Kanovei V.G., Reeken M. — Nonstandard Analysis: Axiomatically :: Электронная библиотека попечительского совета мехмата МГУ

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Kanovei V.G., Reeken M. — Nonstandard Analysis: Axiomatically

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Название: Nonstandard Analysis: Axiomatically

Авторы: Kanovei V.G., Reeken M.

Аннотация:

The book is devoted to nonstandard set theories that serve as foundational basis for nonstandard mathematics. Several popular and some less known nonstandard theories are considered, including internal set theory IST, Hrbacek set theory HST, and others. The book presents the basic structure of the set universe of these theories and methods to effectively develop "applied" nonstandard analysis, metamathematical properties and interrelations of these nonstandard theories between each other and with ZFC and some variants of ZFC, foundational problems of the theories, including the problem of external sets and the Power Set problem, and methods of their solution. The book is oriented towards a reader having some experience in foundations (set theory, model theory) and in nonstandard analysis.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 432

Добавлена в каталог: 11.04.2008

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Предметный указатель

Base      325
Base Base $(2^{I})$       27 321
Base, $B(\Pi^{0}_{\epsilon})$       9 331
Base, $B(\Pi^{1}_{n)$       331
Bijection, internal-preserving      272
Bijection, locally internal      272
Bisimulation      198
Borel cardinality      363
Borel class      321
Borel hierarchy      321
Borel, map      362
Borel, set      321
Card      24
Card X      10 29
Cardinal      10 24
Cardinal, $\mathbb{I}$ -cardinal      25
Cardinal, $\mathbb{S}$ -cardinal      25
Cardinal, *-cardinal      25
Cardinal, *-cardinality      25
Cardinal, regular      106
Cardinal, singular      106
Cardinality      10 29
Cardinality of continuum      371
CD (countably determined)      327
CD-smooth      381
Chain, elementary continuous      144
Characteristic function      325
class      10 45
Class, $0rd^{H}$ of all $\mathbb{H}$ -ordinals      258
Class, $A(\textit{f})$       245
Class, $CD(\mathcal{A})$       327
Class, $\Delta^{0}_{0}[\mathcal{X}]$       321
Class, $\Delta^{0}_{\epsilon(H)$       9
Class, $\Delta^{0}_{\epsilon[H]$       321
Class, $\Delta^{0}_{\epsilon}[\mathcal{X}]$       322
Class, $\Delta^{1}_{n}[H]$       322
Class, $\Delta^{1}_{n}[\mathcal{X}]$       321
Class, $\Delta^{\blacksquare\blacksquare}_{1}$       34
Class, $\Delta^{\blacksquare\blacksquare}_{2}$       34
Class, $\equiv$ -class      47
Class, $\kappa$ --deep saturated      230
Class, $\kappa$ -size saturated      230
Class, $\mathbb{A}$ of all atoms      304
Class, $\mathbb{B}$ of all bounded sets $\textbf{IST}$       111
Class, $\mathbb{E}$ , elementary external sets in external theories      186
Class, $\mathbb{E}$ , “external sets” in $\textbf{BST}$       102
Class, $\mathbb{E}[\textit{f}]$       245
Class, $\mathbb{H}$ of all sets in a nonstandard universe      12
Class, $\mathbb{H}$ -class      260
Class, $\mathbb{H}[\textbf{G}]$ , generic extension      261
Class, $\mathbb{I}$ of all internal sets      12 258
Class, $\mathbb{I}$ of all internal sets in $\textbf{IST}$       111
Class, $\mathbb{I}$ of all internal sets in internal theories      84
Class, $\mathbb{I}_{\kappa}$       230
