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Kunen K. — The Foundations of Mathematics
Kunen K. — The Foundations of Mathematics

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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: en

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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Предметный указатель
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, $\kappa$-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 $(\emptyset)$      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 $(1-1 \rightarrow onto)$      25
Function, composition $(G \circ F)$      27
Function, injection $(1-1 \rightarrow)$      25
Function, restriction of $(\uparrow)$      25
Function, surjection $(onto \rightarrow)$      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 $(R^{-1})$      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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