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Rockingham G.R. — Deducibility and Decidability

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Название: Deducibility and Decidability

Автор: Rockingham G.R.

Аннотация:

The classic results obtained by Gödel, Tarski, Kleene, and Church in the early thirties are the finest flowers of symbolic logic. They are of fundamental importance to those investigations of the foundations of mathematics via the concept of a formal system that were inaugurated by Frege, and of obvious significance to the mathematical disciplines, such as computability theory, that developed from them.

Derived from courses taught by the author over several years, this new exposition presents all of the results with their original proofs and central concepts in a manner that is unified by a systematic grounding of the notion of effectiveness in the semantics of the existential quantifier. Logicians and non-mathematicians, repelled by detail which is not obviously relevant in the standard textbooks, will be able to reach the heart of the matter with a minimum of fuss.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1990

Количество страниц: 162

Добавлена в каталог: 05.04.2008

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Предметный указатель

$Cl \Sigma tb$       69
$Tr_{B}$       64—66 68 69 107
$\Delta fm$       68
$\imath$       68 84
$\mathbb{F}$       140
$\mathbb{T}$       140
$\mathbf{P}$       15 72 89—98 105—108 114 128—131
$\mathbf{P}^{a,m}$       130
$\mathbf{Q}$       75 22 72 108 114 117—118 129
$\mathbf{Q}^{+}$ , $\mathbf{Q}!$       21
$\mathbf{Q}^{O}$ , $\mathbf{Q}^{O}!$ , $\mathbf{Q}'$       22
$\mathbf{R}$       75 72 98—99 110 116—117 121 129
$\mathbf{S}$       1 2—4 72 98
$\mathbf{S}^{+}$       4—5
$\mathbf{S}_{1}$       see “Kleene”
$\mathcal{R}$       93
$\mathcal{T}_{1}$       see “Kleene”
$\omega$ -completeness      see “Property
$\omega.2$       96 fn.
$\Sigma cl$ , $\Sigma fm$ , $\Sigma tb$       68—69
$\Sigma$       2 61—62
$\Sigma$ -provable      26 46
$\Sigma$ -provable, $\Sigma pr$       69 86—87 106—108 114 121
$\Sigma$ -provable, $\Sigma_{n} pr$ , $\Pi_{n} pr$       126—127 128
$\Sigma$ -provable, $\Sigma_{n}$ -, $\Pi_{n}$ -provable      125 126
$\varepsilon_{0}$       96 fn. 97 104
$^{2} \mathbf{P}$ , $^{2} \mathbf{P}^{+}$       114
A      8 137 142
Ackermann, W.      87 112 119
Adequacy Theorem      131 148
ap,       68
Application (formal), [ ]      139
Application (formal), [ ], ap      68
Application (formal), [ ], Application Principle      142 149
Application (formal), [ ], pap      84
Application (formal), [ ], partial application (formal), [ ]      85
Application (informal)      73—74 79—80 134—135
Arbib, M.A.      122
Argument      135
Arithmetical concepts      8 89 97 111
Arithmetization      59—72 esp. 65—66 106 107 108
Arkadencafe (Vienna)      101
Ars Combinatoria      see “Leibniz”
Assign/ment, | |      140 f. 148
Assign/ment, | |, being standardly assigned T      90 109 111—115 119 142
Assign/ment, | |, Putnam assign/ment, | |      131
Assign/ment, | |, standard assign/ment, | |      142 (see also “Model”)
Axiomatization      8 89
B      11
BD      55
Begriffsschrift      67 100
