DOCSLIB.ORG

Categorical Semantics of Constructive Set Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Categorical semantics of constructive set theory Beim Fachbereich Mathematik der Technischen Universit¨atDarmstadt eingereichte Habilitationsschrift von Benno van den Berg, PhD aus Emmen, die Niederlande 2 Contents 1 Introduction to the thesis 7 1.1 Logic and metamathematics . 7 1.2 Historical intermezzo . 8 1.3 Constructivity . 9 1.4 Constructive set theory . 11 1.5 Algebraic set theory . 15 1.6 Contents . 17 1.7 Warning concerning terminology . 18 1.8 Acknowledgements . 19 2 A unified approach to algebraic set theory 21 2.1 Introduction . 21 2.2 Constructive set theories . 24 2.3 Categories with small maps . 25 2.3.1 Axioms . 25 2.3.2 Consequences . 29 2.3.3 Strengthenings . 31 2.3.4 Relation to other settings . 32 2.4 Models of set theory . 33 2.5 Examples . 35 2.6 Predicative sheaf theory . 36 2.7 Predicative realizability . 37 3 Exact completion 41 3.1 Introduction . 41 3 4 CONTENTS 3.2 Categories with small maps . 45 3.2.1 Classes of small maps . 46 3.2.2 Classes of display maps . 51 3.3 Axioms for classes of small maps . 55 3.3.1 Representability . 55 3.3.2 Separation . 55 3.3.3 Power types . 55 3.3.4 Function types . 57 3.3.5 Inductive types . 58 3.3.6 Infinity . 60 3.3.7 Fullness . 61 3.4 Exactness and its applications . 63 3.5 Exact completion . 66 3.6 Stability properties of axioms for small maps . 73 3.6.1 Representability . 74 3.6.2 Separation . 74 3.6.3 Power types . 74 3.6.4 Function types . 78 3.6.5 Inductive types . 80 3.6.6 Infinity . 87 3.6.7 Fullness . 87 3.7 Soundness . 91 3.8 Completeness . 96 3.9 Set-theoretic axioms . 100 3.10 Positive Heyting categories . 101 4 Realizability 103 4.1 Introduction . 103 4.2 The category of assemblies . 109 4.3 The predicative realizability category . 117 4.4 Additional axioms . 120 4.5 Realizability models for set theory . 127 CONTENTS 5 4.6 A model of CZF in which all sets are subcountable . 131 4.7 Set-theoretic axioms . 137 4.7.1 Axioms of IZF . 137 4.7.2 Axioms of CZF . 138 4.7.3 Constructivist principles . 138 4.8 Predicative categories with small maps . 139 5 Sheaves 143 5.1 Introduction . 143 5.2 Preliminaries . 147 5.2.1 Review of Algebraic Set Theory . 147 5.2.2 W-types . 152 5.2.3 Fullness . 155 5.2.4 Axiom of multiple choice . 157 5.2.5 Main results . 158 5.3 Presheaves . 159 5.3.1 Pointwise small maps in presheaves . 159 5.3.2 Locally small maps in presheaves . 163 5.4 Sheaves . 172 5.4.1 Sheafification . 173 5.4.2 Small maps in sheaves . 175 5.4.3 Collection and fullness in sheaves . 181 5.4.4 W-types in sheaves . 186 5.5 Sheaf models of constructive set theory . 192 6 Applications: derived rules 195 6.1 Introduction . 195 6.2 Constructive set theory . 197 6.2.1 CZF . 197 6.2.2 Inductive definitions in CZF . 198 6.3 Predicative locale theory . 199 6.3.1 Formal spaces . 199 6 CONTENTS 6.3.2 Inductively generated formal topologies . 200 6.3.3 Formal Cantor space . 201 6.3.4 Formal Baire space . 202 6.3.5 Points of a formal space . 204 6.3.6 Morphisms of formal spaces . 206 6.3.7 Double construction . 207 6.4 Sheaf models . 208 6.4.1 Choice principles . 212 6.4.2 Brouwer ordinals . 213 6.5 Main results . 217 7 Note on the Axiom of Multiple Choice 221 7.1 Introduction . 221 7.2 The Set Compactness Theorem . 223 7.3 Stability under exact completion . 227 7.4 Stability under realizability . 230 7.5 Stability under presheaves . 