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Index of Definitions
INDEX OF DEFINITIONS
Abelian group 40 absolute formula 562 absorption laws 38 abstract logic 128 additive function 386 additive number theory 43 admit elimination of quantifiers 202 admit a pair of cardinals 154 admit a pair of cardinals omitting a type 532 admit a triple of cardinals 532 algebraic in a theory 496, 529 algebraically closed field 41 algebraically independent 348 algebraically prime model 195 amalgamate 196 amalgamation property 195 antisymmetric 580 antisymmetric property 37 Archimedean ordered field 70 arithmetical predicate 415 arithmetical set 125 arithmetical statement 415 arithmetically saturated model 125 Aronsajn tree 525 arrangement 51, 55 Artin’s Conjecture 368 assignment 7 associativity 39 atom of Boolean algebra 39 atomic Boolean algebra 39 atomic element of a Boolean algebra 373 atomic formula 22 atomic model 97 atomic theory 97 atomless Boolean algebra 39 atomless element of a Boolean algebra 373 Automorphism Theorem 172
automorphism type 176 autonomous set of formulas 426 Axiom of Choice 584, 593 Axiom of Constructibility 238 Axiom of Regularity 588 axioms of a theory 12, 37 a-categorical 2, 482, 507 a-complete filter 227 a-enlargement 283 a-good ultrafilter 386 a-homogeneous model 297 a-isomorphism property 305 a-like model 557 a-omit 87 a-realize 87 a-regular filter 248 a-saturated model 256, 283, 292 a-saturated over V ( X ) 283 a-standard model 529 a-universal model 255, 297 a-universal with respect to K 297 back and forth condition 114 back and forth construction 98 Back and Forth Lemma 298 base set 263 basic elementary class 220 basic formula 50 basic Horn formula 407 Bernays set theory 594 Bernays-Morse set theory 595 Beth’s Theorem 90 bit of a tuple in a Boolean algebra 379 Boolean algebra 38 Boolean combination 50 bound variable 22 bounded elementary chain theorem 287
641
642
INDEX OF DEFINITIONS
bounded elementary embedding 266 bounded elementary submodel 266 bounded limit ultrapower 457 bounded quantifier 265 bounded quantifier formula 265, 562 bounded ultrapower 269 branch of a tree 244 built-in Skolem functions 165 built up from constants 66 Cantor diagonal method 592 cardinal 586 cardinal exponent 586 cardinal of a language 19 cardinal of a model 21 cardinal power 586 cardinality of a set 586 Cartesian product 579, 580 categorical in power a 2, 482, 507 Cauchy sequence 351 CH 586 chain 580 chain, elementary 140 chain of models 140 Chang’s Conjecture 156, 534 Chang’s Two Cardinal Theorem 520 Chang-Makkai Theorem 333 characteristic of a field 41 Choice Axiom 593 Church’s Thesis 110 classification theory 502 closed set of ordinals 246 closed theory 12, 36 closed unbounded set 246 closed under directed unions 146 closed under elementary equivalence 220 closed under reduced roots 446 closed under roots 354 closed under ultraproducts 220 closed under union of well ordered chains 146 Closure Property 128 cofinal 589 cofinality 589 collection scheme 593 commutative ring 40 compact class of models 446 Compactness Theorem 11, 33 companion operator 205 completable formula 97 complete arithmetic 42 complete, a-(filter) 227
complete embedding 448 complete extension 448 complete formula 97 complete number theory 42 complete, w - (theory) 82 complete theory 17, 37 Completeness Theorem 8, 32 Completeness Theorem (Extended) 11, 33 Completeness Theorem for w-logic 82 completion of a model 448 completion, valued field 351 comprehension scheme 595 Comprehensiveness Theorem 284 concatenation 584 concurrent relation 290 condition, forcing 209 conditional sentence 14 congruence relation 64 connected partial ordering 580 consequence 11, 17, 32, 50 consistent 9, 25 consistent with T 79 constant 20 constant symbol 18 Constructibility, Axiom of 238 constructible set 559 constructible universe 559 continuum hypothesis 586 core model 253 corresponding relation 20 cotheory 192 countably compact logic 130 countably homogeneous model 113 countably incomplete filter 226 countably prime model 99 countably saturated model 100 countably universal model 103 cover of a model 404 Craig Interpolant 87 Craig Interpolation Theorem 87 cross section axiom 351 De Morgan laws 38 deducible 9, 25 deduction 9 Deduction Theorem 10, 25 definable element 35, 185 definable function 461 definable subset 259, 559 define explicitly 90, 329 define implicitly 90 definitional extension 185
