PART I ORIENTATION
1. Terms and questions .......................................... l
1.1. Orientation ............................................. 1
1.2. What is the issue? ..................................... 10
1.3. Sets and properties .................................... 16
2. Foundationalism and foundations of mathematics .............. 25
2.1. Variations and metaphors ............................... 26
2.2. Foundations and psychologism ........................... 31
2.3. Two conceptions of logic ............................... 35
2.4. Marriage: Can there be harmony? ........................ 40
2.5. Divorce: Life without completeness ..................... 43
2.6. Logic and computation .................................. 49
PART II LOGIC AND MATHEMATICS
3. Theory ...................................................... 61
3.1. Language ............................................... 61
3.2. Deductive systems ...................................... 65
3.3. Semantics .............................................. 70
4. Metatheory .................................................. 79
4.1. First-order theories ................................... 79
4.2. Second-order—standard semantics ........................ 80
4.3. Non-standard semantics—Henkin and first-order .......... 88
5. Second-order logic and mathematics .......................... 96
5.1. Mathematical notions ................................... 96
5.2. First-order theories—what goes wrong .................. 110
5.3. Second-order languages and the practice of
mathematics ........................................... 116
5.4. Set theory ............................................ 127
6. Advanced metatheory ........................................ 134
6.1. A word on semantic theory ............................. 134
6.2. Reductions ............................................ 137
6.3. Reflection: small large cardinals ..................... 141
6.4. Lowenheim-Skolem analogues: large large cardinals ..... 147
6.5. Characterizations of first-order logic ................ 157
6.6. Definability and other odds and ends .................. 161
PART III HISTORY AND PHILOSOPHY
7. The historical 'triumph' of first-order languages .......... 173
7.1. Introduction .......................................... 173
7.2. Narrative ............................................. 177
7.3. To the present ........................................ 193
8. Second-order logic and rule-following ...................... 203
8.1. The regress ........................................... 204
8.2. Options ............................................... 208
8.3. Rules and logic ....................................... 210
9. The competition ............................................ 220
9.1. Other logics .......................................... 220
9.2. Free relation variables ............................... 246
9.3. First-order set theory ................................ 250
References .................................................... 263
Index ......................................................... 273
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