Главная    Ex Libris    Книги    Журналы    Статьи    Серии    Каталог    Wanted    Загрузка    ХудЛит    Справка    Поиск по индексам    Поиск    Форум   
blank
Авторизация

       
blank
Поиск по указателям

blank
blank
blank
Красота
blank
Kaye R. — Models of Peano Arithmetic
Kaye R. — Models of Peano Arithmetic

Читать книгу
бесплатно

Скачать книгу с нашего сайта нельзя

Обсудите книгу на научном форуме



Нашли опечатку?
Выделите ее мышкой и нажмите Ctrl+Enter


Название: Models of Peano Arithmetic

Автор: Kaye R.

Аннотация:

Non-standard models of arithmetic are of interest to mathematicians through the presence of infinite integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s, they have come to play an important role in model theory, and in combinatorics through independence results such as the Paris-Harrington theorem. This book is an introduction to these developments, and stresses the interplay between the first-order theory, recursion-theoretic aspects, and the structural properties of these models. Prerequisites for an understanding of the text have been kept to a minimum, these being a basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets. Consequently, the book is suitable for postgraduate students coming to the subject for the first time, and a number of exercises of varying degrees of difficulty will help to further the reader's understanding.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 292

Добавлена в каталог: 02.07.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
blank
Предметный указатель
$(\frac{x}{y})(x/y)$      54
$(\mux)(...)$ (mimmalization)      6 30
$({\mu}y\leq z)(...)$ (bounded minimalization)      7 30
$<_{r}, <_{\Delta_{0}}, <_{\Sigma_{1}}$      24
$a>\mathbb{N}$ (a is nonstandaid)      12
$a^{1/\mathbb{N}}$ (initial segment)      72
$a^{\mathbb{N}}$ (initial segment)      69
$I_{x}\varphi, I\varphi$ (induction axiom)      13
$L_{x}\varphi, L\varphi$(least-number principle)      44
$PA(\mathfrak{L}_{A})$ (extension of PA)      47
$PA^{-}$      16
$Th(\mathbb{N})$ (true arithmetic)      10
$t^{M}$ (value of a closed term)      3
$T_{DIS}$ (discrete linear order)      73
$T_{x}\varphi, T\varphi$ (axiom of total induction)      45
$U^{n}_{l}$ (projection function)      5
$[\frac{x}{y}][x/y]$      54
$\aleph_{a}$ (cardinal)      1
$\beta$-function      64 273
$\Box$ (end of proof)      1
$\chi_{A}(\overline{x}), \chi_{\theta}(\overline{x})$ (characteristic function)      7 29
$\Delta_{0}$ induction      70 73 83 269 271 273
$\Delta_{0}, \Sigma_{1}, \Pi_{1}$(formula classes)      23 78-9
$\forall_{1}-Th(\mathbb{N})$      20
$\Gamma_{\textit{f}}$ (graph of $\textit{f}$)      31
$\Im$ (paitial recursive functions)      5
$\kappa$-like model      101 (see also $\omega_{1}$-like model)
$\mathbb{N}$ (standard model)      10
$\mathcal{L}_{A}$ (language of arithmetic)      10
$\mathcal{PR}$ (primitive recursive (unctions)      6
$\mathcal{X}$-saturated      181—5 263 268
$\omega$-consistent      39
$\omega$-homogeneous      259
$\omega_{1}$-like model      90 101 165 247 251
$\omega_{a}$ (initial ordinal)      1
$\Pi_{1}$ formula      23 78—9 218 275 7
$\Pi_{1}$-theory of $\mathbb{N}$      166 185
$\Pi_{n}$ Formula      79 116 126 7
$\Pi_{n}$-theory of $\mathbb{N}$      128—9
$\Pi_{n}$-theory of M      167
$\prod_{1}/\prod_{n}$      (see also Satisfaction short Type)
$\rightarrow, \leftrightarrow, \exists!$      2
$\Sigma_{1}-Th(\mathbb{N})$      25
$\subseteq_{e}$ (end-extension/mitial segment)      12 21
$\sum_{1}$ formula      23 78—9 274
$\sum_{1}$ formula, sentence      248 274
$\sum_{1}$ theory of N      25 27
$\sum_{1}$-representation of functions and sets      36 40
$\sum_{n}$      (see Satisfaction Satisfaction Type)
$\sum_{n}$ ormula      79 116 126 2
$\sum_{n}$ recursive saturation      146—52 223 269 2
$\sum_{n}$ tallness      152 2
$\sum_{n}$-definable elements      96 130—4 2
$\sum_{n}$-theory of $\mathbb{N}$      128—9 2
$\sum_{n}$-theory of M      166 269 2
$\sum_{n}, \prod_{n}, \Delta_{n}$ (formula classes)      79
$\textit{I}\Gamma, \textit{I}\Delta_{0}, \textit{I}\Sigma_{1}, \textit{I}\Pi_{1}$ (theories)      43
$\underline{0}, \underline{1}, \underline{n}$      10 17
$\wedge, \vee, \square, =, \forall, \exists$      2
* (against exercise numbers)      1
1-consistency (of PA, $\textit{I}\Sigma_{n}$)      218