INDEX OF DEFINITIONS
degree of a field extension 348 degree of a formula 496, 497 dense order 38 descendingly a-complete ultrafilter 258 Detachment, Rule of 9 determined formula 421 diagonal intersection 246 diagram 68 direct factor 321 direct limit 322 direct product 224, 321 direct product of formulas 428 direct system 322 directed relation 146 discrete simple ordering 182 distributive laws 38, 40 divisible torsion-free Abelian group 40 Downward Lowenheim-Skolem-Tarski Theorem 66 A,, formula 110 effective process 423 elementarily equivalent 32 elementarily equivalent, abstract 129 elementary chain 140 Elementary Chain Theorem 140 elementary class 199, 220 elementary diagram 137 elementary embedding 84 Elementary Embedding Theorem 172 elementary extension 84, 136 elementary submodel 84, 136 elimination of quantifiers 50, 202 end extension 85, 163 enlargement 283 enumeration 584 Enumeration Principle 585 equivalence relation 59, 580 equivalent 17 equivalent formulas 50 equivalent logics 129 equivalent over Y 174 Erdos cardinal 555 Erdos notation 554 Erdos-Rado Theorem 510 Ershov’s Theorem 440 essentially existential 125, 322 essentially finite direct power 431 existence of nonstandard universes 268 Existence Theorem for Countably Saturated Models 102
643
Existence Theorem for Recursively Saturated Models 112 Existence Theorem for Special Models 296 existential sentence 34 existential-positive formula 319 existentially closed 192 expansion of a language 19 expansion of a model 20 Expansion Property 128 Expansion Theorem 216, 279 Extended Completeness Theorem 11, 33 Extended Omitting Types Theorem 84 extension of a model 22 extension of a theory 37 external set 276 false in a model 7, 32 field 41 field of characteristic zero 41 field of prime characteristic 41 field of sets 39 filter 46, 211 filter generated by E 212 finite cover property 404 finite direct power sentence 412 finite direct product 406 finite direct product of formulas 428 finite forcing companion 210 finite forcing relation 209 finite intersection property 46, 166, 212 Finite Iteration Theorem 463 finite ordinal 582 finite spectrum problem 597 finitely axiomatizable 12, 37 finitely generated model 73 finitely generic model 210 finitely generic set 209 finitely satisfiable 11 first-order predicate language 18 first-order theory 34 forking 505 formal power series 352 formula 22 Frayne’s Theorem 256 Frechet filter 211 free variable 22 fully compact logic 129 function 20, 579 function symbol 18 Fundamental Theorem of Ultraproducts 217
gap n conjecture 524 GCH 586
644
INDEX OF DEFINITIONS
generalized continuum hypothesis 586 generalized w-logic 83 generated from a set of indiscernibles 171 germ of a meromorphic function 371 Godel Completeness Theorem 33 good ultrafilter 257, 386 group 40 Hanf number 134 Hausdorff Maximal Principle 585 Hensel field 353 Hensel’s lemma 353 henselization 354 hereditary set 340 homogeneous model 113, 297 homomorphic 70 homomorphic image 71 homomorphically embedded 71 homomorphism 71 Horn formula 407 huge cardinal 537 hull, Skolem 171 hypothesis of inaccessibility 590 ideal in a Boolean algebra 373 Idealization Scheme 285 idempotent laws 38 identity axioms 25 identity formula 24 identity symbol 22 implicitly define 90 improper filter 211 inaccessible cardinal 590 incompletable formula 97 inconsistent 9, 25 increasing sentence 13 independent set of sentences 18 independent unary relations 179 indescribable cardinal 247 indiscernible 169 individual relative to V ( X ) 263 induction scheme 42 inductive order, theory of 382 infimum of a set of ordinals 582 infinite forcing companion 209 infinite forcing relation 209 infinite order in a group 78 infinitely generic 209 infinity axiom 592 initial ordinal 586 initial segment 583 inner model 537