Benacerraf, P.      7
Bernays, P.      131
Berry, G.G.      4
Beth numbers      50 53
Beth, E.W.      6
Binary (notation, representation)      95
Branch      32—34 143
Brouwer — Koenig Infinity Lemma      33—36
Brouwer, L.E.J.      33 35 101
Buridan, J.      78 84
Cantor's argument      10 37 73 81 110
Cantor, G.      10 60 96
Cargile, J.      76
Carnap on definite descriptions      141—142
Carnap on syntax      65—66
Carnap's Fixed-point Lemma      83—84 89—98 130
Carnap's fixed-point, C      82—85 89—98 109 115 117 126 130
Carnap, carn      85
Carnap, R.      101—102
casus      86
Category-mistake      75
Chain      32—33
Chandra, A.K.      39
Charles II      113
Choice      see “Function”
Chomsky, A.N.      67
Church's Theorem      117—119
Church's thesis      40—43 47—49 57 72 84 115 116
Church, A.      45 60 66—67 79 115
Cichon, E.A.      95
Classification      78
Clos/ed, closure $\Sigma cl$ , $Cl \Sigma tb$       69
Clos/ed, closure of $\Sigma$ -tableaux      24
Clos/ed, closure of regular tableaux      145
Clos/ed, closure, Rgcl, Clrgtb      71
Closure condition      147
Clrgtb      71
cnjn      64
Co-denoting      135
Co-extensive      8 43f. 55 57 114 135
Cock Robin      35 43
Code number      see “Encoding”
Cohen, L.J.      86
Compatibility, formal      117 118 130
Complement      44 46 135
Complete/ness      4—5 14 18—19 19—22 67 89 148—149 110
composition      85
Composition, comp      85
Concatenation      58 136
Concatenation, $^{\frown}$       58
Confirm/ation, confirm/ation ability      40 43—44 “Procedure”)
Congruen/t, congruence      83 138
conjugate      143
Conjugate, $\bar{ }$       144
Connected/ness      1 5 6 93
Connectives      136
Consisten/t, consistency      90 91 93 110 147 148
Constructive choice function      121
Constructive techniques      20—22 90—91
Constructive truth      35—36 110
Constructivists      90 93 121
Continuum, uncountability of      10
Convention T      114
Corners, $\ulcorner \urcorner$       1 (see also “Quotation”)
Correctness Theorem      131 148
Cretan/s      see “Paradox Liar”
Cross-reference      83—84
Curry, H.B.      79
Cut-free      145 148
Cut-Rule      144 148
Dash/ing      2 61—62
Dash/ing, dash      62 68
Davidson, D.      67 112 113
Davis, M.      40
Dbl      89
Ddn      89
de Fermat, P.      35 43
Deci/sion, lability      67 72 115—118 120 “Procedure”)
Dedekind, R.      4 15 50—51
Deduc/tion, cf.      115—118
Deduc/tion, deducibility, formal, $\vdash$       3 16 19 101 145
Deduc/tion, deducibility, formal, $\vdash$ , Deducible, Dbl      89—98 105—108 126 128—131
Deduc/tion, deducibility, formal, $\vdash$ , Fmddn, Fmdbl      70 72
Definition      8 105 114—115
Definition, recursive      49 f. 86—87 111
Degree      see “Sentence-form”
denote      78 135
Descend/ant, descendance-chain      see “Sentence-form”
Description operator      14 136 142 149
Descriptions, definite      141—142
Descriptor      135
Descriptor-form      139
Descriptor-form, $\Delta$ -descriptor-form      23 56
Descriptor-form, $\Sigma$ -descriptor-form      23
Descriptor-form, Dscfm      67
Descriptor-letter      see “Letter/s”
diag      82
diagg      85
Diagonal, diagonal argument, diagonal procedure      73 102