232 7.6 Stability under sheaves . 234 7.7 Relation of AMC to other axioms . 238 7.8 Conclusion . 240 8 Ideas on constructive set theory 241 8.1 Constructive set theory: the very idea . 241 8.2 Constructive set theory as a constructive formal theory . 244 8.3 Constructive set theory as a foundation for formal topology . 245 8.4 Constructive set theory as a predicative formal theory . 247 8.5 The proof theory of constructive set theory . 249 Chapter 1 Introduction to the thesis 1.1 Logic and metamathematics Many different topics are studied within mathematical logic, but for the purposes of this thesis we may say that logic is concerned with mathematical proofs. Although every mathematician knows what a mathematical proof is, it might be good to spend a few more words on the topic. When a mathematician publishes a proof, he1 typically takes a few known the- orems and uses generally accepted proof principles to derive from them a new, un- known, result. The theorems on which he bases his argument have been obtained in the same way and if one would start to trace back what the assumptions are on which these in turn are based, then ultimately one will find a few fundamental axioms con- cerning sets on which all mathematical results are based. It has been laid down very precisely which fundamental set-theoretic axioms are allowed to occur at the end, and which proof principles may be used to derive results from them. So precisely in fact, that they can be captured in a formal system and be studied mathematically: this is what is done in “metamathematics.” The formal system which captures the commonly accepted axioms and methods of proof has been called ZFC, for Zermelo-Fraenkel set theory with the Axiom of Choice. Since this system is an (almost) universally acknowledged foundation for mathematics, one obtains insights into the strengths and limitations of the usual methods of proof by studying ZFC. Two classic results in this respect are: • G¨odel’sIncompleteness Theorems (1931). G¨odel’sFirst Incompleteness Theo- rem implies that ZFC is incomplete, meaning that there are statements that ZFC can neither prove nor refute (unless it is inconsistent, i.e., proves con- tradictory statements). G¨odel’sSecond Incompleteness Theorem shows that a formal statement expressing the consistency of ZFC in ZFC is an example of a statement which ZFC cannot decide. 1Throughout this introduction, please read “she or he” for “he” and “her or his” for “his”. 7 8 CHAPTER 1. INTRODUCTION TO THE THESIS • The independence of the continuum hypothesis. The work of G¨odel(published in 1940) on the constructible hierarchy and that of Cohen (published in 1963) on forcing together imply that the continuum hypothesis is another example of a statement which cannot be decided in ZFC. (The continuum hypothesis says that every subset of the real line is either countable or has the same cardinality as the real line.) 1.2 Historical intermezzo Nowadays, the idea that all of mathematics can be captured in a formal system and that there are a few basic set-theoretic assumptions from which it all follows using only logic, may seem obvious. Nevertheless, it was not always like this and this state of affairs was the result of a long and complicated development which took place during the second half of the nineteenth and the first half of.
Recommended publications
  • Categories of Sets with a Group Action