inseparable theories 88, 93 integral domain 41 Internal Definition Principle 277 internal formula 285 internal set 276 Internal Set Theory 285 Internalization Scheme 285 Interpolation Theorem 17 interpretation function 20 invariants of a Boolean algebra 375 inverse limit 322 inverse system 322 irreducible formula 496, 498 isomorphic chains 580 isomorphic embedding 22 isomorphic models 21 isomorphism 21 isomorphism of well orderings 583 isomorphism, partial 114 Isomorphism Property 128 Isomorphism Theorem for Ultrapowers 398 IST 285 iterated ultrapower 467, 468 joint embedding property 195 J6nsson algebra 554 Jonsson cardinal 538 Jonsson model 534 Keisler Sandwich Theorem 31 1 Keisler-Rabin Theorem 450 Kueker Finite Definability Theorem 330 Kueker’s conjecture 506 Kurepa family 527 Kurepa Hypothesis 527 Kurepa tree 259 language 18 limit cardinal 586 limit ordinal 582 limit ultrapower 451, 452 limited formula 265, 562 Lindenbaum algebra 47, 74 Lindenbaum’s Theorem 10, 26 Lindstrom’s Characterization 132 Lindstrom’s Second Theorem 135 linear order 38 Local Ultrapower Theorem 281 locally omit 80 locally realize 79 logic, w 82 logical symbols 22
INDEX OF DEFINITIONS
hi-Vaught Test 139 Lowenheim number 129 Lowenheim-Skolem-Tarski Theorem 67 Lyndon Interpolation Theorem 92 Mahlo number 246 Mahlo’s operation 247 main gap 507 maximal consistent 9, 25 measurable cardinal 229 medium compactness 245 method of diagrams 68 minimal nonstandard universe 291 model companion 197 model complete 186 model completion 201 model, first-order logic 20 model of a sentence 32 model pair 115 model, sentential logic 4, 10 Modus Ponens 9, 25 monotonic function 386 Morley Categoricity Theorem 494 Morley Omitting Types Theorem 512 Morley rank 497, 498 Mostowski collapse 272 Mostowski Collapsing Theorem 271 narrowing a gap 254 n-ary operation 580 n-ary relation 579 natural embedding 221 natural model of set theory 45 natural number 582 neat elementary extension 156 negation normal form (nnf) 93 negative formula 313 negative Occurrence 92 nonprincipal filter 166 nonstandard analysis 70 nonstandard element, Peano arithmetic 78 nonstandard model of number theory 42 nonstandard universe 266 normal ultrafilter 239 n-tuple 579 null set axiom 592 number theory 42
0 sharp 566 0 sharp exists 566 occur negatively 92 occur positively 92
Occurrence Property 128 omit a set of formulas 78 Omitting Types Theorem 80 onto function 580 open formula 51 operation over X 580 order of a group element 40 order of a tree 244 order type 583 order type of a model of set theory 238 ordered field 41 ordered pair 579 ordered tuple 579 ordermorphic embedding 169 ordermorphism 169 ordinal 580 ordinal sum 584 outer extension 125, 322 Overspill Principle 289 w-categorical theory 104 w-complete 82 w-consistent 81 o-homogeneous 113 w-logic 82 w-model 81, 515 w-rule 82 w-saturated 100 p-adic numbers 366 p-adic valuation 366 pair, ordered 579 pairs axiom 592 partial isomorphism 114 partial order, theory of 37 partial ordering 580 Peano arithmetic 42 pin down 134 positive diagram 70 positive occurrence 82 positive sentence 13, 72 power of a language 19 power of a model 21 power of a set 586 power set 588 power set axiom 592 preservation theorem 147 preserved by reduced factors 419 preserved under finite intersections 14 preserved under homomorphisms 147 preserved under intersections 15 preserved under submodels 147 preserved under unions of chains 147
645
646
INDEX OF DEFlNlTlONS
Pressburger arithmetic 43 prime model 99 principal filter 46 principal ideal in a Boolean algebra 374 product of ultrafilters 463 proof 25 proper filter 212 pseudo-elementary class 224 Quantifier Axioms 24 Quantifier Property 128 quotient Boolean algebra 374 r.e. 110 Ramsey cardinal 539 Ramsey’s Theorem 168 rank function 588 rank, Morley 497, 498 rank of a set 588 rank 1 valued field 371 real closed field 41 real closure 344 realize a set of formulas 77 Realizing and Omitting Types Theorem 172 recursive language 110 recursive set 110 recursively enumerable 110 recursively saturated model 111 reduced power 215 reduced power sentence 412 reduced product 215, 216 reduced product of formulas 428 reduced product sentence 412 reduct of a model 20 reduction of a language 19 reflexive property 37 reflexive relation 580 regular cardinal 589 regular filter 248 regularity axiom 592 relation 20, 579 relation symbol 118 relative algebraic closure 354 relatively compact class of models 260 relativization of a formula 125, 321 Relativization Property 129 relativization to V 594 Renaming Property 128 replacement scheme 593, 595 Representation Theorem for Boolean Algebras 39
