Nonstandard Analysis: Axiomatically

Front cover......Page 1
Title......Page 2
Title page......Page 3
Date-line......Page 4
Preface......Page 5
Table of Contents......Page 11
Introduction......Page 17
Basic notation......Page 26
1 Getting started......Page 27
1.1a The universe of HST......Page 28
1.1c Axioms for standard and internal sets......Page 30
1.1d Well-founded sets......Page 32
1.1e The $\\in$-structure of internal and well-founded sets......Page 33
1.1f Axioms for sets of standard size......Page 35
1.1h Zermelo - Fraenkel theory ZFC......Page 36
1.2b Closure properties and absoluteness......Page 38
1.2c Ordinals and cardinals......Page 40
1.2d Natural numbers, finite and *-finite sets......Page 41
1.2e Hereditarily finite sets......Page 44
1.3a Cardinalities of sets of standard size......Page 45
1.3b Saturation and the Hrbacek paradox......Page 46
1.3c The principle of Extension......Page 48
1.4a Basic properties of $\\Delta_2^{ss}$......Page 50
1.4b Cuts (initial segments) of $\\ast$-ordinals......Page 51
1.4c Monads and transversals......Page 53
1.4d On non-well-founded cardinalities......Page 54
1.4e Small and large sets......Page 56
1.5a Von Neumann hierarchy and Reflection in ZFC......Page 58
1.5b Von Neumann hierarchy over internal sets in HST......Page 60
1.5c Classes and structures......Page 61
1.5d Interpretations......Page 63
1.5e Models......Page 64
1.5f Simulation of models of ZFC......Page 65
1.5g Asterisk is an elementary embedding......Page 66
Historical and other notes to Chapter 1......Page 68
2 Elementary real analysis in the nonstandard universe......Page 69
2.1a Hyperreals......Page 70
2.1b Fundamentals of nonstandard real analysis......Page 72
2.1c Directed Saturation......Page 73
2.1d Nonstandard characterization of closed and compact sets......Page 74
2.2 Sequences and functions......Page 75
2.2a Limits......Page 76
2.2b Continuous functions......Page 77
2.2d Robinson\'s lemma and uniform limits......Page 78
2.3a Shadows and equivalences......Page 80
2.3b Near-standard elements......Page 82
2.3c Topology......Page 85
2.4a Euler factorization of the sine function......Page 89
2.4b Jordan curve theorem......Page 92
Historical and other notes to Chapter 2......Page 97
3 Theories of internal sets......Page 99
3.1a Internal set theory......Page 100
3.1b Bounded set theory......Page 102
3.1c Internal sets interpret BST in the external universe......Page 103
3.1d Basic internal set theory......Page 104
3.1e Standard natural numbers and standard finite sets......Page 106
3.1f Remarks on Basic Idealization and Saturation......Page 108
3.2a Half-bounded forms of Idealization......Page 109
3.2b Reduction to two \"external\" quantifiers......Page 110
3.2c Finite axiomatizability of BST and other corollaries......Page 111
3.2d Collection in BST......Page 113
3.2e Other basic theorems of BST......Page 115
3.2f Introduction to the problem of external sets......Page 117
3.2g More on \"external sets\" in BST......Page 120
3.3a Two schemes of partially saturated internal theories......Page 121
3.3b $\\kappa$-deep Basic Idealization scheme......Page 122
3.3c $\\kappa$-size Basic Idealization scheme......Page 125
3.4a Bounded sets in IST......Page 127
3.4b Bounded formulas: reduction to two \"external\" quantifiers......Page 129
3.4c Collection in IST......Page 130
3.4d Uniqueness in IST......Page 133
3.5a Truth definition for the standard universe......Page 134
3.5b Connection with the ordinary truth......Page 136
3.5c Extension of the definition of formal truth......Page 138
3.6a Standard and nonstandard theories of Nelson\'s system......Page 140
3.6b The background nonstandard universe......Page 141
3.6c Three \"myths\" of IST......Page 143
Historical and other notes to Chapter 3......Page 145
4 Metamathematics of internal theories......Page 147
4.1a Nonstandard extensions of structures......Page 148
4.1b Nonstandard extensions of theories......Page 149
