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Kunen K. — The Foundations of Mathematics
 Обсудите книгу на научном форуме Нашли опечатку?Выделите ее мышкой и нажмите Ctrl+Enter Название: The Foundations of Mathematics Автор: Kunen K. Аннотация: Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general. This book is designed for students who plan to specialize in logic, as well as for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginning graduate-level course. There are three main chapters: Set Theory, Model Theory, and Recursion Theory. The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms. It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals. The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and L?wenheim-Skolem Theorems, elementary submodels, model completeness, and applications to algebra. This chapter also continues the foundational issues begun in the set theory chapter. Mathematics can now be viewed as formal proofs from ZFC. Also, model theory leads to models of set theory. This includes a discussion of absoluteness, and an analysis of models such as H(?) and R(?). The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of G?del, and Tarski's theorem on the non-definability of truth. Язык: Рубрика: Математика/ Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2009 Количество страниц: 262 Добавлена в каталог: 06.06.2014 Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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 Atomic formula      97 Automorphism      156 Axioms of field theory      116 Axioms of group theory      4 96 Axioms of set theory      9 Axioms of set theory, Axiom of Choice      10 58 Axioms of set theory, Comprehension Axiom      10 18 Axioms of set theory, Extensionality Axiom      10 16 Axioms of set theory, Foundation Axiom      10 69 Axioms of set theory, Infinity Axiom      10 37 Axioms of set theory, Pairing Axiom      10 21 Axioms of set theory, Power Set Axiom      10 48 Axioms of set theory, Replacement Axiom      10 26 Axioms of set theory, Union Axiom      10 22 Bernstein set      82 Bound variable      98 Cardinal      49 Cardinal, arithmetic      63 Cardinal, von Neumann      53 Cartesian product      26 Categorical, -categorical      142 ch      see "Continuum Hypothesis" Choice function      58 Choice set      58 class      19 Cofinality      66 Compactness Theorem      107 140 Complete theory      115 Completeness theorem      130-140 Computably enumerable      200 213 Conservative extension      137 141 150 Consistent      106 123 Continuum Hypothesis      7 13 15 52 65 Countable      52 Countably infinite      52 Counting      14 21 23 CST (Core Set Theory)      172 Dedekind-complete      81 89 90 Diagonal argument      51 67 178 218 Divisible abelian groups      143 Ducks      14 16 69 Empty set       19 Empty structure      114 Equational theory      97 146 Essentially undecidable      228 Field      116 finite      52 Finitist      28 129 187 Formal theory      29 Formula      2 98 Free variable      2 98 Function      25 29 Function, bisection       25 Function, composition       27 Function, injection       25 Function, restriction of       25 Function, surjection       25 GCH      see "Generalized Continuum Hypothesis" Generalized Continuum Hypothesis      65 Goedel number      176 198 Halting problem      200 Hartogs      54 Hausdorff maximal principle      61 HC (the hereditarily countable sets)      76 HF (the hereditarily finite sets)      74 Hyper-exponential      57 208 Inaccessible cardinal      68 78 167 Incompleteness Theorem, First      229 Incompleteness Theorem, Second      241 Inconsistent      106 123 Induction, ordinary      37 Induction, transfinite      39 43 Infinite      52 Isomorphism      28 115 Kleene T Predicate      221 Koenig      67 Lattice      152 Lexicographic order      27 Lexicon      91 Liar paradox      21 241 Loewenheim — Skolem theorem      89 90 107 140 Loewenheim — Skolem — Tarski Theorem      5 153 154 Logical consequence (|=)      106 Logical symbols      95 Logically equivalent      109 Luzin set      82 Maximal      31 Meta-variable      84 100 Metatheory      28 190 191 Minimal      31 modus ponens      5 119 Natural number      36 Non-standard analysis      160 179 Nonlogical symbols      95 Ordinal      15 33 Ordinal, arithmetic      41 Ordinal, limit      36 Ordinal, successor      36 PA      see "Peano Arithmetic" Paradox, Burali — Forti's      36 Paradox, Cantor's      51 Paradox, Russell's      18 51 PAS      174 Peano arithmetic      174-176 Polish notation      90 Precedence      100 Proper class      9 19 29 34 Quantifier elimination      144 Recursion      43 Recursion theorem, first      225 Recursion theorem, second      244 Recursively enumerable      200 213 Register machine      197 Relation      24 29 Relation, equivalence      24 Relation, inverse       27 Relation, irreflexive      24 Relation, partial order      24 Relation, reflexive      24 Relation, total order      24 Relation, transitive      24 Relation, well-founded      31 Relation, well-order      32 Representable      233 Satisfiable      106 Schroeder — Bernstein theorem      50 Scope      93 98 Semantic consequence (|=)      106 Sentence      2 98 Sierpinski set      82 Structure      102 Substitution      110-113 Successor, function      10 23 Successor, ordinal      36 Tautology      118-119 Transitive closure      47 Transitive relation      24 Transitive set      33 Troll      20 70 Truth in a model      105 Truth table      3 Truth, non-definability of      237 Tukey's Lemma      60 Turing machine      197 Turnstile      4 86 103 106 Uncountable      52 Unique readability      92 Universal closure      99 110 Universal set (V)      18 Universe of a model      2 Universe of set theory (V)      18 Vaught's Conjecture      145 Venn diagram      22 Well-founded relation      see "Relation well-founded" Well-founded set      70 Well-order      see "Welation well-order" Zorn's lemma      61
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