Representation Theorem for Nonstandard Universes 457 residue class field 352 Restricted Omitting Types Theorem 94 restriction 580 restriction of a closed theory 37 rigid model 598 ring 40 Robinson Consistency Theorem 91, 119 Robinsonian analysis 262 Rowbottom cardinal 546 Rowbottom’s Theorem 545 Rule of Detachment 25 Rule of Generalization 25 rule of inference 25 satisfaction relation, abstract 128 satisfiable 10, 16, 32 satisfy a formula 29 satisfy a set of formulas 77 saturated model 100, 292 saturated over V ( X ) 283 saturated relative to A 511 Scott’s Theorem 238 selective filter 481 selective ultrafilter 262 self-determining set of formulas 425 semantically equivalent 17 semantics 3 semigroup 558 sentence, first-order logic 24 sentence, sentential logic 5 sentenial axioms 24 sequence 584 set relative to V ( X ) 263 set-theoretic arithmetic 594 Silver’s Theorem 547 simple expansion 19 simple model 320 simple order 38 simple ordering 580 singular cardinal 589 Skolem expansion 164 Skolem function 164 Skolem hull 165 Skolem Normal Form Theorem 175 Skolem theory 164 special Aronsajn tree 525 special Horn sentence 419 special model 295 specializing chain 295 spectrum problem 505
INDEX OF DEFINITIONS
stability theory 502 stable in power a 485 stable theory 485 stand representation 172 standard model of number theory 42 standard set 276 Standardization Scheme 285 star 01 A 267 Stretching Theorem 172 strict basic Horn formula 418 strict Horn formula 418 strict well ordering 580 strong homomorphism 321 strong limit cardinal 586 strongly compact cardinal 245 subdirect product of models 419 subformula 23 submodel 21 submodel complete theory 302 submodel generated by X 34 Subset Theorem 172 subsets scheme 592 substitution 22 subtheory 37 successor cardinal 586 successor ordinal 582 sum, ordinal 584 supercompact cardinal 526 superstable theory 505 superstructure 263 supremum of a set of ordinals 582 Svenonius' Theorem 329 symmetric relation 580 syntax 3 2,-formula 110 Tarski's Theorem 346 tautology 8, 24 term 22 theorem 25 theory 12, 36 theory of a model 37 theory of a type 100 three-cardinal problem 598 torsion group 78 torsion-free group 40 totally indiscernible set 495 totally transcendental theory 500 transcendence basis 181 transcendence rank of a field extension 181 transcendence rank of a formula 497 transcendence rank of a model 497
647
transcendence rank zero 496 transcendental set in a field extension 348 Transfer Scheme 285 Transfinite Induction 582 transitive model 45 transitive property 37 transitive realization 49 transitive relation 580 transitive submodel 125, 270 tree 244 tree property 244 trivial Boolean algebra 369 trivial filter 211 true in a model 7, 32 truncation 271 truth definition 1 truth table 8 tuple 579 two-cardinal model 533 type 78 type of a model 100 type of a tuple 100 ultrafilter 46, 166, 213 Ultrafilter Theorem 214 ultrapower chain 260 ultrapower extension 260, 448 ultraproduct 215 unbounded set 246 uniform filter 252 union of a chain of models 140 union of a set of models 145, 321 unions axiom 592 Uniqueness Theorem for Countably Saturated Models 103 Uniqueness Theorem for Special Models 300 universal model 103, 297 universal sentence 34 universal-existential sentence 146 universe 20 universe class axiom 595 unstable theory 505 Upward Lowenheim-Skolem-Tarski Theorem 67 Upward Morley Theorem 488 valid sentence 7, 32 valuation ring 352 value group 351 value of a sentence 8 value of a term 27 valued field 351
648
INDEX OF DEFINITIONS
valued subfield 354 Vaught's conjecture 506, 597
Well-Ordering Principle 585 witness 61
Weak Compactness Theorem 232 weakly compact cardinal 233 weakly forces 209 weakly homogeneous model 303 weakly Horn formula 418 well-founded partial order 78 well-founded relation 237 well-ordered chain 580 well-ordered model 147, 535 well-ordering 580
Zermelo set theory 593 Zermelo-Fraenkel set theory 592 zero sharp 566 zero sharp exists 566 Z F 592 ZF - P 594 ZFC 593 Z-group 370 Zorn's Lemma 585 Z-valued field 371
