TECHNICAL ALGEBRA HANDBOOK

By MATTHEW MOORE

CONTENTS PART I INTRODUCTORY AN INTRODUCTION TO INTUITIONISTIC LOGIC INTRODUCTION CONSTRUCTIVE EXISTENCE THE BROUWER–HEYTING–KOLMOGOROV INTERPRETATION NATURAL DEDUCTIONS A HILBERT-STYLE SYSTEM FOR INTUITIONISTIC LOGIC REALIZABILITY THE CURRY–HOWARD CORRESPONDENCE REFERENCES AN INTRODUCTION TO CONSTRUCTIVE SET THEORY: AN APPETIZER THE AXIOMATIC FRAMEWORK ELEMENTARY MATHEMATICS IN CZF THE DEVELOPMENT OF SET THEORY IN CZF LARGE SETS IN CZF AXIOMS OF CHOICE IN CONSTRUCTIVE SET THEORY CZF AND THE LIMITED PRINCIPLE OF OMNISCIENCE MODELS OF CZF AND AXIOMATIC FREEDOM P BLI H D LI B C B ID U I I P BISHOP’S MATHEMATICS: A PHILOSOPHICAL PERSPECTIVE BISHOP ON BROUWER BROUWER’S MATHEMATICS PERSUASION AND DIALOGUE FORMALISATION PHILOSOPHY TRADITIONAL PHILOSOPHICAL ARGUMENTS FOR INTUITIONISTIC LOGIC PHILOSOPHICAL OBJECTIONS TOO STRONG CONCLUDING REMARKS ACKNOWLEDGEMENTS PART II ALGEBRA AND GEOMETRY ALGEBRA IN BISHOP’S STYLE: A COURSE IN CONSTRUCTIVE ALGEBRA REVISITING BISHOP’S SET THEORY THE CORPUS OF CLASSICAL ABSTRACT ALGEBRA TREATED IN THE BOOK PRINCIPAL IDEAL DOMAINS FACTORIZATION PROBLEMS NOETHERIAN RINGS, PRIMARY DECOMPOSITIONS AND THE PRINCIPAL IDEAL THEOREM WEDDERBURN STRUCTURE THEOREM FOR FINITE-DIMENSIONAL K-ALGEBRAS DEDEKIND DOMAINS CONSTRUCTIVE ALGEBRA: THE QUILLEN–SUSLIN THEOREM QUILLEN’S PROOF OF SERRE’S PROBLEM SUSLIN’S PROOF OF SERRE’S PROBLEM CONSTRUCTIVE ALGEBRA AND POINT-FREE TOPOLOGY THIERRY COQUAND ZARISKI SPECTRUM MINIMAL AND MAXIMAL PRIMES FORCING OVER A SITE CONSTRUCTIVE PROJECTIVE GEOMETRY MARK MANDELKERN REAL PROJECTIVE PLANE PROJECTIVE EXTENSIONS PART III ANALYSIS ELEMENTS OF CONSTRUCTIVE ANALYSIS HAJIME ISHIHARA REAL NUMBERS METRIC SPACES NORMED LINEAR SPACES CONSTRUCTIVE FUNCTIONAL ANALYSIS PRELIMINARIES COMPLETENESS CONVEXITY DUALITY IN HILBERT SPACES CONSTRUCTIVE BANACH ALGEBRA THEORY ROBIN S HAVEA AND DOUGLAS BRIDGES THE SPECTRAL MAPPING THEOREM APPROXIMATING THE STATE SPACE HERMITIAN AND POSITIVE ELEMENTS CONSTRUCTIVE CONVEX OPTIMISATION SOME DE?NITIONS AND NOTATION CONVEXITY AND EXISTENCE OF IN?MA AND MINIMA CONVEXITY AND BROUWER’S FAN THEOREM LEMMAS OF THE ALTERNATIVE AND CONSEQUENCES CONSTRUCTIVE MATHEMATICAL ECONOMICS PREFERENCE AND UTILITY DEMAND FUNCTIONS ECONOMIC EQUILIBRIUM GAME THEORY A LEISURELY RANDOM WALK DOWN THE LANE OF A CONSTRUCTIVE THEORY OF STOCHASTIC PROCESSES STOCHASTIC PROCESS, IN A NUTSHELL CONSTRUCTIVE MATHEMATICS, IN A NUTSHELL STOCHASTIC PROCESSES, IN A BIGGER NUTSHELL CONSTRUCTIVE THEORY OF STOCHASTIC PROCESSES, IN AN EVEN BIGGER NUTSHELL PART IV TOPOLOGY BASES OF PSEUDOCOMPACT BISHOP SPACES THE PROBLEM OF CONSTRUCTIVISING GENERAL TOPOLOGY OVERVIEW OF RECENT WORK ON BISHOP SPACES STRUCTURE OF THE TECHNICAL PART OF THIS CHAPTER BASIC NOTIONS IN THE THEORY OF BISHOP SPACES BASES OF BISHOP SPACES THE FIRST BASE THEOREM THE SECOND BASE THEOREM APPLICATIONS OF THE SECOND BASE THEOREM P BLI H D LI B C B ID U I I P ACKNOWLEDGEMENTS BISHOP METRIC SPACES IN FORMAL TOPOLOGY TATSUJI KAWAI FORMAL TOPOLOGY FUNCTORIAL EMBEDDING OF LOCALLY COMPACT METRIC SPACES LOCATED SUBSETS IN FORMAL TOPOLOGY POINTFREE CHARACTERISATION OF COMPACT METRIC SPACES POINTFREE CHARACTERISATION OF LOCALLY COMPACT METRIC SPACES BEYOND LOCALLY COMPACT METRIC SPACES RELATED WORKS SUBSPACES IN POINTFREE TOPOLOGY: TOWARDS A NEW APPROACH TO MEASURE THEORY FRANCESCO CIRAULO POINTFREE PARTS OF THE REAL LINE A MEASURE ON ?