    Categories of Sets with a Group Action

    Categories of sets with a group action Bachelor Thesis of Joris Weimar under supervision of Professor S.J. Edixhoven Mathematisch Instituut, Universiteit Leiden Leiden, 13 June 2008 Contents 1 Introduction 1 1.1 Abstract . .1 1.2 Working method . .1 1.2.1 Notation . .1 2 Categories 3 2.1 Basics . .3 2.1.1 Functors . .4 2.1.2 Natural transformations . .5 2.2 Categorical constructions . .6 2.2.1 Products and coproducts . .6 2.2.2 Fibered products and fibered coproducts . .9 3 An equivalence of categories 13 3.1 G-sets . 13 3.2 Covering spaces . 15 3.2.1 The fundamental group . 15 3.2.2 Covering spaces and the homotopy lifting property . 16 3.2.3 Induced homomorphisms . 18 3.2.4 Classifying covering spaces through the fundamental group . 19 3.3 The equivalence . 24 3.3.1 The functors . 25 4 Applications and examples 31 4.1 Automorphisms and recovering the fundamental group . 31 4.2 The Seifert-van Kampen theorem . 32 4.2.1 The categories C1, C2, and πP -Set ................... 33 4.2.2 The functors . 34 4.2.3 Example . 36 Bibliography 38 Index 40 iii 1 Introduction 1.1 Abstract In the 40s, Mac Lane and Eilenberg introduced categories. Although by some referred to as abstract nonsense, the idea of categories allows one to talk about mathematical objects and their relationions in a general setting. Its origins lie in the field of algebraic topology, one of the topics that will be explored in this thesis. First, a concise introduction to categories will be given.
  • Sets in Homotopy Type Theory

    Sets in Homotopy Type Theory

    Predicative topos Sets in Homotopy type theory Bas Spitters Aarhus Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos About me I PhD thesis on constructive analysis I Connecting Bishop's pointwise mathematics w/topos theory (w/Coquand) I Formalization of effective real analysis in Coq O'Connor's PhD part EU ForMath project I Topos theory and quantum theory I Univalent foundations as a combination of the strands co-author of the book and the Coq library I guarded homotopy type theory: applications to CS Bas Spitters Aarhus Sets in Homotopy type theory Most of the presentation is based on the book and Sets in HoTT (with Rijke). CC-BY-SA Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Bas Spitters Aarhus Sets in Homotopy type theory Predicative topos Homotopy type theory Towards a new practical foundation for mathematics. I Modern ((higher) categorical) mathematics I Formalization I Constructive mathematics Closer to mathematical practice, inherent treatment of equivalences. Towards a new design of proof assistants: Proof assistant with a clear (denotational) semantics, guiding the addition of new features. E.g. guarded cubical type theory Bas Spitters Aarhus Sets in Homotopy type theory Formalization of discrete mathematics: four color theorem, Feit Thompson, ... computational interpretation was crucial. Can this be extended to non-discrete types? Predicative topos Challenges pre-HoTT: Sets as Types no quotients (setoids), no unique choice (in Coq), ..
  • A Taste of Set Theory for Philosophers

    A Taste of Set Theory for Philosophers

    Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. A taste of set theory for philosophers Jouko Va¨an¨ anen¨ ∗ Department of Mathematics and Statistics University of Helsinki and Institute for Logic, Language and Computation University of Amsterdam November 17, 2010 Contents 1 Introduction 1 2 Elementary set theory 2 3 Cardinal and ordinal numbers 3 3.1 Equipollence . 4 3.2 Countable sets . 6 3.3 Ordinals . 7 3.4 Cardinals . 8 4 Axiomatic set theory 9 5 Axiom of Choice 12 6 Independence results 13 7 Some recent work 14 7.1 Descriptive Set Theory . 14 7.2 Non well-founded set theory . 14 7.3 Constructive set theory . 15 8 Historical Remarks and Further Reading 15 ∗Research partially supported by grant 40734 of the Academy of Finland and by the EUROCORES LogICCC LINT programme. I Journal of the Indian Council of Philosophical Research, Vol. XXVII, No. 2. A Special Issue on "Logic and Philosophy Today", 143-163, 2010. Reprinted in "Logic and Philosophy Today" (edited by A. Gupta ans J.v.Benthem), College Publications vol 29, 141-162, 2011. 1 Introduction Originally set theory was a theory of infinity, an attempt to understand infinity in ex- act terms. Later it became a universal language for mathematics and an attempt to give a foundation for all of mathematics, and thereby to all sciences that are based on mathematics.
  • An Introduction to Category Theory and Categorical Logic