4.1c Comments......Page 150
4.1d Metamathematics of internal theories: the main results......Page 152
4.2a Saturated structures and nonstandard set theories......Page 154
4.2b Quotient power extensions......Page 156
4.2c Adequate and good uitrafilters and ultrapowers......Page 158
4.2d Elementary chains of structures......Page 160
4.3a Warmup: several examples......Page 162
4.3b Infinite Fubini products of adequate uitrafilters......Page 164
4.3c Standard core interpretation of BST in ZFC......Page 166
4.3d Saturated standard core interpretation......Page 168
4.4a Good extensions of von Neumann sets in ZFC universe......Page 170
4.4b Iterated adequate extensions of von Neumann sets......Page 171
4.4d Long iterated quotient power chains......Page 172
4.4e Conservativity of IST by inner models......Page 173
4.5a The minimality axiom......Page 175
4.5b The source of counterexamples......Page 176
4.5c The ultrafilter......Page 177
4.5d \"Definable\" adequate quotient power......Page 179
4.5e Corollaries and remarks......Page 180
4.6a Standard theory with a global choice and a truth predicate......Page 182
4.6b Formally definable classes......Page 184
4.6c A nonstandard theory extending IST......Page 185
4.6d The ultrafilter......Page 186
4.6e The interpretation......Page 189
4.6f Extendibility of standard models......Page 191
Historical and other notes to Chapter 4......Page 192
5 Definable external sets and metamathematics of HST......Page 195
5.1a Internal core embeddings and interpretability......Page 196
5.1b Metamathematics of HST : an overview......Page 197
5.2a Interpretation of EEST in BST......Page 200
5.2b Elementary external sets in external theories......Page 202
5.2c Some basic theorems of EEST......Page 204
5.2d Standard size, natural numbers, finiteness in EEST......Page 205
5.3a Well-founded trees......Page 207
5.3b Coding of the assembling construction......Page 208
5.3c Examples of codes......Page 209
5.3d Regular codes......Page 211
5.4a The domain of the interpretation......Page 212
5.4b Basic relations between codes......Page 214
5.4c The structure of basic relations......Page 216
5.4d The interpretation and the embedding......Page 218
5.4e Verification of the HST axioms......Page 220
5.4f Superposition of interpretations......Page 223
5.4g The problem of external sets revisited......Page 225
5.5a Sets constructible from internal sets......Page 227
5.5b Proof of the theorem on $\\mathbb{I}$-constructible sets......Page 228
5.5c The axiom of $\\mathbb{I}$-constructibility......Page 230
5.5d Transfinite constructions in $\\mathbb{L}[\\mathbb{I}]$......Page 231
Historical and other notes to Chapter 5......Page 233
6 Partially saturated universes and the Power Set problem......Page 235
6.1a Some basic definitions and results......Page 236
6.1b Relative standardness......Page 237
6.1c Simple relative standardness......Page 238
6.1d Gordon classes......Page 240
6.1e Associated structures......Page 241
6.1f More on internal subuniverses......Page 244
6.1g Appendix: Kunen\'s theorem......Page 245
6.2a Partially saturated classes $\\mathbb{I}_\\kappa$......Page 246
6.2b Good internal subuniverses......Page 248
6.2c Internal universes over complete sets......Page 249
6.3a External universes and internal core extensions......Page 253
6.3b Von Neumann construction over non-transitive classes......Page 255
6.3c Absoluteness for external subuniverses......Page 256
6.4a Partially saturated external theories......Page 257
6.4b Extensions of thin classes......Page 259
6.4c Constructible extensions......Page 260
6.4d Constructible extensions of self-definable classes......Page 262
6.4e The classes $\\mathbb{L}[\\mathbb{I}_\\kappa]$......Page 264
6.4f External universes over complete sets......Page 265
6.4g Collapse onto a transitive class......Page 267
6.4h Outline of applications: subuniverses satisfying Power Set......Page 268
Historical and other notes to Chapter 6......Page 270
