citation     Bookmark   8 citations What are Implicit Definitions?Eduardo N. Giovannini & Georg Schiemer - 2019 - Erkenntnis 86 (6):1661-1691.The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) Download     Export citation     Bookmark   2 citations Second-order characterizable cardinals and ordinals.Benjamin R. George - 2006 - Studia Logica 84 (3):425 - 449.The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed. Download     Export citation     Bookmark   2 citations Second-Order Characterizable Cardinals and Ordinals.Benjamin R. George - 2006 - Studia Logica 84 (3):425-449.The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the second-order theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordinal characterizability and ordinal arithmetic. The broader significance of cardinal characterizability and the relationships between different notions of characterizability are also discussed. Download     Export citation     Bookmark   2 citations Pasch entre Klein et Peano.Sébastien Gandon - 2005 - Dialogue 44 (4):653-692.RÉSUMÉ: Pasch est généralement considéré comme le premier à avoir proposé une axiomatisation de la géométrie. Mais ses Vorlesungen über neure Geometrie (1882) contiennent plusieurs éléments étrangers au paradigme hilbertien. Pasch soutient ainsi que la « géométrie élémentaire », dont il propose une axiomatisation complète, est une théorie empiriquement vraie. Les commentateurs considèrent généralement les différences entre la méthode de Pasch et celle qui deviendra standard après Hilbert comme autant de défauts affectant une pensée encore inaboutie. Notre but consiste au (...) Download     Export citation     Bookmark   6 citations On arbitrary sets and ZFC.José Ferreirós - 2011 - Bulletin of Symbolic Logic 17 (3):361-393.Set theory deals with the most fundamental existence questions in mathematics—questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels of quasi-combinatorialism or combinatorial maximality. After explaining what (...) Download     Export citation     Bookmark   7 citations Projective duality and the rise of modern logic.Günther Eder - 2021 - Bulletin of Symbolic Logic 27 (4):351-384.The symmetries between points and lines in planar projective geometry and between points and planes in solid projective geometry are striking features of these geometries that were extensively discussed during the nineteenth century under the labels “duality” or “reciprocity.” The aims of this article are, first, to provide a systematic analysis of duality from a modern point of view, and, second, based on this, to give a historical overview of how discussions about duality evolved during the nineteenth century. Specifically, we (...) Download     Export citation     Bookmark Axiomatizations of arithmetic and the first-order/second-order divide.Catarina Dutilh Novaes - 2019 - Synthese 196 (7):2583-2597.It is often remarked that first-order Peano Arithmetic is non-categorical but deductively well-behaved, while second-order Peano Arithmetic is categorical but deductively ill-behaved. This suggests that, when it comes to axiomatizations of mathematical theories, expressive power and deductive power may be orthogonal, mutually exclusive desiderata. In this paper, I turn to Hintikka’s :69–90, 1989) distinction between descriptive and deductive approaches in the foundations of mathematics to discuss the implications of this observation for the first-order logic versus second-order logic divide. The descriptive (...) Download     Export citation     Bookmark   1 citation Structure and Categoricity: Determinacy of Reference and Truth Value in the Philosophy of Mathematics.Tim Button & Sean Walsh - 2016 - Philosophia Mathematica 24 (3):283-307.This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory. Download     Export citation     Bookmark   8 citations The Scope of Gödel’s First Incompleteness Theorem.Bernd Buldt - 2014 - Logica Universalis 8 (3-4):499-552.Guided by questions of scope, this paper provides an overview of what is known about both the scope and, consequently, the limits of Gödel’s famous first incompleteness theorem. Download     Export citation     Bookmark   1 citation Peano’s structuralism and the birth of formal languages.Joan Bertran-San-Millán - 2022 - Synthese 200 (4):1-34.Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) Download     Export citation     Bookmark Husserl, Model Theory, and Formal Essences.Kyle Banick - 2020 - Husserl Studies 37 (2):103-125.Husserl’s philosophy of mathematics, his metatheory, and his transcendental phenomenology have a sophisticated and systematic interrelation that remains relevant for questions of ontology today. It is well established that Husserl anticipated many aspects of model theory. I focus on this aspect of Husserl’s philosophy in order to argue that Thomasson’s recent pleonastic reconstruction of Husserl’s approach to essences is incompatible with Husserl’s philosophy as a whole. According to the pleonastic approach, Husserl can appeal to essences in the absence of a (...) Download     Export citation     Bookmark   1 citation Completeness and categoricity (in power): Formalization without foundationalism.John T. Baldwin - 2014 - Bulletin of Symbolic Logic 20 (1):39-79.We propose a criterion to regard a property of a theory as virtuous: the property must have significant mathematical consequences for the theory. We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of (...) Download     Export citation     Bookmark   2 citations Axiomatizing Changing Conceptions of the Geometric Continuum II: Archimedes-Descartes-Hilbert-Tarski†.John T. Baldwin - 2019 - Philosophia Mathematica 27 (1):33-60.In Part I of this paper we argued that the first-order systems HP5 and EG are modest complete descriptive axiomatization of most of Euclidean geometry. In this paper we discuss two further modest complete descriptive axiomatizations: Tarksi’s for Cartesian geometry and new systems for adding $$\pi$$. In contrast we find Hilbert’s full second-order system immodest for geometrical purposes but appropriate as a foundation for mathematical analysis. Download     Export citation     Bookmark   2 citations Completeness: From Husserl to Carnap.Víctor Aranda - 2022 - Logica Universalis 16 (1):57-83.In his Doppelvortrag, Edmund Husserl introduced two concepts of “definiteness” which have been interpreted as a vindication of his role in the history of completeness. Some commentators defended that the meaning of these notions should be understood as categoricity, while other scholars believed that it is closer to syntactic completeness. A detailed study of the early twentieth-century axiomatics and Husserl’s Doppelvortrag shows, however, that many concepts of completeness were conflated as equivalent. Although “absolute definiteness” was principally an attempt to characterize (...) Download     Export citation     Bookmark Alfred Tarski: philosophy of language and logic.Douglas Patterson - 2012 - New York: Palgrave-Macmillan.This study looks to the work of Tarski's mentors Stanislaw Lesniewski and Tadeusz Kotarbinski, and reconsiders all of the major issues in Tarski scholarship in light of the conception of Intuitionistic Formalism developed: semantics, truth, paradox, logical consequence. Download     Export citation     Bookmark   19 citations Logic and Philosophy of Mathematics in the Early Husserl.Stefania Centrone - 2009 - Dordrecht, Netherland: Springer.This volume will be of particular interest to researchers working in the history, and in the philosophy, of logic and mathematics, and more generally, to ... Download     Export citation     Bookmark   23 citations We show how removing faith-based beliefs in current philosophies of classical and constructive mathematics admits formal, evidence-based, definitions of constructive mathematics; of a constructively well-defined logic of a formal mathematical language; and of a constructively well-defined model of such a language. -/- We argue that, from an evidence-based perspective, classical approaches which follow Hilbert's formal definitions of quantification can be labelled theistic'; whilst constructive approaches based on Brouwer's philosophy of Intuitionism can be labelled atheistic'. -/- We then adopt what may (...) Download     Export citation     Bookmark Hermann Cohen’s History and Philosophy of Science.Lydia Patton - 2004 - Dissertation, Mcgill UniversityIn my dissertation, I present Hermann Cohen's foundation for the history and philosophy of science. My investigation begins with Cohen's formulation of a neo-Kantian epistemology. I analyze Cohen's early work, especially his contributions to 19th century debates about the theory of knowledge. I conclude by examining Cohen's mature theory of science in two works, The Principle of the Infinitesimal Method and its History of 1883, and Cohen's extensive 1914 Introduction to Friedrich Lange's History of Materialism. In the former, Cohen gives (...) Download     Export citation     Bookmark   9 citations Objects and objectivity : Alternatives to mathematical realism.Ebba Gullberg - 2011 - Dissertation, Umeå UniversitetThis dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes (...) Download     Export citation     Bookmark Hilbert, completeness and geometry.Giorgio Venturi - 2011 - Rivista Italiana di Filosofia Analitica Junior 2 (2):80-102.This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how (...) Download     Export citation     Bookmark