Class, $\mathbb{I}_{\kappa}$ in $\textbf{IST}$ and $\textbf{BST}$       111
Class, $\mathbb{I}_{\kappa}^{’}$       255
Class, $\mathbb{L}[\mathbb{I}]$ of all sets constructible from internal sets      202 211
Class, $\mathbb{L}[\mathbb{I}_{\kappa}]$       248
Class, $\mathbb{L}[\textit{f}]$       245
Class, $\mathbb{P}$ of all sub-internal sets      203
Class, $\mathbb{S}$ of all standard sets      12 258
Class, $\mathbb{V}$ , the universe of $\textbf{ZFBC}^{-}$       304
Class, $\mathbb{WF}$ in Boffa’s theory      304
Class, $\mathbb{WF}$ of all well-founded sets      16
Class, $\mathbb{WF}[\mathcal{f}]$       239
Class, $\prod^{0}_{0}[\mathcal{X}]$       321
Class, $\prod^{0}_{<\xi}$       348
Class, $\prod^{0}_{\epsilon}(H)$       321
Class, $\prod^{0}_{\epsilon}[H]$       322
Class, $\prod^{0}_{\epsilon}[\mathcal{X}]$       321
Class, $\prod^{1}_{n}[H]$       322
Class, $\prod^{1}_{n}[\mathcal{X}]$       321
Class, $\prod^{\blacksquare\blacksquare}_{1}$       34
Class, $\prod^{\blacksquare\blacksquare}_{2}$       34
Class, $\subseteq$ -complete      16
Class, $\sum^{0}_{0}[\mathcal{X}]$       321
Class, $\sum^{0}_{\epsilon}[H]$       321
Class, $\sum^{0}_{\epsilon}[\mathcal{X}]$       321
Class, $\sum^{0}_{\xi}[H]$       322
Class, $\sum^{1}_{n}[H]$       322
Class, $\sum^{1}_{n}[\mathcal{X}]$       321
Class, $\sum^{\blacksquare\blacksquare}_{1}$       34
Class, $\sum^{\blacksquare\blacksquare}_{2}$       34
Class, $\textbf{E}$       184
Class, $\textbf{Proj}[H]$       322
Class, $\underline{A(\textit{f})}$       245
Class, $\underline{A}$       197
Class, 0rd of all *-ordinals      25
Class, 0rd of all ordinals      10 24
Class, A      197
Class, almost universal      306
Class, Borel      321
Class, Borel $(\mathcal{A})$       321
Class, Borel [H]      322
Class, Card of all *-cardinals      25
Class, Card of all cardinals      24
Class, CD[H]      327
Class, complete over      237
Class, extensional      220 237
Class, external subuniverse      237
Class, formally $\mathfrak{L}_{\epsilon, G}$ -definable      168
Class, internal subuniverse      220
Class, Nms(P) of names      261
Class, projective      322
Class, proper      45
Class, S0rd of all $\mathbb{S}$ -ordinals      190
Class, self-definable      246
Class, thin      222
Class, transitive      16
Class, transitive over      237
Class, universal class V      304
Clopen sets      321
Code, $\mathcal{f}$ -regular      245
Code, A-code      192
Code, E-code      102
Code, regular      195
Coded formula      118 166
Collection, set-size collection      104
Complement, $X^{C}$       321
Complementary base      326
Complementary operation      326
Complete, $\subseteq$ -complete      16
Complete, over      237
Concatenation      191
Core, internal      180
Core, standard      132
Core, sub-internal      203
Core, well-founded      134
Countably determined, map      362
Countably determined, set      327
Cross-section      340
Cut      35 364
Cut (initial segment)      37
Cut, additive      37 364
Cut, countably cofinal      35 364
Cut, countably coinitial      35 364
Cut, minimal      368
Cut, standard size cofinal      35
Cut, standard size coinitial      35
C[y]      194
Descriptive set theory, Polish      317
Descriptive set theory, “hyperfinite”      317
Direct limit      145
Dom P      322
Domain      317 322
Dual operation      326
E-code      102
Elementary continuous chain      144
Embedding, $\epsilon$ -embedding      132
Embedding, elementary      133

