2162:
1498:
1955:
1988:
2707:
75:
As a language with infinitely long formulae is being presented, it is not possible to write such formulae down explicitly. To get around this problem a number of notational conveniences, which, strictly speaking, are not part of the formal language, are used.
1777:
1704:
1371:
1831:
1598:
1236:
may not always be possible, however extra constant symbols may be added for each variable with the resulting satisfiability relation remaining the same. To avoid this, some authors use a different definition of the language
55:
infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied.
2214:
1155:
482:
2157:{\displaystyle ((\land _{\mu <\gamma }{(\lor _{\delta <\gamma }{A_{\mu ,\delta }})})\implies (\lor _{\epsilon <\gamma ^{\gamma }}{(\land _{\mu <\gamma }{A_{\mu ,\gamma _{\epsilon }(\mu )})}}))}
668:
214:
139:
825:
760:
1200:
96:
is used to point out an expression that is infinitely long. Where it is unclear, the length of the sequence is noted afterwards. Where this notation becomes ambiguous or confusing, suffixes such as
2773:
1826:
2613:
3121:
586:
419:
534:
2258:
A theory is any set of sentences. The truth of statements in models are defined by recursion and will agree with the definition for finitary logic where both are defined. Given a theory
2933:
2854:
1363:
2250:. The last axiom schema is strictly speaking unnecessary, as Chang's distributivity laws imply it, however it is included as a natural way to allow natural weakenings to the logic.
885:
2973:
2890:
2814:
2539:
2498:
2402:
2361:
980:
The concepts of free and bound variables apply in the same manner to infinite formulae. Just as in finitary logic, a formula all of whose variables are bound is referred to as a
2304:
1983:
1526:
1268:
1048:
338:
267:
2241:
1230:
911:
1709:
955:
933:
241:
2562:
2445:
387:
3013:
2993:
2585:
2468:
2422:
1623:
1331:
1311:
1094:, or is deduced from previous statements using a rule of inference. As before, all rules of inference in finitary logic can be used, together with an additional one:
845:
606:
358:
290:
166:
94:
51:
sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses
1288:
695:
975:
1493:{\displaystyle ((\land _{\epsilon <\delta }{(A_{\delta }\implies A_{\epsilon })})\implies (A_{\delta }\implies \land _{\epsilon <\delta }{A_{\epsilon }}))}
1628:
1092:
1068:
1015:
421:, has the same set of symbols as a finitary logic and may use all the rules for formation of formulae of a finitary logic together with some additional ones:
1950:{\displaystyle \forall g\in \gamma ^{\gamma }\exists \epsilon <\gamma :\{A_{\epsilon },\neg A_{\epsilon }\}\subseteq \{A_{\mu ,g(\mu )}:\mu <\gamma \}}
1531:
3255:
3151:
2716:
can only be expressed in a logic that allows infinitely many quantifiers in an individual statement. As a consequence many theories, including
2167:
2856:. The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size.
2720:, which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic. Other examples include the theories of
1101:
428:
614:
171:
3366:
99:
765:
700:
1160:
3412:
3145:
3057:
2702:{\displaystyle \forall _{\gamma <\omega }{V_{\gamma }:}\neg \land _{\gamma <\omega }{V_{\gamma +}\in V_{\gamma }}.\,}
2734:
299:
is assumed (as is often done when discussing infinitary logic) as this is necessary to have sensible distributivity laws.
1782:
3325:
2728:. These three theories can be defined without the use of infinite quantification; only infinite junctions are needed.
