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Ebbinghaus H.-D., Flum J., Thomas W. — Mathematical logic
Ebbinghaus H.-D., Flum J., Thomas W. — Mathematical logic

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Название: Mathematical logic

Авторы: Ebbinghaus H.-D., Flum J., Thomas W.

Аннотация:

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete — it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic.


Язык: en

Рубрика: Математика/Алгебра/Математическая логика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1984

Количество страниц: 216

Добавлена в каталог: 21.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\beta$-function lemma      169
$\Delta$-elementary class      92
Accessible      207
Add-instruction      151
Algol      155
Algorithm      146
Alphabet      10
Alphabet of a first-order language      13
AND      13
Antecedent      59
Aristotle      3
Arithmetic      27 96
Arithmetic, Peano      165
Arithmetic, undecidability of      168
Arithmetical relation      172
ASSIGNMENT      27 132
Automorphism      40
Axiomatizable      166
Axioms, system of      7
Back-property      182
Barwise compactness theorem      139
Bolzano      35
Boole      3
Bound occurrence      23
Calculus      14 17
Calculus, induction over a      18
Calculus, rule of a      17
Calculus, sequent      59
Church      154 162
Church's thesis      155
Class of models      91
Class of models, $\Delta$-elementary      92
Class of models, elementary      92
Classical mathematics      103
Coincidence lemma      35
Compactness Theorem      90 134 138 143 195
Complete theory      166
Completeness theorem      87
Computable function      149
Configuration      160
Conjunction      15
Conjunctive normal form      127
Connective      8 28
Consequence relation      5 8 32
Consistency of mathematics      177
Consistency of ZFC      177
Consistent      72
Consistent, maximally      77
Constant      13
Containing witnesses      72
Continuum Hypothesis      110
Correct rule      59
Correct sequent      59
Correctness of $\mathfrak{S}$, theorem on      70
Countable      1
Countable, at most      11
Decidable      146 154
Decision procedure      146
Dedekind"s theorem      47 104
Definition by induction on formulas      21
Definition by induction on terms      21
Definition in a set of sentences      116
Definitions, theorem on      118
Dense ordering      183
Derivable      59 69
Derivable rule      61
Derivation      14
Descending chain      98
Direct product      40
Disjunction      15
Disjunctive normal form      126
Domain      27
Effective procedure      146
Ehrenfeucht game      191
Elementarily equivalent      194
Elementary class      92
Enumerable      147 157
Enumeration procedure      147
Equality symbol      13
Equivalence relation      5
Equivalence relation, axioms for      5
Equivalence structure      6
expansion      37
Extensional      29
Field      44 92
Field, Archimedean      96
Field, ordered      44
Fin-satisfiable      162
Fin-valid      162
Finite structure      93 162
Finitely axiomatizable      166
Finitely isomorphic      182
First-order language      15
First-order language, alphabet of a      13
First-order object      9
Fixed point theorem      174
Follows from      5 7 8
For all      13
Formally provable      59 69
Formula      7 15
Formula Horn      41
Formula in Polish notation      23
Formula, atomic      15
Forth-property      182
FORTRAN      155
Fraisse's theorem      185
Free occurrence      23
Frege      3 17
Function      26
Function, partial      45
Godel      87
Godel completeness theorem      8 87
Godel number      156
Godel numbering      156
Godel's first incompleteness theorem      176
Godel's second incompleteness theorem      177
Group      4
Group theory      4
Group theory, axioms of      4
Group, torsion      46 93
Halt-instruction      151
Halting problem      156
Henkin      78
Henkin's theorem      79
Herbrand's theorem      84
hilbert      3
Hilbert's program      3 112 177
Identilas indiscernibilium      133
If and only if      13
If-then      13
Implication      15
Incompleteness of second-order logic      164
Inconsistent      72
Independent set of sentences      41 94
Induction over a calculus      18
Inductive definition on terms or formulas      21
Infinitary language      137
Instruction      14
Intentional      29
Interpretation      27
Intuitionism      103