7 Forcing extensions of the nonstandard universe......Page 273
7.1a Ground model......Page 274
7.1b Regular extensions......Page 275
7.1c Forcing notions and names......Page 276
7.1d Adding a set......Page 277
7.1e Forcing relation......Page 279
7.1f Generic extensions and the truth lemma......Page 282
7.1g The extension models HST......Page 283
7.2a Making two internal sets equinumerous......Page 286
7.2b Internal preserving bijections......Page 288
7.2c Making elementarily equivalent structures isomorphic......Page 289
7.2d The forcing notion......Page 290
7.2e Key lemma......Page 292
7.2f Generic isomorphisms......Page 294
7.3 Consistency of the isomorphism property......Page 295
7.3a The product forcing notion......Page 296
7.3b Externalization......Page 297
7.3c Restricted forcing relations......Page 298
7.3d Automorphisms and the restriction property......Page 299
7.3e The product generic extension......Page 300
Historical and other notes to Chapter 7......Page 303
8 Other nonstandard theories......Page 305
8.1a The axioms of Kawai\'s theory......Page 306
8.1b Metamathematical properties......Page 308
8.1c Special model axiom......Page 309
8.2a Axioms......Page 311
8.2b Additional axioms of Collection......Page 313
8.2c Conservativity and consistency......Page 314
8.2d Remarks and exercises......Page 317
8.3a Boffa\'s non-well-founded set theory......Page 319
8.3b Extensions of proper classes......Page 321
8.3c Applications to nonstandard analysis......Page 322
8.3d Alpha theory......Page 323
8.3e Interpretation of Alpha theory in ZFBC......Page 326
8.4a A theory with \"definable\" Saturation......Page 327
8.4b Stratified nonstandard set theories......Page 328
8.4c Nonstandard class theories......Page 329
Historical and other notes to Chapter 8......Page 331
9 \"Hyperfinite\" descriptive set theory......Page 333
9.1a General set-up......Page 335
9.1b Comments on notation......Page 336
9.1c Borel and projective sets in a nonstandard domain......Page 337
9.1d Some applications of countable Saturation......Page 339
9.1e Operation A and Souslin sets......Page 340
9.2a Operations and quantifiers......Page 341
9.2b Countably determined sets......Page 343
9.2c Shadows or standard part maps......Page 345
9.3a Operations associated with Borel and projective classes......Page 347
9.3b The \"shadow\" theorem......Page 348
9.3c Closure properties of the classes......Page 351
9.4a Separation and reduction......Page 354
9.4b Countably determined sets with countable cross-sections......Page 356
9.4c Countably determined sets with internal and $\\Sigma_1^0$ cross-sections......Page 359
9.4d Uniformization......Page 360
9.4e Variations on Louveau\'s theme......Page 363
9.4f On sets with $\\Pi_1^0$ cross-sections......Page 366
9.5a Definitions and examples......Page 367
9.5b Loeb measurability of projective sets......Page 369
9.5c Approximations almost everywhere......Page 370
9.5d Randomness in a hyperfinite domain......Page 372
9.5e Law of Large Numbers......Page 374
9.5f Random sequences and hyperfinite gambling......Page 375
9.6a Preliminaries......Page 378
9.6b Borel cardinals and cuts......Page 380
9.6c Proof of the theorem on Borel cardinalities......Page 382
9.6e Countably determined cardinalities......Page 384
9.7 Equivalence relations and quotients......Page 386
9.7a Silver\'s theorem for countably determined relations......Page 387
9.7b Application: nonstandard partition calculus......Page 389
9.7c Generalization......Page 391
9.7d Transversals of \"countable\" equivalence relations......Page 392
9.7e Equivalence relations of monad partitions......Page 394
9.7f Borel and countably determined reducibility......Page 396
9.7g Reducibility structure of monad partitions......Page 398
Historical and other notes to Chapter 9......Page 402
References......Page 405
Index......Page 413
Springer Monographs in Mathematics Series......Page 425
Back cover......Page 427