539:
392:
3465:
487:
851:
The language may also have function, relation, and predicate symbols of finite arity. Karp also defined languages
2935:
fails to be compact, but it is complete (under the axioms given above). Moreover, it satisfies a variant of the
3427:
2898:
2819:
1336:
982:
854:
3470:
3037:
1074:
of statements that obeys the following conditions: Each statement is either a logical axiom, an element of
2945:
2862:
2786:
2511:
2477:
2374:
2340:
995:
2276:
1962:
1505:
1293:
The logical axiom schemata specific to infinitary logic are presented below. Global schemata variables:
1240:
1020:
310:
246:
2219:
1209:
2712:
Unlike the axiom of foundation, this statement admits no non-standard interpretations. The concept of
1772:{\displaystyle \forall \mu \forall \delta \exists \epsilon <\gamma :A_{\mu ,\delta }=A_{\epsilon }}
890:
2501:
3304:
3033:
938:
916:
2364:
219:
2547:
2430:
3299:
3355:
366:
2998:
2978:
2721:
2570:
2453:
2407:
1608:
1316:
1296:
830:
591:
343:
275:
151:
79:
21:
1273:
673:
64:
44:
1699:{\displaystyle (\lor _{\mu <\gamma }{(\land _{\delta <\gamma }{A_{\mu ,\delta }})})}
960:
8:
2936:
2604:
142:
40:
216:. This is meant to represent an infinite sequence of quantifiers: a quantifier for each
3444:
3289:
3233:
2725:
1077:
1053:
1000:
29:
3404:
3269:
3250:
3408:
3141:
3053:
2246:
The last two axiom schemata require the axiom of choice because certain sets must be
1233:
36:
25:
1593:{\displaystyle ((\land _{\epsilon <\delta }{A_{\epsilon }})\implies A_{\gamma })}
3436:
3400:
3334:
3264:
3246:
3133:
3125:
3092:
3073:
3045:
2717:
2713:
1602:
360:
3137:
296:
3122:"The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture"
3049:
3339:
3320:
3117:
2783:
Two infinitary logics stand out in their completeness. These are the logics of
52:
48:
3130:
Set Theory, Arithmetic, and
Foundations of Mathematics: Theorems, Philosophies
3032:
Moore, Gregory H. (1997). "The prehistory of infinitary logic: 1885–1955". In
1050:
is a set of sentences in the logic. A proof in infinitary logic from a theory
3459:
3096:
3236:". Stanford Encyclopedia of Philosophy, revised 2023. Accessed 26 July 2024.
35:
Some infinitary logics may have different properties from those of standard
3422:
146:
3077:
3448:
3392:
2600:
2334:. An infinitary logic can be complete without being strongly complete.
2247:
2209:{\displaystyle \{\gamma _{\epsilon }:\epsilon <\gamma ^{\gamma }\}}
935:
that allow for function and predicate symbols of infinite arity, with
60:
2253:
3440:
1071:
827:
are formulae. (In each case the sequence of quantifiers has length
3294:
3285:
47:. Notions of compactness and completeness that are equivalent in
2995:
is strongly compact (because proofs in these logics cannot use
1150:{\displaystyle A=\{A_{\gamma }|\gamma <\delta <\alpha \}}
989:
913:
an infinite cardinal and some more complicated restrictions on
477:{\displaystyle A=\{A_{\gamma }|\gamma <\delta <\alpha \}}
168:. The same notation may be applied to quantifiers, for example
1157:
that have occurred previously in the proof then the statement
663:{\displaystyle V=\{V_{\gamma }|\gamma <\delta <\beta \}}
70:
2594:
209:{\displaystyle \forall _{\gamma <\delta }{V_{\gamma }:}}
2731:
Truth predicates for countable languages are definable in
134:{\displaystyle \bigvee _{\gamma <\delta }{A_{\gamma }}}
2975:
is strongly complete (under the axioms given above) then
2892:
is also strongly complete, compact and strongly compact.
820:{\displaystyle \exists V_{0}:\exists V_{1}\cdots (A_{0})}
755:{\displaystyle \forall V_{0}:\forall V_{1}\cdots (A_{0})}
1195:{\displaystyle \land _{\gamma <\delta }{A_{\gamma }}}
957:
controlling the maximum arity of a function symbol and
3251:"On the representation of α-complete Boolean algebras"
3190:
32:. The concept was introduced by Zermelo in the 1930s.
3044:. Springer-Science+Business Media. pp. 105–123.
3001:
2981:
2948:
2901:
2865:
2822:
2789:
2737:
2616:
2573:
2550:
2514:
2480:
2456:
2433:
2410:
2377:
2343:
2279:
2222:
2170:
1991:
1965:
1834:
1785:
1712:
1631:
1611:
1534:
1508:
1374:
1339:
1319:
1299:
1276:
1243:
1212:
1163:
1104:
1080:
1056:
1023:
1003:
963:
941:
919:
893:
857:
833:
768:
703:
676:
617:
594:
542:
490:
431:
395:
369:
346:
313:
278:
249:
222:
174:
154:
102:
82:
59:
Considering whether a certain infinitary logic named
3202:
2768:{\displaystyle {\mathcal {L}}_{\omega _{1},\omega }}
588:
are formulae. (In each case the sequence has length
1821:{\displaystyle A_{\mu ,\delta }=\neg A_{\epsilon }}
3214:
3166:
3007:
2987:
2967:
2927:
2884:
2848:
2808:
2767:
2701:
2579:
2556:
2533:
2492:
2462:
2439:
2416:
2396:
2355:
2298:
2254:Completeness, compactness, and strong completeness
2235:
2208:
2156:
1977:
1949:
1820:
1771:
1698:
1617:
1592:
1520:
1492:
1357:
1325:
1305:
1282:
1262:
1224:
1194:
1149:
1086:
1062:
1042:
1009:
969:
949:
927:
905:
879:
839:
819:
754:
689:
662:
600:
580:
528:
476:
413:
381:
352:
332:
284:
261:
235:
208:
160:
133:
88:
39:. In particular, infinitary logics may fail to be
3256:Transactions of the American Mathematical Society
3457:
3425:(1969). "Infinitary logic and admissible sets".