Abelian group 40 absolute formula 562 absorption laws 38 abstract logic 128 additive function 386 additive number theory 43 admit elimination of quantifiers 202 admit a pair of cardinals 154 admit a pair of cardinals omitting a type 532 admit a triple of cardinals 532 algebraic in a theory 496, 529 algebraically closed field 41 algebraically independent 348 algebraically prime model 195 amalgamate 196 amalgamation property 195 antisymmetric 580 antisymmetric property 37 Archimedean ordered field 70 arithmetical predicate 415 arithmetical set 125 arithmetical statement 415 arithmetically saturated model 125 Aronsajn tree 525 arrangement 51, 55 Artin’s Conjecture 368 assignment 7 associativity 39 atom of Boolean algebra 39 atomic Boolean algebra 39 atomic element of a Boolean algebra 373 atomic formula 22 atomic model 97 atomic theory 97 atomless Boolean algebra 39 atomless element of a Boolean algebra 373 Automorphism Theorem 172
automorphism type 176 autonomous set of formulas 426 Axiom of Choice 584, 593 Axiom of Constructibility 238 Axiom of Regularity 588 axioms of a theory 12, 37 a-categorical 2, 482, 507 a-complete filter 227 a-enlargement 283 a-good ultrafilter 386 a-homogeneous model 297 a-isomorphism property 305 a-like model 557 a-omit 87 a-realize 87 a-regular filter 248 a-saturated model 256, 283, 292 a-saturated over V ( X ) 283 a-standard model 529 a-universal model 255, 297 a-universal with respect to K 297 back and forth condition 114 back and forth construction 98 Back and Forth Lemma 298 base set 263 basic elementary class 220 basic formula 50 basic Horn formula 407 Bernays set theory 594 Bernays-Morse set theory 595 Beth’s Theorem 90 bit of a tuple in a Boolean algebra 379 Boolean algebra 38 Boolean combination 50 bound variable 22 bounded elementary chain theorem 287
641
642
INDEX OF DEFINITIONS
bounded elementary embedding 266 bounded elementary submodel 266 bounded limit ultrapower 457 bounded quantifier 265 bounded quantifier formula 265, 562 bounded ultrapower 269 branch of a tree 244 built-in Skolem functions 165 built up from constants 66 Cantor diagonal method 592 cardinal 586 cardinal exponent 586 cardinal of a language 19 cardinal of a model 21 cardinal power 586 cardinality of a set 586 Cartesian product 579, 580 categorical in power a 2, 482, 507 Cauchy sequence 351 CH 586 chain 580 chain, elementary 140 chain of models 140 Chang’s Conjecture 156, 534 Chang’s Two Cardinal Theorem 520 Chang-Makkai Theorem 333 characteristic of a field 41 Choice Axiom 593 Church’s Thesis 110 classification theory 502 closed set of ordinals 246 closed theory 12, 36 closed unbounded set 246 closed under directed unions 146 closed under elementary equivalence 220 closed under reduced roots 446 closed under roots 354 closed under ultraproducts 220 closed under union of well ordered chains 146 Closure Property 128 cofinal 589 cofinality 589 collection scheme 593 commutative ring 40 compact class of models 446 Compactness Theorem 11, 33 companion operator 205 completable formula 97 complete arithmetic 42 complete, a-(filter) 227
complete embedding 448 complete extension 448 complete formula 97 complete number theory 42 complete, w - (theory) 82 complete theory 17, 37 Completeness Theorem 8, 32 Completeness Theorem (Extended) 11, 33 Completeness Theorem for w-logic 82 completion of a model 448 completion, valued field 351 comprehension scheme 595 Comprehensiveness Theorem 284 concatenation 584 concurrent relation 290 condition, forcing 209 conditional sentence 14 congruence relation 64 connected partial ordering 580 consequence 11, 17, 32, 50 consistent 9, 25 consistent with T 79 constant 20 constant symbol 18 Constructibility, Axiom of 238 constructible set 559 constructible universe 559 continuum hypothesis 586 core model 253 corresponding relation 20 cotheory 192 countably compact logic 130 countably homogeneous model 113 countably incomplete filter 226 countably prime model 99 countably saturated model 100 countably universal model 103 cover of a model 404 Craig Interpolant 87 Craig Interpolation Theorem 87 cross section axiom 351 De Morgan laws 38 deducible 9, 25 deduction 9 Deduction Theorem 10, 25 definable element 35, 185 definable function 461 definable subset 259, 559 define explicitly 90, 329 define implicitly 90 definitional extension 185