Isomorphic      38
Isomorphic, finitely      182
Isomorphic, partially      182
Isomorphism      38
Isomorphism lemma      38
Isomorphism, partial      180
Label      151
Language, first-order      9 15
Language, formal      7
Language, many-sorted      45
Language, object      17
Language, second-order      9 132
Law of the excluded middle      62
Leibniz      3 87 162
Length of a string      10
Liar's paradox      173
Lindstrom      193
Lindstrom's first theorem      199
Lindstrom's second theorem      204
Llull      87 162
Logic, mathematical      1
Logical system      193
Logical system, effective      203
Logical system, effectively regular      203
Logical system, regular      195
Logically equivalent      33
Lowenheim — Skolem theorem      88 134 139 175
Lowenheim — Skolem theorem, downward      89
Lowenheim — Skolem theorem, upward      90
Lowenheim — Skolem — Tarski theorem      91
Matrix      126
Metalanguage      17
Model      31 37
Model theory      98
modus ponens      63
Negation      15
Nonstandard analysis      98
Nonstandard model of arithmetic      96
NOT      13
Object, first-order      9
Object, second-order      9
OR      13 29
Ordering      43
Ordering, partially defined      43
PARAMETER      23
Partial isomorphism      180
Partially isomorphic      182
Peano      17
Peano arithmetic      165
Peano axiom system      47
Peano structure      104
Philosophy      3
Predicate calculus, first-order      15
Prefix      126
Prenex normal form      126
Print-instruction      151
Proof by induction on formulas      19
Proof by induction on terms      19
Proof by induction over a calculus      18
Proof, formal      8 101
Proposition      7
Propositional calculus      30 87
Quantifier      8
Quantifier number      127
Quantifier rank      187
Quantifier, for all      13
Quantifier, there are uncountably many      142
Quantifier, there exists      13
Quantifier-free      126
R-axiomatizable      166
Rank      55
RECURSIVE      155
Recursively enumerable      155
Reduct      37
register      150
Register machine      150
Register program      151
Register-computable (R-computable)      154
Register-decidable (R-decidable)      154
Register-enumerable (R-enumerable)      154
Regular logical system      195
Relation      26
Relation symbol      13
Relation, arithmetical      172
Relation, representable      172
Relational symbol set      120
Relativization      123
Relativization lemma      123
Representable function      172
Representable relation      172
Rule      8 14 17
Rule, connective      60
Rule, derivable      61
Rule, equality      65
Rule, quantifier      65
Rule, structural      60
Russell      3 87
S-closed      122
Satisfaction relation      31
Satisfiable      33 195
Second-order logic      132 164
Second-order object      9
Self-referential      174
Semantic      26
Sentence      24
Sentence of $\mathscr{L}$      193
Sequent      59
Sequent calculus      59 142
Sequent calculus, theorem on the adequacy of the      87
Set theory, background      107
Set theory, object      108
Set theory, system of axioms for      105
Set theory, ZFC axioms for      108
Simple group      141
Skolem"s theorem      96
Skolem's paradox      108 112
Spectrum      44
String      10
String, empty      10
Structure      26
Structure, many-sorted      45
Substitution lemma      52
Substitution, simultaneous      49
Subtract-instruction      151
Succedent      59
Symbol set      13
Symbol, "logical"      13
Symbol, equality      13
Symbol, function      13
Symbol, relation      13
Syntactic      26
Tarski      35
Tarski's theorem      175
Term      14
Term interpretation      79
Term, reduced      121
Term, structure      79
Tertium non datur      62
Theorem on the adequacy of the sequent calculus      87
Theorem on the correctness of the sequent calculus      70
Theory      165
Theory of a structure      94
There exists      13
Trahtenbrot's theorem      163
Truth-value      29
Uncountable      11
Undecidability of arithmetic      168
Undecidability of first-order logic      159
Undecidability of the halting problem      156
Universe      27
Universe, mathematical      104
Urelement      105
Valid      33 195
Variable      13
Variables, set of free      23
Vector space      45 94
Weak second-order logic      136 164
Well-ordering number      207
Whitehead      87
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