Embedding, internal core embedding      180
Embedding, natural, *,      141
Embedding, st- $\epsilon$ —embedding      180
Embedding, standard core embedding      132
Entire part      364
Equinumerous      10 24
Equivalence class, $[p]^{o}$       185
Equivalence class, $[x]^{a}$       211
Equivalence relation, $M_{U}$       378
Equivalence relation, B-smooth      381
Equivalence relation, CD-smooth      381
Equivalence relation, countably cofinal      378
Equivalence relation, countably coinitial      378
Equivalence relation, thin      371
Equivalence relation, “countable”      376
ER (equivalence relation)      371
Exp $\kappa$       151
Extension      132
Extension of an $\epsilon$ —structure      132
Extension, *-extension      16 59
Extension, conservative      49
Extension, elementary      133
Extension, internal core      180 181 237
Extension, internal core, conservative      181
Extension, regular (of a model)      259
Extension, standard core      132 133
Extension, standard core, conservative      133
Extension, transitive      237
Extension, wf-core, conservative      134
Extensional class      237
Extensional relation      303
External subuniverse      237
Filter      140
Filter, C-adequate      143
Finite intersection property, f.i.p.      30
Finite support      149
Forcing relation      263
Forcing, $\textbf{forc}$       262
Forcing, condition      260
Forcing, condition, stronger      260
Forcing, notion      260
Forcing, set-size      260
FORM      118
Formula, $\epsilon$ -formula      12
Formula, $\Pi_{n}$ formula      42
Formula, $\Sigma^{st}_{2}$       94
Formula, $\Sigma_{n}$       42
Formula, absolute      23
Formula, bounded      42 113
Formula, bounded st- $\epsilon$ -formula      126
Formula, coded      118 166
Formula, directed      57
Formula, external      12
Formula, formally false (f. false)      119 169
Formula, formally true (f. true)      119 169
Formula, int y      2 12
Formula, internal      12
Formula, st- $\epsilon$ —formula      12
Formula, subint x      203
Foundations, model-theoretic      V
Fubini product      148 150
Function, $\mathcal{A}$ -measurable      329
Function, $\mathcal{f}$ -extendible      32
Function, Borel      362
Function, continuous, uniformly continuous      61
Function, countably determined      362
Function, of finite support      150
Function, validation function      48
Gap      36
Graph, locally of set size      305
Ground model      258
Halo      56
Height      258
Hierarchy, Borel      321
Hierarchy, projective      322
Hierarchy, von Neumann      42
Hierarchy, von Neumann, relative      44
Hyperrational      54
Hyperreal (*-real)      54
Hyperreal (*-real), appreciable      55
Hyperreal (*-real), bounded      55
Hyperreal (*-real), infinitely large      55
Hyperreal (*-real), infinitesimal      55
Hyperreal (*-real), limited      55
Hyperreal (*-real), near-standard      55
Hyperreal (*-real), standard      55
Hyperreal (*-real), unbounded      55
Hyperreal (*-real), unlimited      55
ind      280
Index set      140
Induction      26
Induction, $\epsilon$ —induction      28
Induction, on the $\prec$ -rank      265
Induction, transfinite      26
Inf. Large Exchange      58
Int y      12
Internal core      180
Internal core embedding      180
Internal core extension      180 237
Internal core interpretability      181
Internal core interpretation      181
Internal power set      23 32
Internal subuniverse      220
Internally presented      273
Interpretability, internal core      181
Interpretability, standard core      133
Interpretation      47
Interpretation, internal core      181
Interpretation, standard core      133
Invariant      45
Invariant $\mathfrak{L}$ -structure      45
Isomorphism, $\epsilon$ -isomorphism      132
Language, $\epsilon$ -language      12
Language, $\mathfrak{L}_{\epsilon, G, T}$       166
Language, $\mathfrak{L}_{\epsilon, G}$       148
Language, $\mathfrak{L}_{\epsilon, st, int}$       290
Language, extended $\mathfrak{L}^{\infty}$       275
Language, of standard size      273
Language, st- $\epsilon$ -language      12
Lifting      355 376
Loeb measure      352
Loeb measure, finite      352
Map, $\mathcal{A}$ -measurable      329
Map, Borel      362
Map, countably determined      362
Maxt      191
Measure, finite Borel      353
Membership predicate $\epsilon$       12
Membership relation $\epsilon_{G}$ on $\mathbb{H}[\textbf{G}]$       261
Mint      191
Model      48
Model of a theory      49
Model, $\mathbb{H}$       258
Model, extendible      134
Model, ground model      258
Model, T-extendible      134
Model-theoretic foundations      V
Mon U      95
Monad      56 95
Monad, U-monad      37 378
NAME      261
Natural embedding, *      141
Natural number      10 26
Natural number in EEST      190
Natural number in “internal” theories      90
Natural number, *-natural number      26
Nelson’s algorithm      94 127
Nrk x      261
Number, infinitely large      27
Operation      325

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