<,...,>      31
<,> (pairing function)      30
Absoluteness      24
Ackermann function      68
Adamowicz, Z.      269
Admissible set      272
Ajtai, M.      274
Algebraic fragments of arithmetic      267
Arithmetic completeness theorem      186 190—2 223 225
Arithmetic hierarchy      78—81 127
Arithmetic models      186
Arithmetic structures      186
Atomic sentence      236
Automorphism      93 259—61 272-3
Axiom      3
Axiom of collection      33 81
Axiom of definition      47
Axiom of induction      43
Axiom of second-order collection      102
Axiom of second-order induction      42 46
Axiom of total induction      45 (see also Least-number principle)
Axiomatizability, $\Pi_{n+1}$ of $\textit{I}\Sigma_{n}/B\Sigma_{n}$      134
Axiomatizability, finite, of $\textit{I}\Upsilon_{n}/B\Sigma_{n}$      134
Axiomatizability, non-finite, of PA      132
Axiomatizability, recursive      39 43
Barwise, J.      184 248 272 273
Bennett, R.      271
Bennett’s $\Delta_{0}$ definition of exponentiation      271 273—4
Bezout’s Theorem      55
Binary tree      (see Tree)
Block of quantifiers      79 81
Boolean operations      2
Bounded minimalization      7 30
Bounded quantifier      23 78 81
Bounded quantifier, closure under      82
Buss, S.      274
Canonical term      128 233 274
Cantor      75
Cantor Normal Form      221
Card(W)      2
Cardinal      1
Cardinality of a Model      2
Cases, definition bv      7
Cegieiski, P.      267—8
Chain of Models      5
Characteristic function      7 29
Chinese remainder theorem      58 65
Chronic resplendency      252 273
Church — Turing thesis      8
Code of 0s and 1s      172—3
Code of a function      143—5
Code of a sequence      31 58 60 64
Code of a set      13—14 141 172 261
Code of a tuple      30—1 (see also Dyadic coding Goedel-numbering type coded)
Cofinal extensions      86—90
Cofinal subset      86
Collection axioms      33 81—6
Collection axioms, in set theory      78
Collection axioms, parameter-free      134
Collection axioms, second-order      102
Compactness Theorem      3
Complete diagram      4
Complete theory      39
Complete type      94 147
Completeness theorem      3 (see also Arithmetized completeness theorem)
Composition (of functions)      6
Computable set      5
Congruence      54
Conservation results      271—2
Conservative extension of a model      100—1
Conservative extension of a theory      51
Conservative extension, $B\Sigma_{n+1}$ over $\textit{I}\Sigma_{n}$      137 272
Consistency      4 41
Consistency of $\textit{I}\Sigma_{n}$      138 140 218
Consistency of PA      41 218 223 233 246
Consistency, 1-consistency      236
Consistency, FA -consistency      236
Consistency, simple consistency      38
Constant      2
Cook, S.      274
Coprimality      55
Craigs trick      155
Cut      22—3 70 76 194 269—70
Cut Elimination Theorem      140
Cut rule      118 140 238
Davis, M.      88
Dedekind, R.      42
Deduction Theorem      237
Definable elements      91—6 (see also $\Sigma_{n}$-definable elements)
Definable type      102
Defining axiom      47
Definition by cases      7
Definition of truth      3 104 223 225—6
Degree of a model      268
Dense linear order      75 265
Diagonalization      37
Diagram      4
Dimitracopoulos, C.      269 271 274
Discrete linear order      18 73 157 265
Divisability      54
Division      53—4
DLO (dense linear order)      75
Domain (of a model)      2
Downward Loewenheim — Skolem theorem      4 223
Dyadic coding      173—4
Ehrenfeucht, A.      261 267 268
Elementary chain (of models)      5
Elementary end-extension      90 96—103 152 165 247
Embedding      159 389
Empty sequence      172
End-extension      12 21 90 192 205 268—9 Initial
Engeler, E.      273
Eq(x, y) (equality function)      7
Equality Axioms      117
Euclidean division      53
Evaluation of terms      119
Existential (      3
Expansion (of a model)      3 248 252
exponentiation      67 73 274
Exponentiation, Bennett’s $\Delta_{0}$ definition of      271 273—4
Extended Grzegorczyk hierarchy      220—1 271
Extension (of a model)      4
Extension (of a model), cofinal      86
Extension (of a model), conservative      100—1
Extension (of a model), elementary for formulas in $\Gamma$      24 (see also Elementary end-extension End-extension)
FA-consistent      236
Factorial      66
Finite approximation      234—5