3178:
581:{\displaystyle (A_{0}\land A_{1}\land \cdots )}
3356:"Inexpressible longing for the intended model"
3132:. Cambridge University Press. pp. 13–42.
2262:a sentence is said to be valid for the theory
414:{\displaystyle \omega \leq \beta \leq \alpha }
3397:Languages with Expressions of Infinite Length
2778:
2310:valid in every model there exists a proof of
529:{\displaystyle (A_{0}\lor A_{1}\lor \cdots )}
292:are not part of formal infinitary languages.
3367:Uniwersytet im. Adama Mickiewicza w Poznaniu
3286:"Four departures in Mathematics and Physics"
2314:. It is strongly complete if for any theory
2203:
2171:
1944:
1904:
1898:
1869:
1144:
1111:
990:Definition of Hilbert-type infinitary logics
657:
624:
471:
438:
63:is complete promises to throw light on the
3353:
2063:
2059:
1576:
1572:
1455:
1451:
1437:
1433:
1415:
1411:
1270:forbidding formulas from having more than
71:A word on notation and the axiom of choice
3338:
3303:
3293:
3268:
2698:
3283:
3072:
2595:Concepts expressible in infinitary logic
2541:, without restriction on size, if every
3421:
3318:
2928:{\displaystyle L_{\omega _{1},\omega }}
2849:{\displaystyle L_{\omega _{1},\omega }}
1358:{\displaystyle 0<\delta <\alpha }
3458:
3116:
880:{\displaystyle L_{\alpha \beta o\pi }}
3245:
3031:
3399:. North-Holland Publishing Company.
3391:
3220:
3208:
3196:
3184:
3172:
2968:{\displaystyle L_{\alpha ,\alpha }}
2885:{\displaystyle L_{\omega ,\omega }}
2809:{\displaystyle L_{\omega ,\omega }}
2534:{\displaystyle L_{\kappa ,\kappa }}
2493:{\displaystyle \kappa \neq \omega }
2397:{\displaystyle L_{\kappa ,\kappa }}
2356:{\displaystyle \kappa \neq \omega }
302:
13:
3354:Pogonowski, Jerzy (10 June 2010).
3326:Notre Dame Journal of Formal Logic
2741:
2648:
2618:
2603:the following statement expresses
2306:is complete if for every sentence
2299:{\displaystyle L_{\alpha ,\beta }}
1978:{\displaystyle \gamma <\alpha }
1885:
1854:
1835:
1805:
1725:
1719:
1713:
1521:{\displaystyle \gamma <\delta }
1263:{\displaystyle L_{\alpha ,\beta }}
1043:{\displaystyle L_{\alpha ,\beta }}
943:
921:
785:
769:
720:
704:
333:{\displaystyle L_{\alpha ,\beta }}
307:A first-order infinitary language
262:{\displaystyle \gamma <\delta }
176:
14:
3482:
3270:10.1090/S0002-9947-1957-0086792-1
3036:; Doets, Kees; Mundici, Daniele;
2236:{\displaystyle \gamma ^{\gamma }}
1605:'s distributivity laws (for each
1225:{\displaystyle \beta <\alpha }
141:are used to indicate an infinite
906:{\displaystyle \pi \leq \alpha }
3347:
3312:
3277:
3239:
3042:Structures and Norms in Science
2266:if it is true in all models of
977:controlling predicate symbols.
3226:
3110:
3085:The Bulletin of Symbolic Logic
3066:
3025:
3015:or more of the given axioms).
2151:
2148:
2143:
2138:
2132:
2091:
2064:
2060:
2056:
2052:
2015:
1995:
1992:
1927:
1921:
1693:
1689:
1652:
1632:
1587:
1573:
1569:
1538:
1535:
1487:
1484:
1452:
1438:
1434:
1430:
1426:
1412:
1398:
1378:
1375:
1125:
814:
801:
749:
736:
638:
575:
543:
523:
491:
452:
1:
3428:The Journal of Symbolic Logic
3405:10.1016/S0049-237X(08)70423-3
3018:
3284:Rosinger, Elemer E. (2010).
3138:10.1017/CBO9780511910616.003
950:{\displaystyle \mathrm {o} }
928:{\displaystyle \mathrm {o} }
24:that allows infinitely long
7:
3050:10.1007/978-94-017-0538-7_7
236:{\displaystyle V_{\gamma }}
10:
3487:
3385:
3319:Bennett, David W. (1980).
