2730:
657:
Some authors define a theory to be categorical if all of its models are isomorphic. This definition makes the inconsistent theory categorical, since it has no models and therefore vacuously meets the criterion.
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1109:
613:
at least equal to the cardinality of its language, then the theory is complete. The reason is that all infinite models are first-order equivalent to some model of cardinal
115:
949:
821:
1784:
1867:
1008:
2181:
401:-categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating in
2339:
934:
729:
1127:
2194:
1517:
1779:
2199:
2189:
1926:
1132:
629:. Therefore, the theory is complete as all models are equivalent. The assumption that the theory have no finite models is necessary.
1677:
1123:
2335:
901:
783:
761:
Hodges, Wilfrid, "First-order Model Theory", The
Stanford Encyclopedia of Philosophy (Summer 2005 Edition), Edward N. Zalta (ed.).
2432:
2176:
1001:
1737:
1430:
1171:
888:, Proceedings of Symposia in Pure Mathematics, vol. 25, Providence, R.I.: American Mathematical Society, pp. 187â203,
620:
223:
575:
425:
There are not many natural examples of theories that are categorical in some uncountable cardinal. The known examples include:
2693:
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2153:
1978:
1399:
1083:
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2471:
2388:
2101:
2032:
1909:
1151:
874:
830:
771:
1759:
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2439:
2125:
1358:
1764:
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2096:
1835:
1093:
994:
869:
46:
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2486:
886:
Proceedings of the Tarski
Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. of California, Berkeley, Calif., 1971)
405:'s famous result that these are in fact the only possibilities. The theory was subsequently extended and refined by
2420:
2010:
1404:
1372:
1063:
531:
433:
54:
1137:
604:
2710:
2659:
2556:
2054:
2015:
1492:
864:
2551:
1166:
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2020:
1872:
1855:
1578:
1058:
738:
440:
410:
361:
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1502:
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28:
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2400:
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1993:
1877:
1812:
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1603:
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1472:
1303:
1280:
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1966:
1702:
1608:
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1308:
717:
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states that if a satisfiable theory has no finite models and is categorical in some infinite cardinal
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2538:
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2219:
2059:
1983:
1961:
1789:
1747:
1646:
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638:
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501:
414:
760:
2705:
2596:
2581:
2561:
2518:
2405:
2355:
2281:
2226:
2163:
1956:
1951:
1899:
1667:
1656:
1328:
1228:
1156:
1147:
1143:
1078:
1073:
436:
95:
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2503:
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2213:
2079:
2005:
1988:
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1497:
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1323:
1298:
1068:
1017:
976:
852:
227:
165:
77:
42:
2231:
1687:
490:
and analytic properties. The theory of algebraically closed fields of a given characteristic is
256:
up to isomorphism? This is a deep question and significant progress was only made in 1954 when
2669:
2476:
2286:
2276:
2168:
2049:
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1860:
1641:
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560:
402:
157:
89:
69:
58:
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1971:
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1255:
958:
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834:
805:
789:
750:
713:
911:
2664:
2654:
2608:
2591:
2546:
2508:
2410:
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2137:
2064:
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2025:
1931:
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1742:
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1345:
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1203:
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907:
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535:
528:
448:
429:
Pure identity theory (with no functions, constants, predicates other than "=", or axioms).
