Cambridge University Press 0521753732 - Lectures in Logic and Set Theory, Volume 1: Mathematical Logic George Tourlakis Index More information Index ∀-introduction, 43 bounded multiplication, 134 absolutely provable, 37 bounded product, 197 AC, 86 bounded quantification, 125 Ackermann function, 198 bounded search, 134 ∀∃-theory, 93 bounded summation, 134, 197 alphabet, 6, 13 Bourbaki, 5, 14, 41 alternating chain, 96 Brouwer, L. E. J., 4 ambiguity, 24 ambiguous, 24 call by value, 144 antisymmetry, 210 cancellation laws, 214 argot,39 categorical, 97 arithmetic, 150 characteristic function, 130, 219 arithmetic hierarchy, 151 characteristic term, 222 arithmetic relations, 150 Chinese remainder theorem, 160 arithmetical relations, 150 Church, Alonzo, 124, 188 arithmetization, 134, 265 Church’s thesis, 264 arity, 8 class, 75 ascending chains, 93 closed form, 201 assignment function, 88 closed formula, 18 associativities, 17 closed interval, 111 automorphism, 196 closed under, 19 axiom, 6 closure, 20 Henkin, 65 coding, 136 logical, 6 collection, 258 nonlogical, 6, 36 commutative diagram, 78 special, 6 compact, 107 axiomatic theories, 38 compactness theorem, 71, 74 axiomatized, 38 complete, 97, 194 axioms, logical, 34 simply complete, 66 complete arithmetic, 39, 170 basic diagram, 88 complete index set, 149, 199 Bennett, J., 162 completeness theorem, 52, 71, 73, 194 Bernays, P., 41 completion, 195 beta function, 159, 235 composition, 127 Boolean, 7 computability, 123 Boolean operations, 133 computable, 128 bound variable, 18 computable function, 124 bounded, 112 computation, 140 323 © Cambridge University Press www.cambridge.org Cambridge University Press 0521753732 - Lectures in Logic and Set Theory, Volume 1: Mathematical Logic George Tourlakis Index More information 324 Index computation model, 126 ∃-introduction, 36 computer program, 264 elementarily equivalent structures, 76 concatenate, 10 elementary chains, 93 concatenation, 13, 137, 243 elementary diagram, 88 formal, 243, 245 elementary embedding, 82 conjunct, 277 elementary extension, 82 conjunction, 17 elementary substructure, 82 conjunctionally, 216 elimination of defined symbols, 116, 118 connectives, 7 empty sequence, 243 conservative, 115, 121, 218 Entscheidungsproblem, 40, 124 conservative extension, 46, 118 enumeration theorem, 153 consistency theorem, 71, 73 ε-term, 123 constant, 8 equivalence theorem, 51, 183 Henkin, Leon, 65 existential axioms, 93 witnessing, 65 existential formula, 113 construction formative,41 expansion, 88 constructive arithmetic, 164, 170 explicit definition, 218 constructive arithmetic predicates, 160 explicit transformations, 160 continuous, 111 expressions, 6 contrapositive, 158 extension, 77 converges, 125 extensionality, 114 correct, 177 extreme value theorem, 197 countable, 62 course-of-values induction, 214 finitary, 4 course-of-values recursion, 137, finite hyperreal, 102 251 finitely satisfiable, 195 Craig, W., 204 first incompleteness theorem, 155 C-theory, 272 fixed point, 203, 310 fixpoint, 203, 310 decision problem, 40, 188 fixpoint theorem, 310, 317 decoding, 136, 240 formal, 1 Dedekind, R., 124, 129 beta function, 235 deduced, 37 Formal Arithmetic, 166, 175 deduction theorem, 48 formal language definability, 60 first order, 6 in a structure, 60 formalize, 3 definable, 170, 171 formally definable, 180 definable in arithmetic, 171 formally functionally definable, 202 definition by cases, 134, 221 formula definition by positive cases, 200 decidable by a theory, 66 definition by recursion, 25 mixed-mode, 61 ∆0 formulas, 172 prime, 28 derivability conditions, 272, 296 propositional, 28 derivation, 20, 254 satisfiable, 30 derived rule, 38, 40 tautology, 30 diagonalization, 125, 145 unsatisfiable, 30 diagonalization lemma, 310 formula form, 34 diagram, 64, 88 formula schema, 34 diagram expansion, 88 free for, 32 diagram language, 64, 88 free variable, 18 Diophantine equation, 201 function disjunct, 276 computable, 128 distributive laws, 193 partial computable, 128 diverges, 125 partial recursive, 128 domain, 53 primitive recursive, 129 dummy renaming, 45 recursive, 128 © Cambridge University Press www.cambridge.org Cambridge University Press 0521753732 - Lectures in Logic and Set Theory, Volume 1: Mathematical Logic George Tourlakis Index More information Index 325 G¨odel number, 155 infinite prime, 197 generalization, 43 infinitely close, 104 Gentzen, Gerhard, 193 infinitesimal 97, 102 G¨odel, Kurt, 124 infix, 14, 98 G¨odel-Malcev compactness theorem, 74 informal, 8 G¨odel-numbering, 155 initial functions, 128 G¨odel-Rosser first incompleteness input, 19 theorem, 189 intermediate value theorem, 197 G¨odel’s first incompleteness theorem, 177, interpretation, 53 202, 317 iota notation, 120 G¨odel’s second incompleteness theorem, 312 irreflexive, 