    An Introduction to Category Theory and Categorical Logic

    An Introduction to Category Theory and Categorical Logic Wolfgang Jeltsch Category theory An Introduction to Category Theory basics Products, coproducts, and and Categorical Logic exponentials Categorical logic Functors and Wolfgang Jeltsch natural transformations Monoidal TTU¨ K¨uberneetika Instituut categories and monoidal functors Monads and Teooriaseminar comonads April 19 and 26, 2012 References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and comonads Monads and comonads References References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and Monads and comonads comonads References References An Introduction to From set theory to universal algebra Category Theory and Categorical Logic Wolfgang Jeltsch I classical set theory (for example, Zermelo{Fraenkel): I sets Category theory basics I functions from sets to sets Products, I composition
  • On the Constructive Elementary Theory of the Category of Sets

    On the Constructive Elementary Theory of the Category of Sets

    On the Constructive Elementary Theory of the Category of Sets Aruchchunan Surendran Ludwig-Maximilians-University Supervision: Dr. I. Petrakis August 14, 2019 1 Contents 1 Introduction 2 2 Elements of basic Category Theory 3 2.1 The category Set ................................3 2.2 Basic definitions . .4 2.3 Basic properties of Set .............................6 2.3.1 Epis and monos . .6 2.3.2 Elements as arrows . .8 2.3.3 Binary relations as monic arrows . .9 2.3.4 Coequalizers as quotient sets . 10 2.4 Membership of elements . 12 2.5 Partial and total arrows . 14 2.6 Cartesian closed categories (CCC) . 16 2.6.1 Products of objects . 16 2.6.2 Application: λ-Calculus . 18 2.6.3 Exponentials . 21 3 Constructive Elementary Theory of the Category of Sets (CETCS) 26 3.1 Constructivism . 26 3.2 Axioms of ETCS . 27 3.3 Axioms of CETCS . 28 3.4 Π-Axiom . 29 3.5 Set-theoretic consequences . 32 3.5.1 Quotient Sets . 32 3.5.2 Induction . 34 3.5.3 Constructing new relations with logical operations . 35 3.6 Correspondence to standard categorical formulations . 42 1 1 Introduction The Elementary Theory of the Category of Sets (ETCS) was first introduced by William Lawvere in [4] in 1964 to give an axiomatization of sets. The goal of this thesis is to describe the Constructive Elementary Theory of the Category of Sets (CETCS), following its presentation by Erik Palmgren in [2]. In chapter 2. we discuss basic elements of Category Theory. Category Theory was first formulated in the year 1945 by Eilenberg and Mac Lane in their paper \General theory of natural equivalences" and is the study of generalized functions, called arrows, in an abstract algebra.
  • Constructive Set Theory

    Constructive Set Theory

    CST Book draft Peter Aczel and Michael Rathjen CST Book Draft 2 August 19, 2010 Contents 1 Introduction 7 2 Intuitionistic Logic 9 2.1 Constructivism . 9 2.2 The Brouwer-Heyting-Kolmogorov interpretation . 11 2.3 Counterexamples . 13 2.4 Natural Deductions . 14 2.5 A Hilbert-style system for intuitionistic logic . 17 2.6 Kripke semantics . 19 2.7 Exercises . 21 3 Some Axiom Systems 23 3.1 Classical Set Theory . 23 3.2 Intuitionistic Set Theory . 24 3.3 Basic Constructive Set Theory . 25 3.4 Elementary Constructive Set Theory . 26 3.5 Constructive Zermelo Fraenkel, CZF ................ 26 3.6 On notations for axiom systems. 27 3.7 Class Notation . 27 3.8 Russell's paradox . 28 4 Basic Set constructions in BCST 31 4.1 Ordered Pairs . 31 4.2 More class notation . 32 4.3 The Union-Replacement Scheme . 35 4.4 Exercises . 37 5 From Function Spaces to Powerset 41 5.1 Subset Collection and Exponentiation . 41 5.2 Appendix: Binary Refinement . 44 5.3 Exercises . 45 3 CST Book Draft 6 The Natural Numbers 47 6.1 Some approaches to the natural numbers . 47 6.1.1 Dedekind's characterization of the natural numbers . 47 6.1.2 The Zermelo and von Neumann natural numbers . 48 6.1.3 Lawv`ere'scharacterization of the natural numbers . 48 6.1.4 The Strong Infinity Axiom . 48 6.1.5 Some possible additional axioms concerning ! . 49 6.2 DP-structures and DP-models . 50 6.3 The von Neumann natural numbers in ECST . 51 6.3.1 The DP-model N! .....................
  • Logic and Categories As Tools for Building Theories

