While intuitionistic (or constructive) set theory IST 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 - IST has been chiefly developed in a tops-theoretic context. In particular, proofs of relative consistency with IST 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 IST of the well-known Boolean-valued models for classical set theory.
In this book I offer a brief but systematic introduction to IST which develops the subject up to and including the use of Heyting-algebra-valued models in relative consistency proofs. I believe that IST, 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.
Author(s): John L. Bell
Series: Studies in Logic 50
Publisher: College Publications
Year: 2014
Language: English
Commentary: Scanned by Envoy
Pages: 132
City: London
John L. Bell “Intuitionistic Set Theory” (2014) ......Page 1
Table of contents ......Page 8
Preface ......Page 10
The natural numbers and countability ......Page 12
Power sets ......Page 16
The Continuum ......Page 18
Axioms and basic definitions ......Page 22
Logical principles in IZ ......Page 26
The axiom of choice ......Page 29
The natural numbers ......Page 36
Models of Peano's axioms ......Page 38
Definitions by recursion ......Page 39
Finite sets ......Page 45
Frege’s construction of the natural numbers ......Page 49
Dedekind real numbers and weak real numbers ......Page 57
Cauchy real numbers ......Page 62
Intuitionistic Zermelo-Fraenkel set theory IZF ......Page 65
Frame-valued models of IZF developed in IZF ......Page 69
The consistency of ZF and ZFC relative to IZF ......Page 81
Frame-valued models of IZF developed in ZFC ......Page 82
A frame-valued model in which NN is subcountable ......Page 88
The axiom of choice in frame-valued extensions ......Page 93
Real numbers and real functions in spatial extensions ......Page 95
Properties of the set of real numbers over R ......Page 102
Properties of the set of real numbers over Baire space ......Page 104
The independence of the fundamental theorem of algebra from IZF ......Page 107
Lattices ......Page 109
Heyting and Boolean algebras ......Page 111
Coverages on partially ordered sets and their associated frames ......Page 116
Connections with logic ......Page 117
Concluding observations ......Page 122
Historical notes ......Page 125
Bibliography ......Page 127
Index ......Page 130
Back cover ......Page 132