210 graph, 98, 161, 163, 199, 202 irreflexivity, 210 greatest common divisor, 157 iteration theorem, 148 gcd, 157 iteratively, 21 Gries, David, 5 group theory, 92 κ-categorical, 97 groups, 93 Keisler, H. Jerome, 88 Grzegorczyk, A., 132 Kleene, S. C., 124, 150 g-term, 296 Kleene normal form theorem, 142 Kleene predicate, 164 halting problem, 145 Kleene schemata, 127, 264 halting set, 145 Kleene T -predicate, 142 Henkin, Leon, 62, 112, 314 Kleene’s recursion theorem, 149 Henkin theory, 66 Kronecker, L., 4 Hermes, H., 41 Heyting, A., 4 λ-notation, 125 hierarchy theorem, 154 language Hilbert, D., 4, 41 extension, 54 history function, 137 first order, 16 hyperreal numbers, 99 restriction, 54 hyperreals, 99 least common multiple, 157, 230 least principle, 216 I.H., 22 left endpoint, 111 i.p., 24 left field, 125 Identities, 128 Leibniz, G. W., 34 iff, 11 Leibniz rule, 51 immediate predecessors, 24 limit ordinal, 73 implied multiplication notation, 212 Lindemann, F., 4 inclusion map, 77 L¨ob, Martin H., 310, 314 incompletable, 177 L¨ob’s derivability condition, 310 incompletableness, 124 L¨ob’s theorem, 314 inconsistent, 265 logical symbols, 7 increasing chains, 93 logically implies, 52 indefinite article, 121 L¨owenheim-Skolem theorem, 71 indexing theorem, 153 individual variables, 7 Manin, Yu. I., 39 induced isomorphism, 78 Markov, A. A., 127 induction axiom, 206 material implication, 17 induction axiom schema, 206 mathematical theory, see theory induction hypothesis, 22 maximum, 230 induction rule, 208 metalanguage, 3 induction schema, 206 metamathematics, 3 inductive definitions, 20 metatheory, 3 inductive theories, 93 metavariable, 207 infinite hyperreal, 102 modus ponens, 36 infinite natural numbers, 196 monotonicity, 50 © Cambridge University Press www.cambridge.org Cambridge University Press 0521753732 - Lectures in Logic and Set Theory, Volume 1: Mathematical Logic George Tourlakis Index More information 326 Index monus, 131 problem, 144 Mostowski, A., 150 productive function, 179 MP, 36 productive sets, 179 µ-definitions, 218 programming language, 264 µ-term, 218 projection, 128 projection function symbols, 254 neighbourhood, 108 projection functions, 156 non-monotone, 265 projection theorem, 147 non-standard hyperreals, 99 proof, 37 non-standard numbers, 99 -, 37 nontotal, 125 proof by auxiliary constant, 52 Num function, 296 proof by cases, 183 number-theoretic functions, 124 proof by contradiction, 50 numeral, 41, 171 proper prefix, 13 proper subtraction, 131, 222 object variables, 7 propositional connectives, 7 occurrence, 13 propositional axioms, 194, 269 occurs in, 13 propositional calculus, 194 ω-consistent, 189 propositional logic, 194 ω-incompleteness, 190 propositional segment, 194 ω-inconsistent, 191 propositional valuations, 29 ω-rule, 190 propositional variable, 28 one point rule, 118 provability predicate, 280, 311 onto, 62 provably equivalent, 42 open, 92 proved from, 37 open formulas, 91 punctured neighbourhood, 108 open theory, 91 operations of substitution, 132 Quine, W. V. O., 156 ordered sequence, 19 ordinary computability, 124 R-closed, 20 ordinary recursion theory, 124 r.e. predicate, 147 output, 19 Rasiowa, H., 41 real function, 108 pairing function, 156 recursion, 19 parsable, 29 recursion theorem, 149 parse, 21 recursion theory, 123 partial computable, 128 recursive, 128, 130, 170 partial function, 125 recursive definition, 20 partial recursive, 128 recursive extension, 220, 262, 265, Peanoarithmetic, 206 273 φ-index, 138 recursively axiomatizable, 39 π-function, 198 recursively axiomatized, 38, 271 pinching lemma, 106 recursively enumerable predicate, 147 polynomial, 201 recursively inseparable, 187 positively strongly definable, 191, 202 recursively unsolvable, 144 Post, E. L., 124, 194 regular, 171 Post’s theorem, 194 regular formula, 171 predicate, 8, 130 regular functions, 204 prefix, 13 relation, see predicate, 8, 19 prime power coding, 136 primitive recursive, 130 primitive recursion, 127 recursive, 130 primitive recursive, 129, 170, 253 relational notation, 125 primitive recursive definitions, 248 relatively prime, 157, 226 primitive recursive functions, 124, 127 remainder function, 224 primitive recursive schema, 127 restricted atomic formulas, 256 priorities, 17 restriction, 78 © Cambridge University Press www.cambridge.org Cambridge University Press 0521753732 - Lectures in Logic and Set Theory, Volume 1: Mathematical Logic George Tourlakis Index More information Index 327 Rice, H. G., 149 underlying set of, 53 right endpoint, 111 universe of, 53 right finite limit, 108 structure embedding, 77 right positive infinite limit, 108 structure isomorphism, 77 Robinson, Abraham, 64 structures, elementarily equivalent, 76 Robinson, R. M., 175 substitutable for, 32 Rosser, J. Barkley, 189 substitution,