The mathematics of metamathematics.

Preface......Page 3
1. Sets, mappings, Cartesian products......Page 9
2. Topological spaces......Page 11
3. Equivalence relations......Page 18
4. Abstract algebras......Page 20
5. Ordered sets......Page 30
6. Lattices......Page 32
7. Infinite joins and meets......Page 36
8. Filters and ideals......Page 42
9. Distributive lattices......Page 46
10. Complement and pseudo-complement......Page 50
11. Relative pseudo-complement. Difference......Page 52
12. Relatively pseudo-complemented lattices. Pseudo-Boolean algebras......Page 56
13. Filters in relatively pseudo-complemented lattices......Page 61
1. Definition and elementary properties......Page 66
2. Subalgebras......Page 71
3. Boolean homomorphisms......Page 73
4. The two-element Boolean algebra......Page 75
5. Filters and ideals......Page 76
6. Relativization......Page 77
7. Products of Boolean algebras......Page 79
8. The Stone spaces of Boolean algebras......Page 81
9. Representations preserving some infinite joins and meets......Page 84
10. Minimal extensions of Boolean algebras......Page 86
11. The Cantor discontinuum......Page 89
1. Definition and elementary properties......Page 91
2. Relativization to principal ideals......Page 93
3. Topological homomorphisms and isomorphisms. Interior mappings......Page 95
4. Extensions and imbeddings of topological Boolean algebras......Page 98
5. Strongly compact spaces......Page 99
6. Metric spaces......Page 100
7. A fundamental lemma on metric spaces......Page 103
8. Finite topological Boolean algebras......Page 107
9. Cartesian products of topological spaces......Page 111
10. A representation theorem for enumerable topological Boolean algebras......Page 115
11. Complete spaces......Page 116
12. Quotient algebras......Page 118
13. Products of topological Boolean algebras. Direct unions of topological spaces......Page 119
1. Preliminaries......Page 121
2. Pseudo-Boolean homomorphisms and isomorphisms......Page 125
3. Representation theorems......Page 126
4. Finite pseudo-Boolean algebras......Page 128
5. Dense elements......Page 129
6. Regular elements......Page 131
7. Infinite joins and meets......Page 133
8. Relativization......Page 136
9. Embeddings and extensions of pseudo-Boolean algebras......Page 137
10. Enumerable pseudo-Boolean algebras......Page 139
11. Products of pseudo-Boolean algebras......Page 140
1. The concept of formalized theories......Page 142
2. Operations on expressions......Page 149
3. Formalized languages of elementary mathematical theories......Page 150
4. Interpretations......Page 158
5. The intuitive concept of propositional tautologies......Page 161
6. Formalized languages of propositional calculi......Page 164
7. The intuitive concept of predicate tautologies......Page 168
8. Rules of inference......Page 171
9. Formal proofs......Page 177
10. The consequence operations. Formalized deductive systems and theories......Page 179
11. The general notion of logic. The classical logic......Page 185
12. Axioms for equality......Page 188
13. Examples of elementary formalized theories based on classical logic......Page 190
14. Some fundamental metamathematical notions......Page 199
15. Definitions in formalized theories......Page 204
1. The algebra of formulas......Page 207
2. The algebra of formulas of a formalized language of zero order......Page 208
3. The algebra of terms. Realizations of terms......Page 213
4. The algebra and Q-algebra of a formalized language of the first-order......Page 217
5. The I-algebra of a formalized language of the first-order......Page 220
6. Realizations of a formalized language of the first-order......Page 223
7. Canonical realizations of a formalized language of the first-order......Page 232
8. Products of realizations......Page 238
9. The algebra of open formulas......Page 241
10. The algebra of a formalized theory......Page 242
11. The Q-algebra of a formalized theory of the first order......Page 249
1. Preliminaries......Page 254
2. The completeness of the propositional calculi......Page 257
3. Examples of propositional tautologies......Page 258
4. The algebra of the two-valued propositional calculus......Page 259
5. Normal forms......Page 261
6. Diagrams of formulas......Page 262
7. Consistency and existence of models......Page 268
8. Deduction theorems......Page 270
9. The connection between theories and filters......Page 271
10. Maximal and prime theories......Page 273
11. Effectivity problems......Page 275
1. Preliminaries......Page 277
2. Models......Page 279
3. Canonical models. The consistency and existence of models......Page 284
4. Semantic models......Page 286
5. The existence of semantic enumerable models for enumerable theories......Page 290
6. The completeness of the predicate calculi. Examples of tautologies......Page 293
7. Diagrams of formulas......Page 297
8. Rich theories......Page 304
9. The existence of semantic models for arbitrary consistent theories......Page 308
10. The deduction theorems......Page 311
11. The connection between theories and filters......Page 312
12. Maximal and prime theories......Page 315
14. The inessentiality of definitions......Page 322
15. Open theories......Page 324
16. The prenex form......Page 328
17. The elimination of quantifiers from the axioms of a theory......Page 332
18. Products of semantic realizations modulo a prime filter......Page 336
19. Cardinals of models......Page 339
20. Non-enumerable arithmetic and enumerable set theory......Page 347
21. Effectivity problems......Page 351
22. Canonical semantic models. Representation problems for Q-algebras of theories......Page 352
23. A topological characterization of open theories......Page 360
24. The algebra of the two-valued predicate calculus......Page 363
25. The deduction theorem for open theories......Page 365
26. Herbrand disjunctions......Page 366
1. Introduction......Page 374
2. Preliminaries......Page 379
3. The completeness theorem......Page 383
4. Examples of intuitionistic propositional tautologies......Page 386
5. Connection between tautologies and intuitionistic tautologies......Page 388
6. A theorem on intuitionistically derivable disjunctions......Page 392
7. The algebra of the intuitionistic propositional calculus......Page 393
8. Consistency and the existence of models......Page 395
9. Deduction theorems......Page 397
10. Connection between theories and filters......Page 398
11. Maximal theories......Page 400
12. Prime theories......Page 401
13. Connection between classical and intuitionistic theories......Page 406
1. Preliminaries......Page 409
2. Models......Page 412
3. Canonical models. Consistency and the existence of models......Page 416
4. The completeness of intuitionistic predicate calculi......Page 420
5. The algebra of the intuitionistic predicate calculus......Page 421
6. Examples of intuitionistic tautologies......Page 423
7. The connection between tautologies and intuitionistic tautologies......Page 426
8. Theorem on intuitionistically derivale disjunctions and existential formulas......Page 428
9. The deduction theorems......Page 430
10. Connection between theories and filters......Page 431
11. Maximal theories......Page 434
12. Prime theories......Page 436
13. Constructive theories......Page 437
14. Cancellation of initial quantifiers in formulas of a ∪-theory......Page 443
15. Theories with the sign of equality......Page 444
16. Open intuitionistic theories......Page 446
17. The deduction theorem for open intuitionistic theories......Page 450
18. A theorem on the extension of topological realizations......Page 451
19. Elimination of initial quantifiers from axioms of an intuitionsitic theory......Page 453
1. Introduction......Page 458
2. The positive logic......Page 459
3. Positive theories of zero order......Page 461
4. The positive propositional calculus......Page 463
5. Positive theories of the first order......Page 464
6. The positive predicate calculus......Page 466
7. The modal logic......Page 467
8. Modal theories of zero order......Page 471
9. The modal propositional calculus......Page 475
10. Modal theories of the first order......Page 479
11. The modal predicate calculus......Page 484
Bibliography......Page 487