2779:Complete infinitary logics
2557:{\displaystyle \subseteq }
2440:{\displaystyle \subseteq }
1098:Given a set of statements
272:All usage of suffixes and
145:over a set of formulae of
3034:Dalla Chiara, Maria Luisa
2567:of cardinality less than
2450:of cardinality less than
1070:is a (possibly infinite)
611:Given a set of variables
3363:Zakład Logiki Stosowanej
3340:10.1305/ndjfl/1093882943
3128:; Kossak, Roman (eds.).
3078:"Zermelo and set theory"
2474:has a model. A cardinal
2424:many formulas, if every
2273:A logic in the language
425:Given a set of formulae
382:{\displaystyle \beta =0}
3466:Systems of formal logic
3008:{\displaystyle \alpha }
2988:{\displaystyle \alpha }
2580:{\displaystyle \kappa }
2463:{\displaystyle \kappa }
2417:{\displaystyle \kappa }
1618:{\displaystyle \gamma }
1326:{\displaystyle \gamma }
1306:{\displaystyle \delta }
1017:in infinitary language
840:{\displaystyle \delta }
601:{\displaystyle \delta }
353:{\displaystyle \alpha }
285:{\displaystyle \cdots }
161:{\displaystyle \delta }
89:{\displaystyle \cdots }
28:and/or infinitely long
3097:10.2178/bsl/1102083759
3009:
2989:
2969:
2929:
2886:
2850:
2810:
2769:
2722:non-archimedean fields
2703:
2581:
2558:
2535:
2504:when for every theory
2494:
2464:
2441:
2418:
2398:
2367:when for every theory
2357:
2300:
2237:
2216:is a well ordering of
2210:
2158:
1979:
1951:
1822:
1773:
1700:
1619:
1594:
1522:
1494:
1359:
1327:
1307:
1284:
1283:{\displaystyle \beta }
1264:
1226:
1196:
1151:
1088:
1064:
1044:
1011:
971:
951:
929:
907:
881:
841:
821:
756:
691:
664:
602:
582:
530:
478:
415:
383:
354:
334:
286:
263:
237:
210:
162:
135:
90:
3010:
2990:
2970:
2930:
2887:
2851:
2811:
2770:
2704:
2582:
2559:
2536:
2495:
2465:
2442:
2419:
2399:
2358:
2301:
2238:
2211:
2159:
1980:
1952:
1823:
1774:
1701:
1620:
1595:
1523:
1495:
1360:
1328:
1308:
1285:
1265:
1227:
1197:
1152:
1089:
1065:
1045:
1012:
972:
952:
930:
908:
882:
842:
822:
757:
692:
690:{\displaystyle A_{0}}
665:
603:
583:
531:
479:
416:
384:
355:
335:
287:
264:
238:
211:
163:
136:
91:
2999:
2979:
2946:
2899:
2863:
2820:
2787:
2735:
2614:
2571:
2548:
2512:
2478:
2454:
2431:
2408:
2375:
2341:
2326:there is a proof of
2277:
2220:
2168:
1989:
1963:
1832:
1783:
1710:
1629:
1609:
1532:
1506:
1372:
1337:
1317:
1297:
1274:
1241:
1210:
1161:
1102:
1078:
1054:
1021:
1001:
970:{\displaystyle \pi }
961:
939:
917:
891:
855:
831:
766:
701:
674:
615:
592:
540:
488:
429:
393:
367:
344:
311:
276:
247:
220:
172:
152:
100:
80:
65:continuum hypothesis
3471:Non-classical logic
3199:, pp. 101–102.
2937:Craig interpolation
2726:torsion-free groups
2599:In the language of
2404:containing at most
2318:for every sentence
3038:van Benthem, Johan
3005:
2985:
2965:
2925:
2882:
2846:
2806:
2765:
2699:
2587:has a model, then
2577:
2554:
2531:
2490:
2470:has a model, then
2460:
2437:
2414:
2394:
2353:
2296:
2233:
2206:
2154:
1975:
1947:
1818:
1769:
1696:
1615:
1590:
1518:
1490:
1355:
1323:
1303:
1280:
1260:
1234:universal closures
1222:
1192:
1147:
1084:
1060:
1040:
1007:
967:
947:
925:
903:
877:
837:
817:
752:
687:
660:
598:
578:
526:
474:
411:
379:
350:
330:
282:
259:
233:
206:
158:
131:
118:
86:
3414:978-0-444-53401-9
3211:, pp. 39–54.
