Table Of ContentIntuitionistic Set Theory
J o h n L. Bell
Studies in Logic
Volume 50
Intuitionistic Set Theory
Volume 39
Non-contradiction
Lawrence H. Powers, with a Foreword by Hans V. Hansen
Volume 40
The Lambda Calculus. Its Syntax and Semantics
Henk P. Barendregt
Volume 41
Symbolic Logic from Leibniz to Husserl
Abel Lassalle Casanave, ed.
Volume 42
Meta-argumentation. An Approach to Logic and Argumentation Theory
Maurice A. Finocchiaro
Volume 43
Logic, Truth and Inquiry
Mark Weinstein
Volume 44
Meta-logical Investigations in Argumentation Networks
Dov M. Gabbay
Volume 45
Errors of Reasoning. Naturalizing the Logic of Inference
John Woods
Volume 46
Questions, Inferences, and Scenarios
Andrzej Wisniewski
Volume 47
Logic Across the University: Foundations and Applications. Proceedings of
the Tsinghua Logic Conference, Beijing, 2013
Johan van Benthem and Fenrong Liu, eds.
Volume 48
Trends in Belief Revision and Argumentation Dynamics
Eduardo L. Ferme, Dov M. Gabbay, and Guillermo R. Simari
Volume 49
Introduction to Propositional Satisfiability
Victor Marek
Volume 50
Intuitionistic Set Theory
John L. Bell
Studies in Logic Series Editor
Dov Gabbay [email protected]
Intuitionistic Set Theory
John L. Bell
© Individual author and College Publications 2014.
All rights reserved.
ISBN 978-1-84890-140-7
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To Sandra, who has given me new life
Table of Contents
Introduction 1
The natural numbers and countability 1
Power sets 4
The Continuum 6
Chapter I. Intuitionistic Zermelo Set Theory 11
Axioms and basic definitions 11
Logical principles in IZ 15
The axiom of choice 18
Chapter II. Natural Numbers and Finite Sets 25
The natural numbers 25
Models of Peano's axioms 27
Definitions by recursion 28
Finite sets 33
Frege’s construction of the natural numbers 37
Chapter III. The Real Numbers 45
Dedekind real numbers and weak real numbers 45
Cauchy real numbers 49
Chapter IV. Intuitionistic Zermelo-Fraenkel Set Theory
and Frame-Valued Models 51
Intuitionistic Zermelo-Fraenkel set theory IZF 51
Frame-valued models of IZF developed in IZF 54
The consistency of ZF and ZFC relative to IZF 65
Frame-valued models of IZF developed in ZFC 66
A frame-valued model in which NN is subcountable 71
The axiom of choice in frame-valued extensions 76
Real numbers and real functions in spatial extensions 77
Properties of the set of real numbers over R 84
Properties of the set of real numbers over Baire space 85
The independence of the fundamental theorem of algebra from IZF 88
Appendix. Heyting Algebras and Frames 91
Lattices 91
Heyting and Boolean algebras 93
Coverages on partially ordered sets and their associated frames 97
Connections with logic 98
Concluding Observations 103
Historical Notes 105
Bibliography 107
Index 111
Preface
While intuitionistic (or constructive) set theory 1ST has received a certain attention
from mathematical logicians, so far as I am aware no book providing a systematic
introduction to the subject has yet been published. This may be the case in part
because, as a form of higher-order intuitionistic logic - the internal logic of a
topos - 1ST has been chiefly developed in a tops-theoretic context. In particular,
proofs of relative consistency with 1ST for mathematical assertions have been
(implicitly) formulated in topos- or sheaf-theoretic terms, rather than in the
framework of Heyting-algebra-valued models, the natural extension to 1ST of the
well-known Boolean-valued models for classical set theory.
In this book I offer a brief but systematic introduction to 1ST which develops the
subject up to and including the use of Heyting-algebra-valued models in relative
consistency proofs. I believe that 1ST, presented as it is in the familiar language of
set theory, will appeal particularly to those logicians, mathematicians and
philosophers who are unacquainted with the methods of topos theory.
The title I originally had in mind for this book was Constructive Set Theory. Then it
occurred to me that the term “constructive” has come to connote not merely the use
of intuitionistic logic, , but also the avoidance of impredicative definitions. This is
the case, for example, with Aczel’s Constructive Set Theory in which the power set
axiom (which permits impredicative definirions of sets) is not postulated. Since the
power set axiom and impredicative definitions are very much a part of 1ST, to avoid
confusion I have decided (with some reluctance) to give the book its present title.
JLB January 2014