312:
268:
261:
231:
124:
35:
17:
257:
555:
but not categorical in uncountable cardinals. The simplest example is the theory of an
2649:
2628:
2586:
2566:
2461:
2316:
1914:
1904:
1894:
1889:
1823:
1697:
1573:
1462:
1457:
1435:
1036:
923:
918:
881:
574:
proved that any such countable linear order is isomorphic to the rational numbers: see
542:
406:
182:
189:) extended Morley's theorem to uncountable languages: if the language has cardinality
2748:
2623:
2301:
1808:
1593:
1583:
1553:
1538:
1208:
944:
524:
517:
215:
2523:
2370:
2271:
2263:
2143:
2091:
2000:
1936:
1919:
1850:
1709:
1568:
1270:
1053:
571:
567:
521:
513:
120:
85:
81:
893:
222:
if all of its models are isomorphic. It follows from the definition above and the
195:
and a theory is categorical in some uncountable cardinal greater than or equal to
2633:
2513:
1692:
1682:
1629:
1313:
1233:
1218:
1098:
1043:
929:, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier,
564:
487:
234:
cannot be categorical. One is then immediately led to the more subtle notion of
176:
173:
1563:
1418:
1389:
1195:
809:
754:
73:
972:
848:
447:
say that all algebraically closed fields of characteristic 0 as large as the
2715:
2618:
1671:
1588:
1548:
1512:
1448:
1260:
1250:
1223:
351:
169:
686:
2700:
2498:
1946:
1651:
1245:
2296:
1088:
980:
856:
389:
In other words, he observed that, in all the cases he could think of,
986:
467:
963:
839:
271:
with at least one infinite model, he could only find three ways for
1840:
1186:
1031:
724:, Studies in Logic and the Foundations of Mathematics, Elsevier,
527:(essentially the same as vector spaces over a finite field) and
123:, the notion of a categorical theory is refined with respect to
990:
563:, both of which are infinite. Another example is the theory of
201:
then it is categorical in all cardinalities greater than
957:(3), American Mathematical Society, Vol. 5, No. 3: 343â384,
925:
Classification theory and the number of nonisomorphic models
623:, and so are all equivalent as the theory is categorical in
549:
There are also examples of theories that are categorical in
27:"Vaught's test" redirects here. Not to be confused with the
179:, then it is categorical in all uncountable cardinalities.
88:
are categorical, having a unique model whose domain is the
65:
its model, uniquely characterizing the model's structure.
500:(the countable infinite cardinal); there are models of
486:, they may (and in fact do) have completely different
395:-categoricity at any one uncountable cardinal implied
603:
is very close to being complete. More precisely, the
98:
2642:
2537:
2369:
2262:
2114:
1807:
1730:
1624:
1528:
1417:
1344:
1279:
1194:
1185:
1107:
1024:
922:
687:"Difference between completeness and categoricity"
109:
950:Transactions of the American Mathematical Society
822:Transactions of the American Mathematical Society
884:(1974), "Categoricity of uncountable theories",
534:(essentially the same as vector spaces over the
240:-categoricity, which asks: for which cardinals
1002:
146:) if it has exactly one model of cardinality
8:
516:over a given countable field. This includes
947:(1904), "A System of Axioms for Geometry",
458:; it only asserts that they are isomorphic
2765:Theorems in the foundations of mathematics
1828:
1423:
1191:
1009:
995:
987:
246:is there exactly one model of cardinality
962:
838:
466:. It follows that although the completed
100:
99:
97:
665:
650:
590:. However, the converse does not hold.
413:and Shelah's more general programme of
597:categorical in some infinite cardinal
186:
161:
84:model. For example, the second-order
80:contains categorical theories with an
432:The classic example is the theory of
7:
672:
409:in the 1970s and beyond, leading to
61:). Such a theory can be viewed as
25:
819:(1965), "Categoricity in Power",
2728:
482:are all isomorphic as fields to
172:language is categorical in some
774:, vol. 217, New York, NY:
743:History and Philosophy of Logic
218:in 1904 defined a theory to be
311:-categorical for all infinite
1:
2689:History of mathematical logic
865:"Categoricity in cardinality"
831:American Mathematical Society
772:Graduate Texts in Mathematics
768:Model theory: An introduction
154:Morley's categoricity theorem
110:{\displaystyle \mathbb {N} .}
2614:Primitive recursive function
833:, Vol. 114, No. 2: 514â538,
685:Mummert, Carl (2014-09-16).
586:Every categorical theory is
576:Cantor's isomorphism theorem
379:-categorical if and only if
344:-categorical if and only if
870:Encyclopedia of Mathematics
532:torsion-free abelian groups
260:noticed that, at least for
76:model can be categorical.