3147:978-0-511-91061-6
3126:Kennedy, Juliette
3074:Kanamori, Akihiro
3059:978-94-017-0538-7
1087:{\displaystyle T}
1063:{\displaystyle T}
1010:{\displaystyle T}
103:
37:first-order logic
3478:
3452:
3418:
3379:
3378:
3376:
3374:
3360:
3351:
3345:
3344:
3342:
3316:
3310:
3309:
3307:
3297:
3281:
3275:
3274:
3272:
3243:
3237:
3234:Infinitary Logic
3230:
3224:
3218:
3212:
3206:
3200:
3194:
3188:
3182:
3176:
3170:
3164:
3163:
3161:
3159:
3150:. Archived from
3114:
3108:
3107:
3105:
3103:
3082:
3070:
3064:
3063:
3029:
3014:
3012:
3011:
3006:
2994:
2992:
2991:
2986:
2974:
2972:
2971:
2966:
2964:
2963:
2942:If the logic of
2934:
2932:
2931:
2926:
2924:
2923:
2916:
2915:
2891:
2889:
2888:
2883:
2881:
2880:
2855:
2853:
2852:
2847:
2845:
2844:
2837:
2836:
2815:
2813:
2812:
2807:
2805:
2804:
2774:
2772:
2771:
2766:
2764:
2763:
2756:
2755:
2745:
2744:
2718:Peano arithmetic
2714:well-foundedness
2708:
2706:
2705:
2700:
2694:
2693:
2692:
2680:
2679:
2666:
2665:
2647:
2643:
2642:
2632:
2631:
2586:
2584:
2583:
2578:
2563:
2561:
2560:
2555:
2540:
2538:
2537:
2532:
2530:
2529:
2502:strongly compact
2499:
2497:
2496:
2491:
2469:
2467:
2466:
2461:
2446:
2444:
2443:
2438:
2423:
2421:
2420:
2415:
2403:
2401:
2400:
2395:
2393:
2392:
2362:
2360:
2359:
2354:
2305:
2303:
2302:
2297:
2295:
2294:
2242:
2240:
2239:
2234:
2232:
2231:
2215:
2213:
2212:
2207:
2202:
2201:
2183:
2182:
2163:
2161:
2160:
2155:
2147:
2146:
2142:
2141:
2131:
2130:
2109:
2108:
2089:
2088:
2087:
2086:
2055:
2051:
2050:
2049:
2033:
2032:
2013:
2012:
1984:
1982:
1981:
1976:
1956:
1954:
1953:
1948:
1931:
1930:
1897:
1896:
1881:
1880:
1853:
1852:
1827:
1825:
1824:
1819:
1817:
1816:
1801:
1800:
1778:
1776:
1775:
1770:
1768:
1767:
1755:
1754:
1705:
1703:
1702:
1697:
1692:
1688:
1687:
1686:
1670:
1669:
1650:
1649:
1624:
1622:
1621:
1616:
1599:
1597:
1596:
1591:
1586:
1585:
1568:
1567:
1566:
1556:
1555:
1527:
1525:
1524:
1519:
1499:
1497:
1496:
1491:
1483:
1482:
1481:
1471:
1470:
1450:
1449:
1429:
1425:
1424:
1410:
1409:
1396:
1395:
1364:
1362:
1361:
1356:
1332:
1330:
1329:
1324:
1312:
1310:
1309:
1304:
1290:free variables.
1289:
1287:
1286:
1281:
1269:
1267:
1266:
1261:
1259:
1258:
1231:
1229:
1228:
1223:
1202:can be inferred.
1201:
1199:
1198:
1193:
1191:
1190:
1189:
1179:
1178:
1156:
1154:
1153:
1148:
1128:
1123:
1122:
1093:
1091:
1090:
1085:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1039:
1038:
1016:
1014:
1013:
1008:
976:
974:
973:
968:
956:
954:
953:
948:
946:
934:
932:
931:
926:
924:
912:
910:
909:
904:
886:
884:
883:
878:
876:
875:
846:
844:
843:
838:
826:
824:
823:
818:
813:
812:
797:
796:
781:
780:
761:
759:
758:
753:
748:
747:
732:
731:
716:
715:
696:
694:
693:
688:
686:
685:
669:
667:
666:
661:
641:
636:
635:
607:
605:
604:
599:
587:
585:
584:
579:
568:
567:
555:
554:
535:
533:
532:
527:
516:
515:
503:
502:
483:
481:
480:
475:
455:
450:
449:
420:
418:
417:
412:
388:
386:
385:
380:
359:
357:
356:
351:
339:
337:
336:
331:
329:
328:
303:Formal languages
291:
289:
288:
283:
268:
266:
265:
260:
242:
240:
239:
234:
232:
231:
215:
213:
212:
207:
205:
201:
200:
190:
189:
167:
165:
164:
159:
140:
138:
137:
132:
130:
129:
128:
117:
95:
93:
92:
87:
18:infinitary logic
3486:
3485:
3481:
3480:
3479:
3477:
3476:
3475:
3456:
3455:
3441:10.2307/2271099
3415:
3388:
3383:
3382:
3372:
3370:
3358:
3352:
3348:
3317:
3313:
3305:10.1.1.760.6726
3282:
3278:
3244:
3240:
3231:
3227:
3219:
3215:
3207:
3203:
3195:
3191:
3183:
3179:
3175:, pp. 1–2.
