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Barnes D., Mack J. — An Algebraic Introduction to Mathematical Logic
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Название: An Algebraic Introduction to Mathematical LogicАвторы: Barnes D., Mack J. Аннотация: This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub stantial course on abstract algebra. Consequently, our treatment ofthe sub ject is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of . the exercises. We also assurne a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model oflogic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics co me only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1975Количество страниц: 130Добавлена в каталог: 21.09.2014Операции: Положить на полку |
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59 61 61 Addition, theory of 61 Adequacy 18 Adequacy Theorem 22 36 Algebra, free 4 Algebra, full first-order 27 Algebra, Lindenbaum 22 Algebra, proposition 12 Algebra, reduced first-order 28 Algebra, relatively free 8 Algebra, universal 2 Algebraic theory 72 Alphabet of Turing machine 85 Alphabet of Turing machine, universal 90 Arity 1 Axiom of Choice 55 57 Axiom of extension 53 56 Axiom of infinity 54 57 Axiom of pairing 53 57 Axiom of Power Set 54 57 Axiom of regularity 56 Axiom of union 54 57 Axiom Schema of Replacement 55 57 Axiom Schema of Restriction 55 57 axiom schema of subsets 53 56 Axioms of a theory 40 Axioms of Pred(V, 30 Axioms of Prop(X) 15 Bounded universal quantifier 108 Cardinal of model 44 Cardinal of theory 45 Categorical 44 47 Categorical in cardinal 45 47 Church's Theorem 102 Church's thesis 105 Compactness Theorem 22 37 Completeness 43 44 47 51 Computation by Turing machine 88 Concurrent relation 76 Conjugacy problem 110 Consequence 13 30 Consequence, proper 39 Consistency 18 Consistency of first-order theory 43 Consistency theorem 19 34 Consistent subset 40 Consistent subset, maximal 40 Constant 40 Continuity 83 Continuity, uniform 83 Decidability 19 94 Decidability of Prop(X) 24 Decidability of Theory of Equality 51 Deduction 15 31 Deduction Theorem 19 34 Definable 43 Definable, strongly 97 Dense linear order 47 Depth of quantification 28 Description of relation 76 Description, instantaneous of Turing machine 87 Description, shortest 87 Diophantine set 107 Domain of relation 76 Dual 42 Effectively axiomatised theory 93 Elementary 41 Elementary Group Theory 41 Elementary theory of fields 45 Enlargement of standard object 77 Enlargement of theory 77 Enlargement, full 77 Extension of model 77 Extension of theory 77 Failure of Turing machine 88 Field, algebraic closure of 66 Fields, Elementary Theory of 45 Filter 63 Filter, existence of 65 Filter, Frechet 63 Finite intersection property 65 Generalisation 31 Godel's Incompleteness Theorem 100 Goedel number 91 92 Goedel's completeness theorem 37 Hilbert's tenth problem 105 Homomorphism 13 Identical relation 7 Identity, axioms of 38 Identity, Predicate Calculus with 39 Incompleteness of recursive arithmetic 100 Infinite real numbers 81 Infinitesimal real numbers 81 Initial state 88 Internal relation 80 Interpretation 30 Interpretation, proper 39 Isomorphism of models 44 Isomorphism of T-algebras 4 Isomorphism problem 110 Kalmar's Theorem 103 kth-order objects 75 Language of theory 40 Law of a variety 7 Law of an algebra 7 Limit of function 83 Limit of sequence 82 Limit, direct 70 Limit, existence of 72 Logic 18 Loуwenheim — Skolem Theorem 46 Minimalisation 106 Model of first-order theory 42 Model of Peano arithmetic 61 Model, non-standard 75 Model, standard 75 modus ponens 15 Monad 82 Natural numbers in ZF 54 Non-standard model 75 Normal form, conjunctive 24 Normal form, disjunctive 24 Normal form, prenex 37 Operation 1 2 pair 53 Pairing function 106 Path 112 Peano axioms 58 59 Predicate 26 Presentation 110 111 Primary proposition 49 Primitive recursion 105 Problem, Conjugacy 110 Problem, decision 94 Problem, Hilbert's Tenth 105 Problem, Isomorphism 110 Problem, Stopping 95 Problem, word 110 Projective geometry 40 proof 15 31 Proper consequence 39 Proper consequence, interpretation 39 Quantifier, bounded universal 108 Quantifier, elimination 48 Quantifier, existential 26 Quantifier, reduction 48 Quantifier, universal 26 Recursive arithmetic 97 Recursive arithmetic, function 89 106 Recursive arithmetic, subset 90 Recursively enumerable 90 94 Recursively enumerable but non-recursive 95 Recursively soluble 94 Relation, concurrent 76 Relation, definable 43 Relation, external 80 Relation, identical 7 Relation, internal 80 Russell paradox 53 Satisfiability Theorem 21 36 39 see Adequacy Theorem 36 Semantic implication 13 30 Semi-homomorphism 32 33 Semigroup of Turing machine 111 Sequence number function 97 Soundness 18 Soundness Theorem 19 33 Special word 112 Standard model 75 Standard relation 78 Standard system 74 Standard system, higher-order 76 Standard system, language of 74 Standard system, theory of 74 State function 97 State of Turing machine 87 State of Turing machine, initial 88 State of Turing machine, terminal 88 STEP 112 Stopping problem 95 Strongly definable 97 Substitution theorem 17 33 Subsystem of standard system 77 Subultraproduct 69 Successor 54 Syntactic implication 15 31 T-algebra 2 t-subalgebra 3 Tautology 14 30 Terminal state 88 Terminal state, word 112 Theorem 15 31 40 100 Theory 40 Transitive set 55 Truth 13 29 Truth, function 23 Turing machine 85 Turing machine, universal 92 TYPE 2 Ultrafilter 63 Ultrafilter, 67 Ultrafilter, existence of 65 Ultrafilter, principal 63 Ultrafilter, restriction of 63 Ultrafilter, uniform 65 Ultrapowers 67 Ultraproducts 64 Undecidability of predicate calculus 101 Undecidability of recursive arithmetic 100 Uniform convergence 84 Uniform ultrafilter 65 Universal algebra 1 Universal algebra, alphabet 90 Universal algebra, proposition 72 Universal algebra, Turing machine 92 Valid 14 30 Valuation 13 Variable 6 Variables involved in element of 27 Variables involved in element of element of P 29 Variety of T-algebras 7 Word 6 Word problem 110 Word, special 112
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