2781:
1678:SchröderâBernstein theorem
1405:Monadic predicate calculus
1064:Foundations of mathematics
545:with a successor function.
281:-categorical at some
33:
26:
18:Categorical (model theory)
2724:
2711:Philosophy of mathematics
2660:Automated theorem proving
1831:
1785:Von NeumannâBernaysâGödel
1426:
894:10.1090/pspum/025/0373874
810:10.1007/978-1-4684-9452-5
755:10.1080/01445348008837010
541:The theory of the set of
230:with a model of infinite
863:Palyutin, E.A. (2001) ,
800:Monk, J. Donald (1976),
741:(1980), "Categoricity",
621:LöwenheimâSkolem theorem
385:is a countable cardinal.
224:LöwenheimâSkolem theorem
34:Not to be confused with
2361:Self-verifying theories
2182:Tarski's axiomatization
1133:Tarski's undefinability
1128:incompleteness theorems
329:uncountably categorical
72:, only theories with a
2735:Mathematics portal
2346:Proof of impossibility
1994:propositional variable
1304:Propositional calculus
766:Marker, David (2002),
211:History and motivation
111:
53:if it has exactly one
2604:Kolmogorov complexity
2557:Computably enumerable
2457:Model complete theory
2249:Principia Mathematica
1309:Propositional formula
1138:BanachâTarski paradox
415:classification theory
363:countably categorical
158:Michael D. Morley
112:
2552:ChurchâTuring thesis
2539:Computability theory
1748:continuum hypothesis
1266:Square of opposition
1124:Gödel's completeness
639:Spectrum of a theory
557:equivalence relation
502:transcendence degree
443:. Categoricity does
434:algebraically closed
252:of the given theory
164:) stating that if a
96:
2706:Mathematical object
2597:P versus NP problem
2562:Computable function
2356:Reverse mathematics
2282:Logical consequence
2159:primitive recursive
2154:elementary function
1927:Free/bound variable
1780:TarskiâGrothendieck
1299:Logical connectives
1229:Logical equivalence
1079:Logical consequence
804:, Springer-Verlag,
699:Marker (2002) p. 42
570:with no endpoints;
561:equivalence classes
296:totally categorical
152:up to isomorphism.
92:of natural numbers
2755:Mathematical logic
2504:Transfer principle
2467:Semantics of logic
2452:Categorical theory
2428:Non-standard model
1942:Logical connective
1069:Information theory
1018:Mathematical logic
802:Mathematical Logic
718:Keisler, H. Jerome
228:first-order theory
183:Saharon Shelah
166:first-order theory
107:
78:Higher-order logic
43:mathematical logic
29:TarskiâVaught test
2742:
2741:
2674:Abstract category
2477:Theories of truth
2287:Rule of inference
2277:Natural deduction
2258:
2257:
1803:
1802:
1508:Cartesian product
1413:
1412:
1319:Many-valued logic
1294:Boolean functions
1177:Russell's paradox
1152:diagonal argument
1049:First-order logic
940:(IX, 1.19, pg.49)
936:978-0-444-70260-9
731:978-0-444-88054-3
714:Chang, Chen Chung
559:with exactly two
262:complete theories
70:first-order logic
59:up to isomorphism
16:(Redirected from
2772:
2733:
2732:
2684:History of logic
2679:Category of sets
2572:Decision problem
2351:Ordinal analysis
2292:Sequent calculus
2190:Boolean algebras
2130:
2129:
2104:
2075:logical/constant
1829:
1815:
1738:ZermeloâFraenkel
1489:Set operations:
1424:
1361:
1192:
1172:LöwenheimâSkolem
1059:Formal semantics
1011:
1004:
997:
988:
983:
966:
939:
928:
914:
877:
859:
842:
812:
796:
757:
749:(1â2): 187â207,
734:
700:
697:
691:
690:
682:
676:
670:
658:
655:
628:
618:
612:
602:
554:
509:
499:
454:are the same as
411:stability theory
400:
394:
384:
378:
349:
343:
320:
310:
286:
280:
251:
245:
239:
206:
200:
194:
156:is a theorem of
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144:
132:
116:
114:
113:
108:
103:
21:
2780:
2779:
2775:
2774:
2773:
2771:
2770:
2769:
2745:
2744:
2743:
2738:
2727:
2720:
2665:Category theory
2655:Algebraic logic
2638:
2609:Lambda calculus
2547:Church encoding
2533:
2509:Truth predicate
2365:
2331:Complete theory
2254:
2123:
2119:
2115:
2110:
2102:
1822: and
1818:
1813:
1799:
1775:New Foundations
1743:axiom of choice
1726:
1688:Gödel numbering
1628: and
1620:
1524:
1409:
1359:
1340:
1289:Boolean algebra
1275:
1239:Equiconsistency
1204:Classical logic
1181:
1162:Halting problem
1150: and
1126: and
1114: and
1113:
1108:Theorems (
1103:
1020:
1015:
964:10.2307/1986462
943:
937:
919:Shelah, Saharon
917:
904:
882:Shelah, Saharon
880:
862:
840:10.2307/1994188
817:Morley, Michael
815:
799:
786:
776:Springer-Verlag
765:
737:
732:
712:
709:
704:
703:
698:
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684:
683:
679:
671:
667:
662:
661:
656:
652:
647:
635:
624:
614:
608:
605:ĆoĆâVaught test
598:
584:
550:
543:natural numbers
505:
495:
494:categorical in
481:
449:complex numbers
423:
396:
390:
380:
374:
345:
339:
316:
306:
282:
276:
267:over countable
247:
241:
235:
213:
202:
196:
190:
147:
140:
139:categorical in
128:
127:. A theory is
94:
93:
39:
36:Category theory
32:
23:
22:
15:
12:
11:
5:
2778:
2776:
2768:
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2762:
2757:
2747:
2746:
2740:
2739:
2725:
2722:
2721:
2719:
2718:
2713:
2708:
2703:
2698:
2697:
2696:
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2681:
2676:
2667:
2662:
2657:
2652:
2650:Abstract logic
2646:
2644:
2640:
2639:
2637:
2636:
2631:
2629:Turing machine
2626:
2621:
2616:
2611:
2606:
2601:
2600:
2599:
2594:
2589:
2584:
2579:
2569:
2567:Computable set
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2559:
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2549:
2543:
2541:
2535:
2534:
2532:
2531:
2526:
2521:
2516:
2511:
2506:
2501:
2496:
2495:
2494:
2489:
2484:
2474:
2469:
2464:
2462:Satisfiability
2459:
2454:
2449:
2448:
2447:
2437:
2436:
2435:
2425:
2424:
2423:
2418:
2413:
2408:
2403:
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2392:
2391:
2386:
2379:Interpretation
2375:
2373:
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2363:
2358:
2353:
2348:
2343:
2333:
2328:
2327:
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2325:
2324:
2314:
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2299:
2294:
2289:
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2279:
2274:
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2259:
2256:
2255:
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2252:
2244:
2243:
2242:
2241:
2236:
2235:
2234:
2229:
2224:
2204:
2203:
2202:
2200:minimal axioms
2197:
2186:
2185:
2184:
2173:
2172:
2171:
2166:
2161:
2156:
2151:
2146:
2133:
2131:
2112:
2111:
2109:
2108:
2107:
2106:
2094:
2089:
2088:
2087:
2082:
2077:
2072:
2062:
2057:
2052:
2047:
2046:
2045:
2040:
2030:
2029:
2028:
2023:
2018:
2013:
2003:
1998:
1997:
1996:
1991:
1986:
1976:
1975:
1974:
1969:
1964:
1959:
1954:
1949:
1939:
1934:
1929:
1924:
1923:
1922:
1917:
1912:
1907:
1897:
1892:
1890:Formation rule
1887:
1882:
1881:
1880:
1875:
1865:
1864:
1863:
1853:
1848:
1843:
1838:
1832:
1826:
1809:Formal systems
1805:
1804:
1801:
1800:
1798:
1797:
1792:
1787:
1782:
1777:
1772:
1767:
1762:
1757:
1752:
1751:
1750:
1745:
1734:
1732:
1728:
1727:
1725:
1724:
1723:
1722:
1712:
1707:
1706:
1705:
1698:Large cardinal
1695:
1690:
1685:
1680:
1675:
1661:
1660:
1659:
1654:
1649:
1634:
1632:
1622:
1621:
1619:
1618:
1617:
1616:
1611:
1606:
1596:
1591:
1586:
1581:
1576:
1571:
1566:
1561:
1556:
1551:
1546:
1541:
1535:
1533:
1526:
1525:
1523:
1522:
1521:
1520:
1515:
1510:
1505:
1500:
1495:
1487:
1486:
1485:
1480:
1470:
1465:
1463:Extensionality
1460:
1458:Ordinal number
1455:
1445:
1440:
1439:
1438:
1427:
1421:
1415:
1414:
1411:
1410:
1408:
1407:
1402:
1397:
1392:
1387:
1382:
1377:
1376:
1375:
1365:
1364:
1363:
1350:
1348:
1342:
1341:
1339:
1338:
1337:
1336:
1331:
1326:
1316:
1311:
1306:
1301:
1296:
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1283:
1277:
1276:
1274:
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1242:
1241:
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1221:
1216:
1211:
1206:
1200:
1198:
1189:
1183:
1182:
1180:
1179:
1174:
1169:
1164:
1159:
1154:
1142:Cantor's
1140:
1135:
1130:
1120:
1118:
1105:
1104:
1102:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1061:
1056:
1051:
1046:
1041:
1040:
1039:
1028:
1026:
1022:
1021:
1016:
1014:
1013:
1006:
999:
991:
985:
984:
945:Veblen, Oswald
941:
935:
915:
902:
878:
860:
813:
797:
784:
763:
758:
739:Corcoran, John
735:
730:
708:
705:
702:
701:
692:
677:
675:, p. 349.
664:
663:
660:
659:
649:
648:
646:
643:
642:
641:
634:
631:
583:
580:
547:
546:
539:
518:abelian groups
511:
504:0, 1, 2, ...,
477:
441:characteristic
430:
422:
419:
407:Saharon Shelah
403:Michael Morley
387:
386:
355:
322:
212:
209:
106:
102:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2777:
2766:
2763:
2761:
2758:
2756:
2753:
2752:
2750:
2737:
2736:
2731:
2723:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2695:
2692:
2691:
2690:
2687:
2685:
2682:
2680:
2677:
2675:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2647:
2645:
2641:
2635:
2632:
2630:
2627:
2625:
2624:Recursive set
2622:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2574:
2573:
2570:
2568:
2565:
2563:
2560:
2558:
2555:
2553:
2550:
2548:
2545:
2544:
2542:
2540:
2536:
2530:
2527:
2525:
2522:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2502:
2500:
2497:
2493:
2490:
2488:
2485:
2483:
2480:
2479:
2478:
2475:
2473:
2470:
2468:
2465:
2463:
2460:
2458:
2455:
2453:
2450:
2446:
2443:
2442:
2441:
2438:
2434:
2433:of arithmetic
2431:
2430:
2429:
2426:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2398:
2397:
2394:
2390:
2387:
2385:
2382:
2381:
2380:
2377:
2376:
2374:
2372:
2368:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2341:
2340:from ZFC
2337:
2334:
2332:
2329:
2323:
2320:
2319:
2318:
2315:
2313:
2310:
2308:
2305:
2304:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2269:
2267:
2265:
2261:
2251:
2250:
2246:
2245:
2240:
2239:non-Euclidean
2237:
2233:
2230:
2228:
2225:
2223:
2222:
2218:
2217:
2215:
2212:
2211:
2209:
2205:
2201:
2198:
2196:
2193:
2192:
2191:
2187:
2183:
2180:
2179:
2178:
2174:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2141:
2139:
2135:
2134:
2132:
2127:
2121:
2116:Example
2113:
2105:
2100:
2099:
2098:
2095:
2093:
2090:
2086:
2083:
2081:
2078:
2076:
2073:
2071:
2068:
2067:
2066:
2063:
2061:
2058:
2056:
2053:
2051:
2048:
2044:
2041:
2039:
2036:
2035:
2034:
2031:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2008:
2007:
2004:
2002:
1999:
1995:
1992:
1990:
1987:
1985:
1982:
1981:
1980:
1977:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1944:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1921:
1918:
1916:
1913:
1911:
1908:
1906:
1903:
1902:
1901:
1898:
1896:
1893:
1891:
1888:
1886:
1883:
1879:
1876:
1874:
1873:by definition
1871:
1870:
1869:
1866:
1862:
1859:
1858:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1833:
1830:
1827:
1825:
1821:
1816:
1810:
1806:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
1771:
1768:
1766:
1763:
1761:
1760:KripkeâPlatek
1758:
1756:
1753:
1749:
1746:
1744:
1741:
1740:
1739:
1736:
1735:
1733:
1729:
1721:
1718:
1717:
1716:
1713:
1711:
1708:
1704:
1701:
1700:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1673:
1669:
1665:
1662:
1658:
1655:
1653:
1650:
1648:
1645:
1644:
1643:
1639:
1636:
1635:
1633:
1631:
1627:
1623:
1615:
1612:
1610:
1607:
1605:
1604:constructible
1602:
1601:
1600:
1597:
1595:
1592:
1590:
1587:
1585:
1582:
1580:
1577:
1575:
1572:
1570:
1567:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1536:
1534:
1532:
1527:
1519:
1516:
1514:
1511:
1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1490:
1488:
1484:
1481:
1479:
1476:
1475:
1474:
1471:
1469:
1466:
1464:
1461:
1459:
1456:
1454:
1450:
1446:
1444:
1441:
1437:
1434:
1433:
1432:
1429:
1428:
1425:
1422:
1420:
1416:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1374:
1371:
1370:
1369:
1366:
1362:
1357:
1356:
1355:
1352:
1351:
1349:
1347:
1343:
1335:
1332:
1330:
1327:
1325:
1322:
1321:
1320:
1317:
1315:
1312:
1310:
1307:
1305:
1302:
1300:
1297:
1295:
1292:
1290:
1287:
1286:
1284:
1282:
1281:Propositional
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1240:
1237:
1236:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1209:Logical truth
1207:
1205:
1202:
1201:
1199:
1197:
1193:
1190:
1188:
1184:
1178:
1175:
1173:
1170:
1168:
1165:
1163:
1160:
1158:
1155:
1153:
1149:
1145:
1141:
1139:
1136:
1134:
1131:
1129:
1125:
1122:
1121:
1119:
1117:
1111:
1106:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1050:
1047:
1045:
1042:
1038:
1035:
1034:
1033:
1030:
1029:
1027:
1023:
1019:
1012:
1007:
1005:
1000:
998:
993:
992:
989:
982:
978:
974:
970:
965:
960:
956:
952:
951:
946:
942:
938:
932:
927:
926:
920:
916:
913:
909:
905:
903:9780821814253
899:
895:
891:
887:
883:
879:
876:
872:
871:
866:
861:
858:
854:
850:
846:
841:
836:
832:
828:
824:
823:
818:
814:
811:
807:
803:
798:
795:
791:
787:
785:0-387-98760-6
781:
777:
773:
769:
764:
762:
759:
756:
752:
748:
744:
740:
736:
733:
727:
723:
719:
715:
711:
710:
706:
696:
693:
688:
681:
678:
674:
669:
666:
654:
651:
644:
640:
637:
636:
632:
630:
627:
622:
617:
611:
606:
601:
596:
591:
589:
581:
579:
577:
573:
569:
568:linear orders
566:
562:
558:
553:
544:
540:
537:
533:
530:
526:
523:
519:
515:
514:Vector spaces
512:
508:
503:
498:
493:
489:
485:
480:
476:
472:
470:
465:
461:
457:
453:
450:
446:
442:
438:
435:
431:
428:
427:
426:
420:
418:
416:
412:
408:
404:
399:
393:
383:
377:
372:
369:
365:
364:
359:
356:
353:
348:
342:
337:
334:
330:
326:
323:
319:
314:
309:
304:
301:
297:
293:
290:
289:
288:
285:
279:
274:
270:
266:
263:
259:
255:
250:
244:
238:
233:
229:
225:
221:
217:
216:Oswald Veblen
210:
208:
205:
199:
193:
188:
184:
180:
178:
175:
171:
167:
163:
159:
155:
150:
145:
143:
136:
131:
126:
122:
117:
104:
91:
87:
83:
79:
75:
71:
66:
64:
60:
56:
52:
48:
44:
37:
30:
19:
2760:Model theory
2726:
2524:Ultraproduct
2451:
2371:Model theory
2336:Independence
2272:Formal proof
2264:Proof theory
2247:
2220:
2177:real numbers
2149:second-order
2060:Substitution
1937:Metalanguage
1878:conservative
1851:Axiom schema
1795:Constructive
1765:MorseâKelley
1731:Set theories
1710:Aleph number
1703:inaccessible
1609:Grothendieck
1493:intersection
1380:Higher-order
1368:Second-order
1314:Truth tables
1271:Venn diagram
1054:Formal proof
954:
948:
924:
885:
868:
826:
820:
801:
767:
746:
742:
722:Model Theory
721:
695:
680:
668:
653:
625:
615:
609:
599:
594:
592:
585:
551:
548:
506:
496:
491:
483:
478:
474:
468:
463:
459:
455:
451:
444:
424:
397:
391:
388:
381:
375:
370:
367:
362:
357:
346:
340:
335:
332:
328:
324:
317:
307:
302:
299:
295:
291:
283:
277:
272:
264:
253:
248:
242:
236:
219:
214:
203:
197:
191:
181:
153:
148:
141:
138:
134:
129:
121:model theory
118:
86:Peano axioms
67:
62:
50:
40:
2634:Type theory
2582:undecidable
2514:Truth value
2401:equivalence
2080:non-logical
1693:Enumeration
1683:Isomorphism
1630:cardinality
1614:Von Neumann
1579:Ultrafilter
1544:Uncountable
1478:equivalence
1395:Quantifiers
1385:Fixed-point
1354:First-order
1234:Consistency
1219:Proposition
1196:Traditional
1167:Lindström's
1157:Compactness
1099:Type theory
1044:Cardinality
593:Any theory
488:topological
439:of a given
352:uncountable
232:cardinality
220:categorical
177:cardinality
174:uncountable
135:categorical
125:cardinality
51:categorical
2749:Categories
2445:elementary
2138:arithmetic
2006:Quantifier
1984:functional
1856:Expression
1574:Transitive
1518:identities
1503:complement
1436:hereditary
1419:Set theory
794:1003.03034
707:References
582:Properties
2716:Supertask
2619:Recursion
2577:decidable
2411:saturated
2389:of models
2312:deductive
2307:axiomatic
2227:Hilbert's
2214:Euclidean
2195:canonical
2118:axiomatic
2050:Signature
1979:Predicate
1868:Extension
1790:Ackermann
1715:Operation
1594:Universal
1584:Recursive
1559:Singleton
1554:Inhabited
1539:Countable
1529:Types of
1513:power set
1483:partition
1400:Predicate
1346:Predicate
1261:Syllogism
1251:Soundness
1224:Inference
1214:Tautology
1116:paradoxes
973:0002-9947
921:(1990) ,
875:EMS Press
849:0002-9947
720:(1990) ,
673:Monk 1976
536:rationals
529:divisible
520:of given
473:closures
460:as fields
354:cardinal.