    Logic and Categories As Tools for Building Theories

    Logic and Categories As Tools For Building Theories Samson Abramsky Oxford University Computing Laboratory 1 Introduction My aim in this short article is to provide an impression of some of the ideas emerging at the interface of logic and computer science, in a form which I hope will be accessible to philosophers. Why is this even a good idea? Because there has been a huge interaction of logic and computer science over the past half-century which has not only played an important r^olein shaping Computer Science, but has also greatly broadened the scope and enriched the content of logic itself.1 This huge effect of Computer Science on Logic over the past five decades has several aspects: new ways of using logic, new attitudes to logic, new questions and methods. These lead to new perspectives on the question: What logic is | and should be! Our main concern is with method and attitude rather than matter; nevertheless, we shall base the general points we wish to make on a case study: Category theory. Many other examples could have been used to illustrate our theme, but this will serve to illustrate some of the points we wish to make. 2 Category Theory Category theory is a vast subject. It has enormous potential for any serious version of `formal philosophy' | and yet this has hardly been realized. We shall begin with introduction to some basic elements of category theory, focussing on the fascinating conceptual issues which arise even at the most elementary level of the subject, and then discuss some its consequences and philosophical ramifications.
  • Notes on Categorical Logic

    Notes on Categorical Logic

    Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour.
  • An Outline of Algebraic Set Theory

    An Outline of Algebraic Set Theory

    An Outline of Algebraic Set Theory Steve Awodey Dedicated to Saunders Mac Lane, 1909–2005 Abstract This survey article is intended to introduce the reader to the field of Algebraic Set Theory, in which models of set theory of a new and fascinating kind are determined algebraically. The method is quite robust, admitting adjustment in several respects to model different theories including classical, intuitionistic, bounded, and predicative ones. Under this scheme some familiar set theoretic properties are related to algebraic ones, like freeness, while others result from logical constraints, like definability. The overall theory is complete in two important respects: conventional elementary set theory axiomatizes the class of algebraic models, and the axioms provided for the abstract algebraic framework itself are also complete with respect to a range of natural models consisting of “ideals” of sets, suitably defined. Some previous results involving realizability, forcing, and sheaf models are subsumed, and the prospects for further such unification seem bright. 1 Contents 1 Introduction 3 2 The category of classes 10 2.1 Smallmaps ............................ 12 2.2 Powerclasses............................ 14 2.3 UniversesandInfinity . 15 2.4 Classcategories .......................... 16 2.5 Thetoposofsets ......................... 17 3 Algebraic models of set theory 18 3.1 ThesettheoryBIST ....................... 18 3.2 Algebraic soundness of BIST . 20 3.3 Algebraic completeness of BIST . 21 4 Classes as ideals of sets 23 4.1 Smallmapsandideals . .. .. 24 4.2 Powerclasses and universes . 26 4.3 Conservativity........................... 29 5 Ideal models 29 5.1 Freealgebras ........................... 29 5.2 Collection ............................. 30 5.3 Idealcompleteness . .. .. 32 6 Variations 33 References 36 2 1 Introduction Algebraic set theory (AST) is a new approach to the construction of models of set theory, invented by Andr´eJoyal and Ieke Moerdijk and first presented in [16].
  • Notes Towards Homology in Characteristic One

