9780521770347

The Foundations of Mathematics in the Theory of Sets

by
  • ISBN13:

    9780521770347

  • ISBN10:

    0521770343

  • Format: Hardcover
  • Copyright: 2001-04-23
  • Publisher: Cambridge University Press
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Summary

This book presents a unified approach to the foundations of mathematics in the theory of sets, covering both conventional and finitary (constructive) mathematics. It is based on a philosophical, historical and mathematical analysis of the relation between the concepts of 'natural number' and 'set'. This leads to an investigation of the logic of quantification over the universe of sets and a discussion of its role in second order logic, as well as in the analysis of proof by induction and definition by recursion. The subject matter of the book falls on the borderline between philosophy and mathematics, and should appeal to both philosophers and mathematicians with an interest in the foundations of mathematics.

Table of Contents

Preface x
Part One: Preliminaries 1(64)
The Idea of Foundations for Mathematics
3(14)
Why mathematics needs foundations
3(5)
What the foundations of mathematics consist in
8(2)
What the foundations of mathematics need not include
10(4)
Platonism
14(3)
Simple Arithmetic
17(48)
The origin of the natural numbers
17(2)
The abstractness of the natural numbers
19(2)
The original conception of number
21(3)
Number words and ascriptions of number
24(5)
The existence of numbers
29(10)
Mathematical numbers and pure units
39(6)
Ascriptions of number: Frege or Aristotle?
45(7)
Simple numerical equations
52(7)
Arithmetica universalis
59(6)
Part Two: Basic Set Theory 65(86)
Semantics, Ontology, and Logic
67(44)
Objects and identity
67(3)
Arithmoi and their units
70(4)
Sets
74(4)
Global functions
78(7)
Species
85(9)
Formalisation
94(5)
Truth and proof in mathematics
99(12)
The Principal Axioms and Definitions of Set Theory
111(40)
The Axiom of Comprehension and Russell's Theorem
111(3)
Singleton selection and description
114(1)
Pair Set, Replacement, Union, and Power Set
115(3)
The status of the principal axioms of set theory
118(6)
Ordered pairs and Cartesian products
124(6)
Local functions and relations
130(4)
Cardinality
134(3)
Partial orderings and equivalence relations
137(2)
Well-orderings and local recursion
139(5)
Von Neumann well-orderings and ordinals
144(2)
The Principle of Regularity
146(5)
Part Three: Cantorian Set Theory 151(108)
Cantorian Finitism
153(38)
Dedekind's axiomatic definition of the natural numbers
153(8)
Cantor's Axiom
161(1)
The Axiom of Choice
162(2)
The extensional analysis of sets
164(6)
The cumulative hierarchy of sets
170(6)
Cantor's Absolute
176(9)
Axioms of strong infinity
185(6)
The Axiomatic Method
191(46)
Mathematics before the advent of the axiomatic method
191(4)
Axiomatic definition
195(12)
Mathematical logic: formal syntax
207(6)
Global semantics and localisation
213(8)
Categoricity and the completeness of theories
221(4)
Mathematical objects
225(12)
Axiomatic Set Theory
237(22)
The Zermelo-Fraenkel axioms
237(5)
Axiomatic set theory and Brouwer's Principle
242(9)
The localisation problem for second order logic
251(8)
Part Four: Euclidean Set Theory 259(137)
Euclidean Finitism
261(39)
The serpent in Cantor's paradise
261(9)
The problem of non-Cantorian foundations
270(6)
The Axiom of Euclidean Finiteness
276(6)
Linear orderings and simple recursion
282(9)
Local cardinals and ordinals
291(3)
Epsilon chains and the Euclidean Axiom of Foundation
294(6)
The Euclidean Theory of Cardinalaity
300(25)
Arithmetical functions and relations
300(10)
Limited recursion
310(7)
S-ary decompositions and numerals
317(8)
The Euclidean Theory of Simply Infinite Systems
325(44)
Simply infinite systems
325(5)
Measures, scales, and elementary arithmetical operations
330(3)
Limited recursion
333(3)
Extending simply infinite systems
336(14)
The hierarchy of S-ary extensions
350(2)
Simply infinite systems that grow slowly in rank
352(10)
Further axioms
362(7)
Euclidean Set Theory from the Cantorian Standpoint
369(12)
Methodology
369(3)
Cumulation Models
372(9)
Envoi
381(15)
Euclid or Cantor?
381(1)
Euclidean simply infinite systems
382(5)
Speculations and unresolved problems
387(9)
Appendix 1 Conceptual Notation 396(15)
A1.1 Setting up a conceptual notation
396(2)
A1.2 Axioms, definitions, and rules of inference
398(8)
A1.3 Global propositional connectives
406(5)
Appendix 2 The Rank of a Set 411(4)
Bibliography 415(6)
Index 421

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