313:cardinals
269:languages
258:Jerzy ĆoĆ
226:that any
170:countable
2701:Logicism
2694:timeline
2670:Concrete
2529:Validity
2499:T-schema
2492:Kripke's
2487:Tarski's
2482:semantic
2472:Strength
2421:submodel
2416:spectrum
2384:function
2232:Tarski's
2221:Elements
2208:geometry
2164:Robinson
2085:variable
2070:function
2043:spectrum
2033:Sentence
1989:variable
1932:Language
1885:Relation
1846:Automata
1836:Alphabet
1820:language
1674:-jection
1652:codomain
1638:Function
1599:Universe
1569:Infinite
1473:Relation
1256:Validity
1246:Argument
1144:theorem,
633:See also
588:complete
525:exponent
421:Examples
82:infinite
63:defining
2643:Related
2440:Diagram
2338: (
2317:Hilbert
2302:Systems
2297:Theorem
2175:of the
2120:systems
1900:Formula
1895:Grammar
1811: (
1755:General
1468:Forcing
1453:Element
1373:Monadic
1148:paradox
1089:Theorem
1025:General
981:1986462
912:0373874
857:1994188
619:by the
185: (
160: (
2406:finite
2169:Skolem
2122:
2097:Theory
2065:Symbol
2055:String
2038:atomic
1915:ground
1910:closed
1905:atomic
1861:ground
1824:syntax
1720:binary
1647:domain
1564:Finite
1329:finite
1187:Logics
1146:
1094:Theory
979:
971:
933:
910:
900:
855:
847:
792:
782:
728:
572:Cantor
437:fields
350:is an
315:
275:to be
74:finite
47:theory
2396:Model
2144:Peano
2001:Proof
1841:Arity
1770:Naive
1657:image
1589:Fuzzy
1549:Empty
1498:union
1443:Class
1084:Model
1074:Lemma
1032:Axiom
977:JSTOR
853:JSTOR
829:(2),
645:Notes
565:dense
522:prime
471:-adic
168:in a
55:model
2519:Type
2322:list
2126:list
2103:list
2092:Term
2026:rank
1920:open
1814:list
1626:Maps
1531:sets
1390:Free
1360:list
1110:list
1037:list
969:ISSN
931:ISBN
898:ISBN
845:ISSN
780:ISBN
726:ISBN
368:i.e.
333:i.e.
300:i.e.
187:1974
162:1965
137:(or
45:, a
2206:of
2188:of
2136:of
1668:Sur
1642:Map
1449:Ur-
1431:Set
959:doi
890:doi
835:doi
827:114
806:doi
790:Zbl
751:doi
492:not
462:to
445:not
373:is
360:is
338:is
327:is
305:is
294:is
119:In
90:set
68:In
49:is
41:In
2751::
2592:NP
2216::
2210::
2140::
1817:),
1672:Bi
1664:In
975:,
967:,
953:,
908:MR
906:,
896:,
873:,
867:,
851:,
843:,
825:,
788:,
778:,
770:,
745:,
716:;
578:.
538:).
417:.
366:,
331:,
298:,
287::
207:.
2672:/
2587:P
2342:)
2128:)
2124:(
2021:â
2016:!
2011:â
1972:=
1967:â
1962:â
1957:â§
1952:âš
1947:ÂŹ
1670:/
1666:/
1640:/
1451:)
1447:(
1334:â
1324:3
1112:)
1010:e
1003:t
996:v
961::
955:5
892::
837::
808::
753::
747:1
689:.
626:Îș
616:Îș
610:Îș
600:Îș
595:T
552:Ï
510:.
507:Ï
497:Ï
484:C
479:p
475:C
469:p
464:C
456:C
452:C
398:Îș
392:Îș
382:Îș
376:Îș
371:T
358:T
347:Îș
341:Îș
336:T
325:T
321:.
318:Îș
308:Îș
303:T
292:T
284:Îș
278:Îș
273:T
265:T
254:T
249:Îș
243:Îș
237:Îș
204:Îș
198:Îș
192:Îș
149:Îș
142:Îș
133:-
130:Îș
105:.
101:N
57:(
38:.
31:.
20:)
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