3171:
3167:
3157:
3155:
3154:on 1 March 2024
3148:
3118:Woodin, W. Hugh
3115:
3111:
3101:
3099:
3080:
3071:
3067:
3060:
3030:
3026:
3021:
3000:
2997:
2996:
2980:
2977:
2976:
2953:
2949:
2947:
2944:
2943:
2911:
2907:
2906:
2902:
2900:
2897:
2896:
2870:
2866:
2864:
2861:
2860:
2832:
2828:
2827:
2823:
2821:
2818:
2817:
2794:
2790:
2788:
2785:
2784:
2781:
2751:
2747:
2746:
2740:
2739:
2738:
2736:
2733:
2732:
2688:
2684:
2672:
2668:
2667:
2655:
2651:
2638:
2634:
2633:
2621:
2617:
2615:
2612:
2611:
2597:
2572:
2569:
2568:
2549:
2546:
2545:
2519:
2515:
2513:
2510:
2509:
2479:
2476:
2475:
2455:
2452:
2451:
2432:
2429:
2428:
2409:
2406:
2405:
2382:
2378:
2376:
2373:
2372:
2342:
2339:
2338:
2284:
2280:
2278:
2275:
2274:
2256:
2227:
2223:
2221:
2218:
2217:
2197:
2193:
2178:
2174:
2169:
2166:
2165:
2126:
2122:
2115:
2111:
2110:
2098:
2094:
2090:
2082:
2078:
2071:
2067:
2039:
2035:
2034:
2022:
2018:
2014:
2002:
1998:
1990:
1987:
1986:
1964:
1961:
1960:
1911:
1907:
1892:
1888:
1876:
1872:
1848:
1844:
1833:
1830:
1829:
1812:
1808:
1790:
1786:
1784:
1781:
1780:
1763:
1759:
1744:
1740:
1711:
1708:
1707:
1676:
1672:
1671:
1659:
1655:
1651:
1639:
1635:
1630:
1627:
1626:
1610:
1607:
1606:
1581:
1577:
1562:
1558:
1557:
1545:
1541:
1533:
1530:
1529:
1507:
1504:
1503:
1477:
1473:
1472:
1460:
1456:
1445:
1441:
1420:
1416:
1405:
1401:
1397:
1385:
1381:
1373:
1370:
1369:
1338:
1335:
1334:
1318:
1315:
1314:
1298:
1295:
1294:
1275:
1272:
1271:
1248:
1244:
1242:
1239:
1238:
1211:
1208:
1207:
1185:
1181:
1180:
1168:
1164:
1162:
1159:
1158:
1124:
1118:
1114:
1103:
1100:
1099:
1079:
1076:
1075:
1055:
1052:
1051:
1028:
1024:
1022:
1019:
1018:
1002:
999:
998:
992:
962:
959:
958:
942:
940:
937:
936:
920:
918:
915:
914:
892:
889:
888:
862:
858:
856:
853:
852:
832:
829:
828:
808:
804:
792:
788:
776:
772:
767:
764:
763:
743:
739:
727:
723:
711:
707:
702:
699:
698:
681:
677:
675:
672:
671:
637:
631:
627:
616:
613:
612:
593:
590:
589:
563:
559:
550:
546:
541:
538:
537:
511:
507:
498:
494:
489:
486:
485:
451:
445:
441:
430:
427:
426:
394:
391:
390:
368:
365:
364:
345:
342:
341:
318:
314:
312:
309:
308:
305:
297:axiom of choice
277:
274:
273:
248:
245:
244:
227:
223:
221:
218:
217:
196:
192:
191:
179:
175:
173:
170:
169:
153:
150:
149:
124:
120:
119:
107:
101:
98:
97:
81:
78:
77:
73:
12:
11:
5:
3484:
3474:
3473:
3468:
3454:
3453:
3435:(2): 226–252.
3419:
3413:
3393:Karp, Carol R.
3387:
3384:
3381:
3380:
3346:
3333:(1): 111–118.
