The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today.
Author(s): Stewart Shapiro
Series: Oxford Logic Guides 17
Year: 1991
Language: English
Pages: 298
Contents......Page 20
PART I: ORIENTATION......Page 22
1.1 Orientation......Page 24
1.2 What is the issue?......Page 31
1.3 Sets and properties......Page 37
2. Foundationalism and foundations of mathematics......Page 46
2.1 Variations and metaphors......Page 47
2.2 Foundations and psychologism......Page 52
2.3 Two conceptions of logic......Page 56
2.4 Marriage: Can there be harmony?......Page 61
2.5 Divorce: Life without completeness......Page 64
2.6 Logic and computation......Page 70
PART II: LOGIC AND MATHEMATICS......Page 80
3.1 Language......Page 82
3.2 Deductive systems......Page 86
3.3 Semantics......Page 91
4.1 First-order theories......Page 100
4.2 Second-order-standard semantics......Page 101
4.3 Non-standard semantics-Henkin and first-order......Page 109
5.1 Mathematical notions......Page 118
5.2 First-order theories-what goes wrong......Page 131
5.3 Second-order languages and the practice of mathematics......Page 137
5.4 Set theory......Page 148
6.1 A word on semantic theory......Page 155
6.2 Reductions......Page 158
6.3 Reflection: small large cardinals......Page 162
6.4 Löwenheim-Skolem analogues: large large cardinals......Page 168
6.5 Characterizations of first-order logic......Page 178
6.6 Definability and other odds and ends......Page 182
PART III: HISTORY AND PHILOSOPHY......Page 192
7.1 Introduction......Page 194
7.2 Narrative......Page 198
7.3 To the present......Page 214
8. Second-order logic and rule-following......Page 224
8.1 The regress......Page 225
8.2 Options......Page 229
8.3 Rules and logic......Page 231
9.1 Other logics......Page 241
9.2 Free relation variables......Page 267
9.3 First-order set theory......Page 271
References......Page 284
C......Page 294
H......Page 295
M......Page 296
S......Page 297
Z......Page 298
