Set Theory and its Philosophy: A Critical Introduction

Front Cover
Clarendon Press, Jan 15, 2004 - Philosophy - 360 pages
Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Introduction to Part I
3
Logic
6
12 The background logic
11
13 Schemes
13
14 The choice of logic
16
15 Definite descriptions
18
Notes
20
Collections
21
94 The axiom of countable choice
161
Notes
165
Basic cardinal arithmetic
167
102 Cardinal arithmetic
168
103 Infinite cardinals
170
104 The power of the continuum
172
Notes
174
Ordinals
175

22 Membership
23
23 Russells paradox
25
24 Is it a paradox?
26
25 Indefinite extensibility
27
26 Collections
30
Notes
32
The hierarchy
34
32 Construction
36
33 Metaphysical dependence
38
34 Levels and histories
40
35 The axiom scheme of separation
42
36 The theory of levels
43
37 Sets
47
38 Purity
50
39 Wellfoundedness
51
Notes
53
The theory of sets
55
42 The initial level
57
43 The empty set
58
44 Cutting things down to size
60
45 The axiom of creation
61
46 Ordered pairs
63
47 Relations
65
48 Functions
67
49 The axiom of infinity
68
410 Structures
72
Notes
75
Conclusion to Part I
76
Numbers
79
Introduction to Part II
81
Arithmetic
88
52 Definition of natural numbers
89
53 Recursion
92
54 Arithmetic
95
55 Peano arithmetic
98
Notes
101
Counting
103
62 The ancestral
106
63 The ordering of the natural numbers
108
64 Counting finite sets
110
65 Counting infinite sets
113
66 Skolems paradox
114
Notes
116
Lines
117
72 Completeness
119
73 The real line
121
74 Souslin lines
125
75 The Baire line
126
Notes
128
Real numbers
129
82 Integral numbers
130
83 Rational numbers
132
84 Real numbers
135
85 The uncountability of the real numbers
136
86 Algebraic real numbers
138
87 Archimedean ordered fields
140
88 Nonstandard ordered fields
144
Notes
147
Conclusion to Part II
149
Cardinals and Ordinals
151
Introduction to Part III
153
Cardinals
155
92 The partial ordering
157
93 Finite and infinite
159
112 Ordinals
179
113 Transfinite induction and recursion
182
114 Cardinality
184
115 Rank
186
Notes
189
Ordinal arithmetic
191
122 Ordinal addition
192
124 Ordinal exponentiation
199
125 Normal form
202
Notes
204
Conclusion to Part III
205
Further Axioms
207
Introduction to Part IV
209
Orders of infinity
211
131 Goodsteins theorem
212
132 The axiom of ordinals
218
133 Reflection
221
134 Replacement
225
135 Limitation of size
227
136 Back to dependency?
230
137 Higher still
231
138 Speedup theorems
234
Notes
236
The axiom of choice
238
142 Skolems paradox again
240
143 The axiom of choice
242
144 The wellordering principle
243
145 Maximal principles
245
146 Regressive arguments
250
147 The axiom of constructibility
252
148 Intuitive arguments
256
Notes
259
Further cardinal arithmetic
261
152 The arithmetic of alephs
262
153 Counting wellorderable sets
263
154 Cardinal arithmetic and the axiom of choice
266
155 The continuum hypothesis
268
156 Is the continuum hypothesis decidable?
270
157 The axiom of determinacy
275
158 The generalized continuum hypothesis
280
Notes
283
Conclusion to Part IV
284
Appendices
289
Traditional axiomatizations
291
A2 Cardinals and ordinals
292
A3 Replacement
296
Notes
298
Classes
299
B1 Virtual classes
300
B2 Classes as new entities
302
B3 Classes and quantification
303
B4 Classes quantified
306
B5 Impredicative classes
307
B6 Impredicativity
308
B7 Using classes to enrich the original theory
310
Sets and classes
312
C2 The difference between sets and classes
313
C3 The metalinguistic perspective
315
Notes
316
References
317
List of symbols
336
Index of definitions
338
Index of names
343
Copyright

Other editions - View all

Common terms and phrases

About the author (2004)

Michael Potter is University Lecturer in Philosophy, and Fellow of Fitzwilliam College, at Cambridge. He is the author of Sets (1990), on which the present work draws but which was written for a more specialist readership, and Reason's Nearest Kin (2000).

Bibliographic information