3311:
3276:
3263:(1): 208–218.
3238:
3225:
3223:, p. 127.
3213:
3201:
3189:
3177:
3165:
3146:
3109:
3091:(4): 487–553.
3065:
3058:
3023:
3022:
3020:
3017:
3004:
2984:
2962:
2959:
2956:
2952:
2922:
2919:
2914:
2910:
2905:
2879:
2876:
2873:
2869:
2843:
2840:
2835:
2831:
2826:
2803:
2800:
2797:
2793:
2780:
2777:
2762:
2759:
2754:
2750:
2743:
2710:
2709:
2697:
2691:
2687:
2683:
2678:
2675:
2671:
2664:
2661:
2658:
2654:
2650:
2646:
2641:
2637:
2630:
2627:
2624:
2620:
2596:
2593:
2576:
2553:
2528:
2525:
2522:
2518:
2489:
2486:
2483:
2459:
2436:
2413:
2391:
2388:
2385:
2381:
2365:weakly compact
2352:
2349:
2346:
2293:
2290:
2287:
2283:
2255:
2252:
2248:well orderable
2244:
2243:
2230:
2226:
2205:
2200:
2196:
2192:
2189:
2186:
2181:
2177:
2173:
2153:
2150:
2145:
2140:
2137:
2134:
2129:
2125:
2121:
2118:
2114:
2107:
2104:
2101:
2097:
2093:
2085:
2081:
2077:
2074:
2070:
2066:
2062:
2058:
2054:
2048:
2045:
2042:
2038:
2031:
2028:
2025:
2021:
2017:
2011:
2008:
2005:
2001:
1997:
1994:
1974:
1971:
1968:
1957:
1946:
1943:
1940:
1937:
1934:
1929:
1926:
1923:
1920:
1917:
1914:
1910:
1906:
1903:
1900:
1895:
1891:
1887:
1884:
1879:
1875:
1871:
1868:
1865:
1862:
1859:
1856:
1851:
1847:
1843:
1840:
1837:
1815:
1811:
1807:
1804:
1799:
1796:
1793:
1789:
1766:
1762:
1758:
1753:
1750:
1747:
1743:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1695:
1691:
1685:
1682:
1679:
1675:
1668:
1665:
1662:
1658:
1654:
1648:
1645:
1642:
1638:
1634:
1614:
1600:
1589:
1584:
1580:
1575:
1571:
1565:
1561:
1554:
1551:
1548:
1544:
1540:
1537:
1517:
1514:
1511:
1500:
1489:
1486:
1480:
1476:
1469:
1466:
1463:
1459:
1454:
1448:
1444:
1440:
1436:
1432:
1428:
1423:
1419:
1414:
1408:
1404:
1400:
1394:
1391:
1388:
1384:
1380:
1377:
1354:
1351:
1348:
1345:
1342:
1322:
1302:
1279:
1257:
1254:
1251:
1247:
1221:
1218:
1215:
1204:
1203:
1188:
1184:
1177:
1174:
1171:
1167:
1146:
1143:
1140:
1137:
1134:
1131:
1127:
1121:
1117:
1113:
1110:
1107:
1083:
1059:
1037:
1034:
1031:
1027:
1006:
991:
988:
966:
945:
923:
902:
899:
896:
874:
871:
868:
865:
861:
849:
848:
836:
816:
811:
807:
803:
800:
795:
791:
787:
784:
779:
775:
771:
751:
746:
742:
738:
735:
730:
726:
722:
719:
714:
710:
706:
684:
680:
670:and a formula
659:
656:
653:
650:
647:
644:
640:
634:
630:
626:
623:
620:
609:
597:
577:
574:
571:
566:
562:
558:
553:
549:
545:
525:
522:
519:
514:
510:
506:
501:
497:
493:
473:
470:
467:
464:
461:
458:
454:
448:
444:
440:
437:
434:
410:
407:
404:
401:
398:
378:
375:
372:
349:
327:
324:
321:
317:
304:
301:
281:
258:
255:
252:
230:
226:
204:
199:
195:
188:
185:
182:
178:
157:
127:
123:
116:
113:
110:
106:
85:
72:
69:
49:finitary logic
9:
6:
4:
3:
2:
3483:
3472:
3469:
3467:
3464:
3463:
3461:
3450:
3446:
3442:
3438:
3434:
3430:
3429:
3424:
3420:
3416:
3410:
3406:
3402:
3398:
3394:
3390:
3389:
3368:
3364:
3357:
3350:
3341:
3336:
3332:
3328:
3327:
3322:
3315:
3306:
3301:
3296:
3291:
3287:
3280:
3271:
3266:
3262:
3258:
3257:
3252:
3248:
3242:
3235:
3232:J. L. Bell, "
3229:
3222:
3217:
3210:
3205:
3198:
3193:
3186:
3181:
3174:
3169:
3153:
3149:
3143:
3139:
3135:
3131:
3127:
3123:
3119:
3113:
3098:
3094:
3090:
3086:
3079:
3075:
3069:
3061:
3055:
3051:
3047:
3043:
3039:
3035:
3028:
3024:
3016:
3002:
2982:
2960:
2957:
2954:
2950:
2940:
2938:
2920:
2917:
2912:
2908:
2903:
2895:The logic of
2893:
2877:
2874:
2871:
2867:
2859:The logic of
2857:
2841:
2838:
2833:
2829:
2824:
2801:
2798:
2795:
2791:
2776:
2760:
2757:
2752:
2748:
2729:
2727:
2723:
2719:
2715:
2695:
2689:
2685:
2681:
2676:
2673:
2669:
2662:
2659:
2656:
2652:
2644:
2639:
2635:
2628:
2625:
2622:
2610:
2609:
2608:
2606:
2602:
2592:
2591:has a model.
