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.

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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

Operation, absolute      23
Operation, complementary operation      326
Operation, dual      326
Operation, Souslin      324 325
Operation, superposition      326
Order, $<_{L}$       161
Ordinal      10 24
Ordinal in $\textbf{EEST}$       190
Ordinal, $\mathbb{I}$ -ordinal      25
Ordinal, $\mathbb{S}$ -ordinal      25 190
Ordinal, *-ordinal      25
Ordinal, good      232
Ordinals $\alpha_{k}$       160
P-forcing relation      263
Polish space      317
Power class, $\mathcal{P}(X)$       10 21
Power class, $\mathcal{P}(X)$ in $\textbf{HST}$       21
Power set in $\textbf{HST}$       21
Power set in $\textbf{ZFC}$       21
Power set internal      23 32
Power set, $\mathcal{P}(X)$       10 21
Power set, $\mathcal{P}(X)$ , “external”      104
Predicate, membership $\epsilon$       12
Predicate, standardness st      12
Predicate, well-foundedness wf      16
Principle, $\Gamma$ -Red      338
Principle, $\Gamma$ -Sep      338
Principle, $\Gamma\textbf{-Unif}$       345
Principle, $\kappa$ -deep      241
Principle, $\kappa$ -size      241
Principle, $\mathbb{S}$ -Separation      122
Principle, $\mathbb{S}$ -Size Choice      122
Principle, $\Pi_{1}_{1}$ -Red multiple      340
Principle, $\Sigma_{1}_{1}$ -Sep multiple      340
Principle, $\Sigma_{n}$ -Collection      43
Principle, *-Transfer in $\textbf{HST}$       17
Principle, Boundedness      18
Principle, Choice c-size      319
Principle, Compactness      30
Principle, countable      319
Principle, extension      32
Principle, Extension, countable      319
Principle, Formal Truth Completeness      122
Principle, Inner Collection      98
Principle, Inner Dependent Choice      99
Principle, Inner Extension      99
Principle, Inner S. S. Choice      99
Principle, Inner S. S. Choice, restricted      127
Principle, Inner Saturation      93
Principle, Internal Definitions      56
Principle, Internal Induction      56
Principle, Isomorphism Property      279
Principle, Local Idealization      93
Principle, Map-Standardization      99
Principle, Overflow      56
Principle, Permanence      57
Principle, Reflection      43
Principle, Saturation      30
Principle, SMA      294
Principle, Special Model Axiom      294
Principle, Underflow      57
Principle, Uniqueness      99
Principle, well-ordering      21 32
Prj U      326
Problem of external sets      5
Problem of external sets in BST      102
Problem of external sets in IST      117
Projection      322 340
Quantifier, $Q_{U}iR(i)$       325
Quantifier, $\exists^{bd}$       111
Quantifier, $\exists^{int}$       14
Quantifier, $\exists^{st}$       14
Quantifier, $\exists^{wf}$       16
Quantifier, $\exists^{\infty lg}$       58
Quantifier, $\forall^{bd}$       111
Quantifier, $\forall^{int}$       14
Quantifier, $\forall^{stfin}$       85
Quantifier, $\forall^{st}$       14
Quantifier, $\forall^{vf}$       16
Quantifier, $\forall^{\infty lg}$       58
Quantifier, bounded      42
Quantifier, U-many      141
Quotient power      141
Quotient power, set-indexed      141
Quotient structure      47
Quotient structure, $\bar{e}=\langle E/ =; \epsilon, st \rangle$       185
Rank in a wf tree      192 197
Rank, irk x      44
Rank, nrk x      261
Rank, rank x      42
Rationals, $\mathbb{Q}$       54
REAL      54
Real, hyperreal      54
Real, hyperreal, appreciable      55
Real, hyperreal, bounded      55
Real, hyperreal, infinitely large      55
Real, hyperreal, infinitesimal      55
Real, hyperreal, limited      55
Real, hyperreal, near-standard      55