-SUBLOCALES THE POINTFREE APPROACH TO THE REAL LINE SYNTHETIC TOPOLOGY DAVORIN LEŠNIK TOPOLOGICAL PROPERTIES PRINCIPLES APARTNESS ON LATTICES AND BETWEEN SETS DOUGLAS BRIDGES LATTICES APARTNESS ON FRAMES FRAME TOPOLOGIES JOIN HOMOMORPHISMS AND CONTINUITY SET–SET PRE-APARTNESS STRONG AND UNIFORM CONTINUITY COMPACTNESS ACKNOWLEDGEMENT PART V LOGIC AND FOUNDATIONS COUNTABLE CHOICE AXIOMS OF CHOICE LIVING WITHOUT COUNTABLE CHOICE THE FUNDAMENTAL THEOREM OF ALGEBRA COMPLETIONS THE ASCENDING TREE CONDITION BISHOP’S PRINCIPLE AND THE ?-TECHNIQUE THE MINIMALIST FOUNDATION AND BISHOP’S CONSTRUCTIVE MATHEMATICS WHY ADOPT A MINIMALIST FOUNDATION? THE MINIMALIST FOUNDATION WHY ADOPTING THE POINTFREE APPROACH TO DEVELOP TOPOLOGY IN MF? EXTENDING MF WITH CHOICE PRINCIPLES IDENTITY, EQUALITY, AND EXTENSIONALITY IN EXPLICIT MATHEMATICS THE BASIC AXIOMATIC OPERATIONAL FRAMEWORK ADDING ELEMENTARY CLASSES ABOUT SOME ONTOLOGICAL ASPECTS OF EC AND EC+ ABSTRACT DATA STRUCTURES THE NUMBER SYSTEMS N, Z, AND Q AS ABSTRACT DATA STRUCTURES REPRESENTING THE REAL NUMBERS INNER AND OUTER MODELS FOR CONSTRUCTIVE SET THEORIES ROBERT S LUBARSKY HEYTING MODELS, OR CONSTRUCTIVE FORCING KRIPKE MODELS HEYTING–KRIPKE MODELS CLASSICAL OUTER MODELS INNER MODELS A FINAL EXAMPLE AN INTRODUCTION TO CONSTRUCTIVE REVERSE MATHEMATICS A FORMAL SYSTEM CONTINUITY PROPERTIES COMPACTNESS PROPERTIES THE MONOTONE COMPLETENESS THEOREM SYSTEMS FOR CONSTRUCTIVE REVERSE MATHEMATICS TAKAKO NEMOTO PRELIMINARY FUNCTION-BASED LANGUAGE AND SYSTEMS BASE THEORY WITH THE STRENGTH OF RCA? BROUWER’S FAN THEOREM JOSEF BERGER NOTATION THE WEAK KÖNIG LEMMA THE FAN THEOREM THE UNIFORM CONTINUITY THEOREM THE FAN THEOREM FOR C-SETS P BLI H D LI B C B ID U I I P PART VI ASPECTS OF COMPUTATION COMPUTATIONAL ASPECTS OF BISHOP’S CONSTRUCTIVE MATHEMATICS PARTIAL CONTINUOUS FUNCTIONALS A TERM LANGUAGE FOR COMPUTABLE FUNCTIONALS A THEORY OF COMPUTABLE FUNCTIONALS COMPUTATIONAL CONTENT OF PROOFS APPLICATIONS APPLICATION OF CONSTRUCTIVE ANALYSIS IN EXACT REAL ARITHMETIC E?CIENT ALGORITHMS FROM PROOFS IN CONSTRUCTIVE ANALYSIS REPRESENTATION OF REAL NUMBERS NUPRL REPRESENTATION OF REAL NUMBERS SOME TYPE THEORY EXTRACTS OF PROOFS BY INDUCTION INVERSE, DIVISION, AND COMPUTATION CONSTRUCTING KTH ROOTS COMPUTING POWER SERIES 28 10 SIN(X), COS(X), AND EX 28 11 LN(X) AND ARCSIN(X) 28 12 COMPUTING ? AND ARCTAN(X) 28 13 CONSTRUCTIVE CONTENT OF BROUWER’S PRINCIPLES 28 14 CONCLUDING REMARKS ON THE COMPUTATIONAL CONTENT OF CHOICE PRINCIPLES A SEMI-CONSTRUCTIVE SYSTEM WITH COMPUTATIONAL CONTENT REALIZABLE AND UNREALIZABLE CHOICE PRINCIPLES COUNTABLE CHOICE AND CLASSICAL LOGIC CONCLUSION