    Notes Towards Homology in Characteristic One

    Notes Towards Homology in Characteristic One David Jaz Myers Abstract In their paper Homological Algebra in Characteristic One, Connes and Consani develop techniques of Grandis to do homological calculations with boolean modules { commutative idempotent monoids. This is a writeup of notes taken during a course given by Consani in Fall of 2017 explaining some of these ideas. This note is in flux, use at your own peril. Though the integers Z are the initial ring, they are rather complex. Their Krull dimension is 1, suggesting that they are the functions on a curve; but each of the epimorphisms out of them has a different characteristic, so they are not a curve over any single field. This basic observation, and many more interesting and less basic observations, leads one to wonder about other possible bases for the algebraic geometry of the integers. Could there be an \absolute base", a field over which all other fields lie? What could such a “field” be? As a first approach, let's consider the characteristic of this would-be absolute base field. Since it must lie under the finite fields, it should also be finite, and therefore its characteristic must divide the order of all finite fields. There is only one number which divides all prime powers (and also divides 0): the number 1 itself. Therefore, we are looking for a “field of characteristic one". Now, the axioms of a ring are too restrictive to allow for such an object. Let's free ourselves up by considering semirings, also known as rigs { rings without negatives.
  • Harish-Chandra 1923–1983

    Harish-Chandra 1923–1983

    NATIONAL ACADEMY OF SCIENCES HARISH- C HANDRA 1 9 2 3 – 1 9 8 3 A Biographical Memoir by R O G E R H O W E Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 2011 NATIONAL ACADEMY OF SCIENCES WASHINGTON, D.C. Photo by Herman Landshoff; Courtesy Archives of the Institute for Advanced Study. HARISH-CHANDRA October 11, 1923–October 12, 1983 BY ROGER HOWE He taught them the Kshatria code of honor: that a warrior may never refuse a challenge…. The Five Sons of Pandu, The Story of the Mahabharata Retold by Elizabeth Seeger ARISH-CHANDRA WAS, if not the exclusive architect, cer- Htainly the chief engineer of harmonic analysis on semisimple Lie groups. This subject, with roots deep in mathematical physics and analysis, is a synthesis of Fou- rier analysis, special functions and invariant theory, and it has become a basic tool in analytic number theory, via the theory of automorphic forms. It essentially did not ex- ist before World War II, but in very large part because of the labors of Harish-Chandra, it became one of the major mathematical edifices of the second half of the twentieth century. Harish-Chandra was born in 1923 in Uttar Pradesh, in northern India. His family belonged to the Kshatria (war- rior) caste. Kshatria traditionally were rulers, landowners, and military leaders, and more recently have commonly been businessmen or civil servants. Harish-Chandra’s father, Chandrakishore, was a civil engineer who monitored and maintained the dikes and irrigation canals that sustain agri- 3 B IOGRA P HICAL MEMOIRS culture on the North Indian plains.
  • On Categorical Theory-Building: Beyond the Formal

    On Categorical Theory-Building: Beyond the Formal

    Andrei Rodin (Ecole Normale Superieure) On Categorical Theory-Building: Beyond the Formal 1. Introduction: Languages, Foundations and Reification of Concepts The term "language" is colloquially used in mathematics to refer to a theory, which grasps common features of a large range (or even all) of other mathematical theories and so can serve as a unifying conceptual framework for these theories. "Set- theoretic language" is a case in point. The systematic work of translation of the whole of mathematics into the set-theoretic language has been endeavoured in 20-th century by a group of mathematicians under the collective name of Nicolas Bourbaki. Bourbakist mathematics proved successful both in research and higher education (albeit not in the school math education) and is practiced until today at mathematical departments worldwide. The project of Bourbaki as well as other attempts to do mathematics "set- theoretically" should be definitely distinguished from the project of reduction of mathematics to set theory defended by Quine and some other philosophers. This latter project is based on the claim that all true mathematical propositions are deducible from axioms of Zermelo-Frenkael set theory with Choice (ZFC) or another appropriate system of axioms for sets, so basically all the mathematics is set theory. A working mathematician usually sees this claim as an example of philosophical absurdity on a par with Zeno's claim that there is no motion, and Bourbaki never tried to put it forward. Actually Bourbaki used set theory (more precisely their own version of axiomatic theory of sets) for doing mathematics in very much the same way in which classical first-order logic is used for doing axiomatic theories like ZFC.