INDEX OF DEFINITIONS
degree of a field extension 348 degree of a formula 496, 497 dense order 38 descendingly a-complete ultrafilter 258 Detachment, Rule of 9 determined formula 421 diagonal intersection 246 diagram 68 direct factor 321 direct limit 322 direct product 224, 321 direct product of formulas 428 direct system 322 directed relation 146 discrete simple ordering 182 distributive laws 38, 40 divisible torsion-free Abelian group 40 Downward Lowenheim-Skolem-Tarski Theorem 66 A,, formula 110 effective process 423 elementarily equivalent 32 elementarily equivalent, abstract 129 elementary chain 140 Elementary Chain Theorem 140 elementary class 199, 220 elementary diagram 137 elementary embedding 84 Elementary Embedding Theorem 172 elementary extension 84, 136 elementary submodel 84, 136 elimination of quantifiers 50, 202 end extension 85, 163 enlargement 283 enumeration 584 Enumeration Principle 585 equivalence relation 59, 580 equivalent 17 equivalent formulas 50 equivalent logics 129 equivalent over Y 174 Erdos cardinal 555 Erdos notation 554 Erdos-Rado Theorem 510 Ershov’s Theorem 440 essentially existential 125, 322 essentially finite direct power 431 existence of nonstandard universes 268 Existence Theorem for Countably Saturated Models 102
643
Existence Theorem for Recursively Saturated Models 112 Existence Theorem for Special Models 296 existential sentence 34 existential-positive formula 319 existentially closed 192 expansion of a language 19 expansion of a model 20 Expansion Property 128 Expansion Theorem 216, 279 Extended Completeness Theorem 11, 33 Extended Omitting Types Theorem 84 extension of a model 22 extension of a theory 37 external set 276 false in a model 7, 32 field 41 field of characteristic zero 41 field of prime characteristic 41 field of sets 39 filter 46, 211 filter generated by E 212 finite cover property 404 finite direct power sentence 412 finite direct product 406 finite direct product of formulas 428 finite forcing companion 210 finite forcing relation 209 finite intersection property 46, 166, 212 Finite Iteration Theorem 463 finite ordinal 582 finite spectrum problem 597 finitely axiomatizable 12, 37 finitely generated model 73 finitely generic model 210 finitely generic set 209 finitely satisfiable 11 first-order predicate language 18 first-order theory 34 forking 505 formal power series 352 formula 22 Frayne’s Theorem 256 Frechet filter 211 free variable 22 fully compact logic 129 function 20, 579 function symbol 18 Fundamental Theorem of Ultraproducts 217
gap n conjecture 524 GCH 586
644
INDEX OF DEFINITIONS
generalized continuum hypothesis 586 generalized w-logic 83 generated from a set of indiscernibles 171 germ of a meromorphic function 371 Godel Completeness Theorem 33 good ultrafilter 257, 386 group 40 Hanf number 134 Hausdorff Maximal Principle 585 Hensel field 353 Hensel’s lemma 353 henselization 354 hereditary set 340 homogeneous model 113, 297 homomorphic 70 homomorphic image 71 homomorphically embedded 71 homomorphism 71 Horn formula 407 huge cardinal 537 hull, Skolem 171 hypothesis of inaccessibility 590 ideal in a Boolean algebra 373 Idealization Scheme 285 idempotent laws 38 identity axioms 25 identity formula 24 identity symbol 22 implicitly define 90 improper filter 211 inaccessible cardinal 590 incompletable formula 97 inconsistent 9, 25 increasing sentence 13 independent set of sentences 18 independent unary relations 179 indescribable cardinal 247 indiscernible 169 individual relative to V ( X ) 263 induction scheme 42 inductive order, theory of 382 infimum of a set of ordinals 582 infinite forcing companion 209 infinite forcing relation 209 infinite order in a group 78 infinitely generic 209 infinity axiom 592 initial ordinal 586 initial segment 583 inner model 537