Finite sequences      31
Finite sequences of 0s and 1s      168 172 Code
First-order language      (see Language)
Formula      88
Formula, $\Delta_{0}$, $\Delta_{1}$, $\Pi_{1}$      23 78—9
Formula, $\Delta_{n}$, $\Delta_{n}$, $\Pi_{n}$      79 116
Formula, finitely satisfied set of      146
Formula, NP      274
Formula, strict $\Sigma_{n}, \Pi_{n}$      82 85 116
Formula, unique readability of      115
Fragments of PA      43 81—6 130—140 271-2 Open
Fraisse, R.      273
Free-variable      3 116—17
Friedman, H. (v)      137 159 184 268 270
Friedman’s theorem      160 164—5 197 247 269
Function      2
Fundamental sequence (of an ordinal)      221
Gaifman, H. (v)      87 89 96 101 268—9 271
Gaifman’s splitting theorm      89 90
Goedel — Rosser incompleteness theorem      38—9 153 154—5 157 178 189-90
Goedel's $\beta$-function      64 273
Goedel's completeness theorem      3
Goedel, K. (v)      3 31 35 39 218
Goedel-numbering      37 108—10 148 153 224—5
Goedel’s compactness theroem      3
Goedel’s first incompleteness theorem      39 218
Goedel’s lemma      31 35 58—64
Goedel’s second incompleteness theorem      41 218
Goodstein sequences      220
Goodstein, R.L.      220
Graph of a function      31
Grzegorczyk hierarchy      220—1 271
Hajek, P.      271
Hardy, G.H.      53
Harrington, L. (v)      193 270 Paris-Harrington
Henkin-style proof      178
Hercules and the hydra      218—19
Hierarchy of $\Sigma_{n}/\Pi_{n}$ formulas      79—80 104-5 127—8
Hilbert’s irreducibility theorem      267
Hilbert’s Tenth Problem      88 267
Hodges, W.      103 266
Homogeneous model      259 265
Homogeneous set      208
Incompleteness theorem      35—41 132 218 incompleteness Goedel's Goedel’s
Indicator      193 270
Indicator for $\Sigma_{n}$-elementary initial segments      197 204 205
Indicator for initial segments isomorphic to the model      197
Indicator for initial segments satisfying coded theories      203
Indicator for intitial segments having certain expansions      264—5
Indicator for recursively saturated initial segments      204—5 265
Indicator, well-behaved      194 197 270
Indiscernible sequence      207 273
Induction axiom, first-order      43
Induction axiom, open      267
Induction axiom, second-order      42 46
Induction axiom, total induction      45
Inference (in a proof)      118
Infinitude of primes      65—7 70
Initial segment      12 21 69 159 264—5
Initial segment, coded      206
Initial segment, elementary      134—40 164 197 204 205 End-extension Indicator)
Initial segment, isomorphic to the model      (see Friedman’s theorem)
Initial segment, recursively saturated      204—5 265
Integer part (after division)      54
Integral domain      21
Interesting number      95
Interpretation      3
Interpretation, strong      190—2 225
Inverse modulo x      56
Irreducible      2 55—8
Jensen, D.      261 267 268
Joint consistency test      253
Kanamori - McAloon principle      208—18 247
Kanamori, A.      208 218 270
Kaufmann, M.      269
Kaye, R.      267 272 273
Ketonen, J.      221 270
Kirby, L. (v)      133 138 193 220 269 270
Kleene's theorem      254
Kleene, S.C.      127 254
Knight, J.F.      268 269
Koenig’s lemma      173
Kossak, R.      272 273
Kotiarski, H. (v)      233 272 273
Krajewski, S. (v)      233 272
Krajfdek, J.      274 275
Lachlan, A. (v)      228 233 272
Lachlan’s theorem      228—33
Language      2
Language of arithmetic      10 16
Language, Godel-numbering of      (see Goedel-numbering)
Language, recursive      147 152—3
Language, recursive extensions of      179 185
Language, satisfaction class for      224 246 252
Language, symbols of      2
Largeness properties      221—2
Larski s theorem on the undelinabilitv of truth      28 40 223
Least-number principle      44 95—6
Length (of a sequence)      105 172
Lesan      11 (see Lessan H.)
Lessan. H.      138 267
Linear order, dense      75 265
Linear order, discrete      18 73 157 265
Lipshitz, L.      267
Loeb. M.H.      220
Loewenheim — Skolem theorems      4 223
Logical symbol      2
Lt(x, y) (less-than function)      7
M-consistent      236
M-finite      64 65
M-rule      247
1 2
blank
Реклама
blank
blank
HR
@Mail.ru
       © Электронная библиотека попечительского совета мехмата МГУ, 2004-2017
Электронная библиотека мехмата МГУ | Valid HTML 4.01! | Valid CSS! О проекте