Diagonalization      73—88
Diagonalization of a predicate      73 90 110
Diagonalization of a predicate-form      82 85 90
Diagonalization, diag      82
Diagonalization, diagg      85
DICTIONARY      20 139
Dictionary, standard dictionary      139—140
Dirichlet, P.G.L.      20 108
Disconfirm/ation, disconfirmability      44 (see also “Predicate” “Procedure”)
Domain      45—48 135
Dreben, B.      120
Dscfm      67
dsjn      64
Dummett, M.A.E.      4 35 103 134
e      8 137 142
Edwards, H.      35 65
Effective/ness      80 (see also “Procedure” “Relation” “Function” “Operation” “Letter/s”)
Ehrenfeucht, A.      122
Elementary      56—57 70
Eliminability      130
Enc      54
Encod/e, enc      54
Encod/e, encoding      52—53
Enderton, H.B.      9
Entscheidungsproblem      117 119—120
Epimenides      see “Paradox Liar”
Equivalence, analytical equivalence      76
Equivalence, elementary equivalence      6
Equivalence, local, global equivalence      107 114
Etchemendy, J.      113
Eubulides      see “Paradox Liar”
Euclid      20—21
Expressions, formal      10 137—139
Extension of predicate      8—9 44—47 115 135
Extension of sequence      33
Extension of set      15 130—131 147
Extension rules for tableaux      26 125 143
Extensionality      4 9 135
Extensionality, cf.      105
Feferman, S.      46 84 92 93 94 129
Finit/ary, fistic      see “Hilbert's Programme”
fit      140
fixed-point      see “Carnap” “Function” “Jeroslow” “Montague”
Fmdbl      72 98—100 108
Fmddn      70
FN      68
FORM      see “Subject- sentence- descriptor-
Form, simple      64 87
Form, top-form      143
Form, tpfm      69
Form, universal-form      21 147
Formal numerals      2
Formal numerals, Fnsq, fn      61—64 68
Formal numerals, theoretically dispensable      14
Formal system      100—104
free      139
Frege, F.L.G.      4 14 21 50 67 78 100—101 103 141 142
Friedman, H.      95 112
Fulfilment      27—28 31
Function      45—48 136
Function, choice function      120 122 128
Function, fixed-point of a function      82 fn.
Function, propositional function      78 134
G      89—92 94 112 126
Geach, P.T.      78 79
Gentzen, G.      97
Geometry      20—21
Goedel on mathematical intuition      102—103
Goedel's First Theorem      79 99 100—103 117
Goedel's second theorem      99 103—104 129
Goedel's witness G to $\mathbf{P}$ 's incompleteness      89—92 94
Goedel, K.      9 20 24 42 54 60 67 76 79 83 89 90 91 93 98—99 101—105 109 110 111 112 117 119 121—123
Goedel-number      60
Goedel-number, **      61 88
Goedel-number, fn      68
Goldfarb, W.      120
Goodstein, R.L., Goodstein's Theorem      95—97
Grelling, K.      110
Grounded (form)      see “Sentence-form” “Induction” “Hierarchy”
Grzegorczyk, A.      56
Hallett, M.      10
halt/ing      30 38
Hardy, G.H.      120 fn.
Harrington, L.      95
Haugeland, J.      37
Henkin, L.      92
Heterologicality      110
Hierarchy, arithmetical      124—128
Hilbert's Programme      91 103—104
Hilbert, D.      60 fn. 87 91 100 103—104 111 112 119 131 142
Hook, hk      64
Hook, hkup      71
Hook-up      145
IFF      10 fh.
Imdsc      106
Incompleteness of $\mathbf{P}$       20 89—98 105 “Goedel”)
Incompleteness of $\mathbf{Q}$       19—22
Incompleteness of $\mathbf{R}$       98
Incompleteness of $\mathbf{S}$       5 98
Inconsistency      see “Consistency”
Independence classic independence proofs re-vamped      98
Independence from $\mathbf{P}$       89—98
Independence from $\mathbf{Q}$       20—22
Independence from $\mathbf{S}$       5
INDEX      see “Sentence-form”
Induction on grounded forms      13
Induction on numbers      3—4 15 96—97 103 126
Induction on ordinals      96—97
Infinit/ary, infinitistic      see “Hilbert's Programme”
Intuition, mathematical      102—103
Intuitionists      see “Constructivists”
Irreflexiv/e, Irreflexivity      1 5 6
Isaacson, D.      97—98
Isomorphism      6
Jeffrey, R.G.      143
Jeroslow's Fixed-point Lemma      88
Jeroslow's fixed-point, J      88
Jeroslow, jer      88
Jeroslow, R.G.      88
Jones, J.P.      22
Kalmar, L.      56—57
Kaplan, D.      79
Kirby, L.      95 96
Kleene, , $\mathfrak{T}_1$       70 (see also “Recursion Theorem”)
Kleene, S.C.      19 42 69 87 115 116 121 124 131 139
Klein, F.      20
Kneale, W.      77
Kneebone, G. T.      101
Kochen, S.      98
Koenig, J.      33 35 82
Kreisel, G.      60 fn. 106 130
Kripke, S.      98
Kronecker, L.      65 fn.
L      1 137 142
Laius      76
Langford, C.H.      5 9
le Veque, W.J.      20
Leibniz, G.W. v.      118—119
Length      136
Length in Mostowski's sense      93 122
Letter/s      66—67 136—137
Levy, A.      106 fn. 130
Lewis, D.      37
Lewis, H.R.      39 120
Limitations by statute      41 f.
Lingua Philosophica      see “Leibniz”
Lob's Theorem      99
Lob, M.H.      92 94
Lobatschewsky, N.I.      20—21
Logic, second-order      87 112 114

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