inseparable theories 88, 93 integral domain 41 Internal Definition Principle 277 internal formula 285 internal set 276 Internal Set Theory 285 Internalization Scheme 285 Interpolation Theorem 17 interpretation function 20 invariants of a Boolean algebra 375 inverse limit 322 inverse system 322 irreducible formula 496, 498 isomorphic chains 580 isomorphic embedding 22 isomorphic models 21 isomorphism 21 isomorphism of well orderings 583 isomorphism, partial 114 Isomorphism Property 128 Isomorphism Theorem for Ultrapowers 398 IST 285 iterated ultrapower 467, 468 joint embedding property 195 J6nsson algebra 554 Jonsson cardinal 538 Jonsson model 534 Keisler Sandwich Theorem 31 1 Keisler-Rabin Theorem 450 Kueker Finite Definability Theorem 330 Kueker’s conjecture 506 Kurepa family 527 Kurepa Hypothesis 527 Kurepa tree 259 language 18 limit cardinal 586 limit ordinal 582 limit ultrapower 451, 452 limited formula 265, 562 Lindenbaum algebra 47, 74 Lindenbaum’s Theorem 10, 26 Lindstrom’s Characterization 132 Lindstrom’s Second Theorem 135 linear order 38 Local Ultrapower Theorem 281 locally omit 80 locally realize 79 logic, w 82 logical symbols 22
INDEX OF DEFINITIONS
hi-Vaught Test 139 Lowenheim number 129 Lowenheim-Skolem-Tarski Theorem 67 Lyndon Interpolation Theorem 92 Mahlo number 246 Mahlo’s operation 247 main gap 507 maximal consistent 9, 25 measurable cardinal 229 medium compactness 245 method of diagrams 68 minimal nonstandard universe 291 model companion 197 model complete 186 model completion 201 model, first-order logic 20 model of a sentence 32 model pair 115 model, sentential logic 4, 10 Modus Ponens 9, 25 monotonic function 386 Morley Categoricity Theorem 494 Morley Omitting Types Theorem 512 Morley rank 497, 498 Mostowski collapse 272 Mostowski Collapsing Theorem 271 narrowing a gap 254 n-ary operation 580 n-ary relation 579 natural embedding 221 natural model of set theory 45 natural number 582 neat elementary extension 156 negation normal form (nnf) 93 negative formula 313 negative Occurrence 92 nonprincipal filter 166 nonstandard analysis 70 nonstandard element, Peano arithmetic 78 nonstandard model of number theory 42 nonstandard universe 266 normal ultrafilter 239 n-tuple 579 null set axiom 592 number theory 42
0 sharp 566 0 sharp exists 566 occur negatively 92 occur positively 92
Occurrence Property 128 omit a set of formulas 78 Omitting Types Theorem 80 onto function 580 open formula 51 operation over X 580 order of a group element 40 order of a tree 244 order type 583 order type of a model of set theory 238 ordered field 41 ordered pair 579 ordered tuple 579 ordermorphic embedding 169 ordermorphism 169 ordinal 580 ordinal sum 584 outer extension 125, 322 Overspill Principle 289 w-categorical theory 104 w-complete 82 w-consistent 81 o-homogeneous 113 w-logic 82 w-model 81, 515 w-rule 82 w-saturated 100 p-adic numbers 366 p-adic valuation 366 pair, ordered 579 pairs axiom 592 partial isomorphism 114 partial order, theory of 37 partial ordering 580 Peano arithmetic 42 pin down 134 positive diagram 70 positive occurrence 82 positive sentence 13, 72 power of a language 19 power of a model 21 power of a set 586 power set 588 power set axiom 592 preservation theorem 147 preserved by reduced factors 419 preserved under finite intersections 14 preserved under homomorphisms 147 preserved under intersections 15 preserved under submodels 147 preserved under unions of chains 147
645
646
INDEX OF DEFlNlTlONS
Pressburger arithmetic 43 prime model 99 principal filter 46 principal ideal in a Boolean algebra 374 product of ultrafilters 463 proof 25 proper filter 212 pseudo-elementary class 224 Quantifier Axioms 24 Quantifier Property 128 quotient Boolean algebra 374 r.e. 110 Ramsey cardinal 539 Ramsey’s Theorem 168 rank function 588 rank, Morley 497, 498 rank of a set 588 rank 1 valued field 371 real closed field 41 real closure 344 realize a set of formulas 77 Realizing and Omitting Types Theorem 172 recursive language 110 recursive set 110 recursively enumerable 110 recursively saturated model 111 reduced power 215 reduced power sentence 412 reduced product 215, 216 reduced product of formulas 428 reduced product sentence 412 reduct of a model 20 reduction of a language 19 reflexive property 37 reflexive relation 580 regular cardinal 589 regular filter 248 regularity axiom 592 relation 20, 579 relation symbol 118 relative algebraic closure 354 relatively compact class of models 260 relativization of a formula 125, 321 Relativization Property 129 relativization to V 594 Renaming Property 128 replacement scheme 593, 595 Representation Theorem for Boolean Algebras 39