Real, hyperreal, standard      55——
Real, hyperreal, unbounded      55
Real, hyperreal, unlimited      55
Reals, $\mathbb{R}$       54
Reduction to true equality      46
Reflects      43
Regular extension      259
Relation, extensional      303
Relation, invariant      45
Relation, transitive      215 303
Relation, well-founded      16
Relation, well-founded, externally      104
Relative standard      89 221 223
Relativization      10 46 141
Relativization to a $\epsilon$ —structure      132
Relativization to a st- $\epsilon$ —structure      132
Relativization, $\Phi^{bd}$       111
Relativization, $\Phi^{int}$ of an $\epsilon$ —formula      14
Relativization, $\Phi^{st}$       14
Relativization, $\Phi^{wf}$       16
Relativization, $\varphi^{V}$       43
Relativization, $^{a}\Phi$       202
Relativization, $^{e}\Phi$       185
Robinson’s Lemma      62
S      296
S0rd      190
Saturation, D-Saturation      57
Scheme, “ $\mathbb{S}\subseteq\mathbb{I}$ ”      83
Scheme, “ $\mathbb{WF}\dot{\rightarrow}\mathbb{I}[in \mathbb{H}]$ ”      22
Scheme, “ $\mathbb{WF}\dot{\rightarrow}\mathbb{I}_{k}[in \mathbb{L}[\mathbb{I}_{k}]}$ ”      253
Scheme, “ $\mathbb{WF}\dot{\rightarrow}\mathbb{I}_{k}[in \mathbb{WF}[\emph{f}]]$ ”      254
Scheme, “ $\mathbb{WF}^{feas}\dot{\rightarrow}\mathbb{I}[in \mathbb{H}]$ ”      291
seq      191
Sequence, $\mathbb{N}$ -sequence      307
Set of S-size      290
Set of standard size      19
Set, $(< \kappa)$ -closed      215
Set, $*\mathbb{N}$ of all *-natural numbers      26
Set, $\cap$ -closed      19
Set, $\emph{f}$ -wrong      237
Set, $\kappa$ -closed      215
Set, $\kappa$ -distributive      215
Set, $\kappa$ -specially distributive      215
Set, $\lambda$ -complete      233
Set, $\mathbb{N}$ of all natural numbers      10 26
Set, $\mathbb{N}$ of all natural numbers in $\textbf{EEST}$       190
Set, $\mathbb{N}$ of all natural numbers in “internal” theories      90
Set, $\textit{w}$ -standard      89 223
Set, $\textit{w}$ -standard in the modified sense      224
Set, *-finite set      26
Set, absolute      23

Set, analytic      322
Set, Borel      321 322
Set, Borel in H      322
Set, Borel, over $\mathcal{A}$       321
Set, bounded      111 363
Set, C-complete      16
Set, CD      327
Set, closed      58
Set, cofinal      35
Set, coinitial      35
Set, compact      58
Set, condensable      291
Set, condensed      17
Set, constructive      159
Set, countable      320
Set, countably determined      327
Set, definable in V      161
Set, dense      266
Set, elementary external      4 186
Set, extendible      134
Set, external      12
Set, feasible well-founded      291
Set, finite set      10 26
Set, finite set in $\textbf{EEST}$       190
Set, finite set in internal theories      90—
Set, generic      266
Set, hereditarily finite      28
Set, hereditarily finite in EEST      190
Set, homogeneous      374
Set, hyperfinite set      26
Set, index set      325
Set, inductive      192
Set, internal      12 84
Set, internal set in $\textbf{IST}$       111
Set, large      40
Set, Loeb measurable      351
Set, open dense      215
Set, projective      322
Set, projective in H      322
Set, small      40
Set, Souslin      324
Set, Souslin in H      324
Set, Souslin over $\mathfrak{B}$       38 324
Set, standard      12
Set, standard size closed      267
Set, standard size distributive      268
Set, sub-internal      203
Set, T-extendible      134
Set, transitive      16
Set, truth set good      122
Set, U-measurable      141
Set, uniform      340
Set, well-founded      16
Set, well-founded over      239
Set, well-founded over U      44
Set, well-founded, externally      104
