Logic for Mathematicians

Cover ......Page 1
Preface ......Page 7
Contents ......Page 9
List of Symbols ......Page 12
1. A hypothetical interview ......Page 15
2. The role of symbolic logic ......Page 17
3. General nature of symbolic logic ......Page 19
4. Advantages and disadvantages of a symbolic logic ......Page 23
1. Statement functions ......Page 26
2. The dot notation ......Page 33
3. The use of truth-value tables ......Page 37
4. Applications to mathematical reasoning ......Page 44
5. Summary of logical principles ......Page 60
Chapter III. The Use of Names ......Page 63
1. The axiomatic method of defining valid statements ......Page 68
2. The truth-value axioms ......Page 69
3. Properties of ├ ......Page 73
4. Preliminary theorems ......Page 74
5. The truth value theorem ......Page 82
6. The deduction theorem ......Page 89
Chapter V. Clarification ......Page 91
1. Variables and unknowns ......Page 96
2. Quantifiers ......Page 101
3. Axioms for the restricted predicate calculus ......Page 115
4. The generalization principle ......Page 117
5. The equivalence and substitution theorems ......Page 121
6. Useful equivalences ......Page 126
7. The formal analogue of an act of choice ......Page 137
8. Restricted quantification ......Page 154
9. Applications to everyday mathematics ......Page 164
10. Church’s theorem ......Page 175
11. A convention concerning bound variables ......Page 176
1. General properties ......Page 177
2. Enumerative quantifiers ......Page 180
3. Applications ......Page 186
1. Axioms for descriptions ......Page 195
2. Definition by cases ......Page 201
3. Uses of descriptions in everyday mathematics ......Page 209
1. The notion of a class ......Page 211
2. Axioms for classes ......Page 221
3. Formalism for classes ......Page 233
4. The calculus of classes ......Page 240
5. Manifold products and sums ......Page 252
6. Unit classes and subclasses ......Page 263
7. Variables over the range 2 ......Page 270
8. Applications ......Page 277
1. The axiom of infinity ......Page 289
2. Ordered pairs and triples ......Page 294
3. The calculus of relations ......Page 298
4. Special properties of relations ......Page 308
5. Functions ......Page 319
6. Ordered sets ......Page 344
7. Equivalence relations ......Page 353
8. Applications ......Page 358
1. Cardinal similarity ......Page 359
2. Elementary properties of cardinal numbers ......Page 385
3. Finite classes and mathematical induction ......Page 404
4. Denumerable classes ......Page 443
5. The cardinal number of the continuum ......Page 458
6. Applications ......Page 465
1. Ordinal similarity ......Page 470
2. Well-ordering relations ......Page 473
3. Elementary properties of ordinal numbers ......Page 484
4. The cardinal number associated with an ordinal number ......Page 491
5. Applications ......Page 495
1. Preliminaries ......Page 497
2. The axiom of counting ......Page 499
3. The pigeonhole principle ......Page 501
4. Applications ......Page 502
1. The general axiom of choice ......Page 504
2. How indispensable is the axiom of choice? ......Page 524
3. The denumerable axiom of choice ......Page 526
Chapter XV. We Rest Our Case ......Page 531
Bibliography ......Page 537
Index ......Page 541