Representation Theorem for Nonstandard Universes 457 residue class field 352 Restricted Omitting Types Theorem 94 restriction 580 restriction of a closed theory 37 rigid model 598 ring 40 Robinson Consistency Theorem 91, 119 Robinsonian analysis 262 Rowbottom cardinal 546 Rowbottom’s Theorem 545 Rule of Detachment 25 Rule of Generalization 25 rule of inference 25 satisfaction relation, abstract 128 satisfiable 10, 16, 32 satisfy a formula 29 satisfy a set of formulas 77 saturated model 100, 292 saturated over V ( X ) 283 saturated relative to A 511 Scott’s Theorem 238 selective filter 481 selective ultrafilter 262 self-determining set of formulas 425 semantically equivalent 17 semantics 3 semigroup 558 sentence, first-order logic 24 sentence, sentential logic 5 sentenial axioms 24 sequence 584 set relative to V ( X ) 263 set-theoretic arithmetic 594 Silver’s Theorem 547 simple expansion 19 simple model 320 simple order 38 simple ordering 580 singular cardinal 589 Skolem expansion 164 Skolem function 164 Skolem hull 165 Skolem Normal Form Theorem 175 Skolem theory 164 special Aronsajn tree 525 special Horn sentence 419 special model 295 specializing chain 295 spectrum problem 505
INDEX OF DEFINITIONS
stability theory 502 stable in power a 485 stable theory 485 stand representation 172 standard model of number theory 42 standard set 276 Standardization Scheme 285 star 01 A 267 Stretching Theorem 172 strict basic Horn formula 418 strict Horn formula 418 strict well ordering 580 strong homomorphism 321 strong limit cardinal 586 strongly compact cardinal 245 subdirect product of models 419 subformula 23 submodel 21 submodel complete theory 302 submodel generated by X 34 Subset Theorem 172 subsets scheme 592 substitution 22 subtheory 37 successor cardinal 586 successor ordinal 582 sum, ordinal 584 supercompact cardinal 526 superstable theory 505 superstructure 263 supremum of a set of ordinals 582 Svenonius' Theorem 329 symmetric relation 580 syntax 3 2,-formula 110 Tarski's Theorem 346 tautology 8, 24 term 22 theorem 25 theory 12, 36 theory of a model 37 theory of a type 100 three-cardinal problem 598 torsion group 78 torsion-free group 40 totally indiscernible set 495 totally transcendental theory 500 transcendence basis 181 transcendence rank of a field extension 181 transcendence rank of a formula 497 transcendence rank of a model 497
647
transcendence rank zero 496 transcendental set in a field extension 348 Transfer Scheme 285 Transfinite Induction 582 transitive model 45 transitive property 37 transitive realization 49 transitive relation 580 transitive submodel 125, 270 tree 244 tree property 244 trivial Boolean algebra 369 trivial filter 211 true in a model 7, 32 truncation 271 truth definition 1 truth table 8 tuple 579 two-cardinal model 533 type 78 type of a model 100 type of a tuple 100 ultrafilter 46, 166, 213 Ultrafilter Theorem 214 ultrapower chain 260 ultrapower extension 260, 448 ultraproduct 215 unbounded set 246 uniform filter 252 union of a chain of models 140 union of a set of models 145, 321 unions axiom 592 Uniqueness Theorem for Countably Saturated Models 103 Uniqueness Theorem for Special Models 300 universal model 103, 297 universal sentence 34 universal-existential sentence 146 universe 20 universe class axiom 595 unstable theory 505 Upward Lowenheim-Skolem-Tarski Theorem 67 Upward Morley Theorem 488 valid sentence 7, 32 valuation ring 352 value group 351 value of a sentence 8 value of a term 27 valued field 351
648
INDEX OF DEFINITIONS
valued subfield 354 Vaught's conjecture 506, 597
Well-Ordering Principle 585 witness 61
Weak Compactness Theorem 232 weakly compact cardinal 233 weakly forces 209 weakly homogeneous model 303 weakly Horn formula 418 well-founded partial order 78 well-founded relation 237 well-ordered chain 580 well-ordered model 147, 535 well-ordering 580
Zermelo set theory 593 Zermelo-Fraenkel set theory 592 zero sharp 566 zero sharp exists 566 Z F 592 ZF - P 594 ZFC 593 Z-group 370 Zorn's Lemma 585 Z-valued field 371