Set, “external”, in BST      101
Set, “planar”      340
Set-like collection      189
Set-size collection      104
Sets $y_{k}$       160
Sets, equinumerous      10 24 270
Sets, external problem of      102
Shadow      56 329
Shadow map      64 329
Shadow, $\mathcal{A}$ -shadow      329
Souslin operation      325
st      12
St- $\epsilon$ -structure      46 132
Standard core      132
Standard core embedding      132
Standard core interpretability      133
Standard core interpretation      133
Standard part      56
Structure      45
Structure with true equality      46
Structure, $\epsilon$ -structure      46 132
Structure, $\kappa$ -saturated      138
Structure, $\kappa$ -saturated, strongly      140
Structure, $\mathfrak{L}$ -structure      45
Structure, $\textit{a} = (A;{ }^{a}\epsilon,{ }^{a}st; { }^{a}=\rangle$       202
Structure, $\textit{e} = \langle E;{ }^{e}\epsilon,{ }^{e}st;{ }^{e}= \rangle$       184
Structure, domain of      45
Structure, internally presented      273
Structure, invariant      45
Structure, quotient      47
Structure, set size      48
Structure, st- $\epsilon$ —structure      46 132
Structure, strongly $\kappa$ -saturated      140
Structure, underlying      47
Structure, universe of      45
Sub-internal core      203
Subint x      203
Submodel elementary      43
Sup X      42
Superposition of operations      326
Support      149 150
TC(x)      44
Theorem, Collection for set-like classes      189
Theorem, Collection in $\textbf{BST}$       98
Theorem, Collection in $\textbf{EEST}$       189
Theorem, Collection in $\textbf{IST}$       114
Theorem, Dependent Choice in $\textbf{EEST}$       190
Theorem, induction      90
Theorem, Inner S. S. Choice in $\textbf{BST}$       100
Theorem, Map-Standardization in $\textbf{BST}$       100
Theorem, parametrization in BST      103
Theorem, Reduction to $\Sigma^{st}_{2}$ in $\textbf{BST}$       94
Theorem, Reduction to $\Sigma^{st}_{2}$ in $\textbf{IST}$       113
Theorem, Saturation in $\textbf{BIST}$       92
Theorem, Saturation in $\textbf{EEST}$       190
Theorem, Separation for set-like classes      189
Theorem, Standard Size Choice in $\textbf{EEST}$       189
Theorem, Standardization in $\textbf{EEST}$       189
Theorem, Uniqueness in $\textbf{BIST}$       89
Theorem, Uniqueness in $\textbf{BST}$       100
Theorem, Uniqueness in $\textbf{IST}$       117
Theory, $\Delta_{0}\textbf{-ZC}$       126
Theory, $\Sigma_{n}-\textbf{ZFC}$       43
Theory, $\textbf{*ZCN}$       125
Theory, $\textbf{*ZC}}$       125
Theory, $\textbf{BST[T]}$       87
Theory, $\textbf{BST}$       86
Theory, $\textbf{BST}\vartheta$       157
Theory, $\textbf{BST}_{\kappa}$       105
Theory, $\textbf{BST}_{\kappa}^{’}$       105
Theory, $\textbf{DNST}$       312
Theory, $\textbf{EEST}$       182
Theory, $\textbf{EST}$       313
Theory, $\textbf{HST}$       13 20
Theory, $\textbf{HST}_{\kappa}$       241
Theory, $\textbf{HST}_{\kappa}^{'}$       7 241
Theory, $\textbf{HST}_{\kappa}^{-}$       241
Theory, $\textbf{IST}$       84
Theory, $\textbf{IST}[T]$       87
Theory, $\textbf{IST}[\textbf{ZC}]$       87
Theory, $\textbf{IST}^{'}$       127
Theory, $\textbf{IST}^{+}$       169
Theory, $\textbf{KSTI}$       300
Theory, $\textbf{KST}$       290
Theory, $\textbf{NBG}$       313
Theory, $\textbf{NCT}$       314
Theory, $\textbf{NST}$       295
Theory, $\textbf{NST}^{+}$       298
Theory, $\textbf{RST}$       313
Theory, $\textbf{SNST}$       312
Theory, $\textbf{THS}$       314
Theory, $\textbf{T}^{st}$       87
Theory, $\textbf{ZCN}$       124
Theory, $\textbf{ZC}$       21

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