2590:
2574:
2566:
2551:
2544:
2526:
2523:
2520:
2516:
2507:
2503:
2487:
2484:
2481:
2473:
2457:
2449:
2434:
2427:
2411:
2389:
2386:
2383:
2379:
2370:
2366:
2350:
2347:
2344:
2335:
2333:
2329:
2325:
2321:
2317:
2313:
2309:
2291:
2288:
2285:
2281:
2271:
2269:
2265:
2261:
2251:
2249:
2228:
2224:
2198:
2194:
2190:
2187:
2184:
2179:
2175:
2135:
2127:
2123:
2119:
2116:
2112:
2105:
2102:
2099:
2095:
2083:
2079:
2075:
2072:
2068:
2046:
2043:
2040:
2036:
2029:
2026:
2023:
2019:
2009:
2006:
2003:
1999:
1972:
1969:
1966:
1958:
1941:
1938:
1935:
1932:
1924:
1918:
1915:
1912:
1908:
1901:
1893:
1889:
1882:
1877:
1873:
1866:
1863:
1860:
1857:
1849:
1845:
1841:
1838:
1813:
1809:
1802:
1797:
1794:
1791:
1787:
1764:
1760:
1756:
1751:
1748:
1745:
1741:
1737:
1734:
1731:
1728:
1722:
1716:
1683:
1680:
1677:
1673:
1666:
1663:
1660:
1656:
1646:
1643:
1640:
1636:
1612:
1604:
1601:
1582:
1578:
1563:
1559:
1552:
1549:
1546:
1542:
1515:
1512:
1509:
1501:
1478:
1474:
1467:
1464:
1461:
1457:
1446:
1442:
1421:
1417:
1406:
1402:
1392:
1389:
1386:
1382:
1368:
1367:
1366:
1352:
1349:
1346:
1343:
1340:
1320:
1300:
1291:
1277:
1255:
1252:
1249:
1245:
1235:
1219:
1216:
1213:
1186:
1182:
1175:
1172:
1169:
1165:
1141:
1138:
1135:
1132:
1129:
1119:
1115:
1108:
1105:
1097:
1096:
1095:
1081:
1073:
1057:
1035:
1032:
1029:
1025:
1004:
997:
987:
985:
984:
978:
964:
900:
897:
894:
872:
869:
866:
863:
859:
834:
809:
805:
798:
793:
789:
782:
777:
773:
744:
740:
733:
728:
724:
717:
712:
708:
682:
678:
654:
651:
648:
645:
642:
632:
628:
621:
618:
610:
595:
572:
569:
564:
560:
556:
551:
547:
520:
517:
512:
508:
504:
499:
495:
468:
465:
462:
459:
456:
446:
442:
435:
432:
424:
423:
422:
408:
405:
402:
399:
396:
376:
373:
370:
362:
347:
325:
322:
319:
315:
300:
298:
293:
279:
270:
256:
253:
250:
228:
224:
202:
197:
193:
186:
183:
180:
155:
148:
144:
125:
121:
114:
111:
108:
104:
83:
68:
66:
62:
57:
54:
50:
46:
42:
38:
33:
31:
27:
23:
19:
3432:
3426:
3423:Barwise, Jon
3396:
3371:. Retrieved
3362:
3349:
3330:
3324:
3314:
3279:
3260:
3254:
3247:Chang, C. C.
3241:
3228:
3216:
3204:
3192:
3187:, p. 1.
3180:
3168:
3156:. Retrieved
3152:the original
3129:
3112:
3100:. Retrieved
3088:
3084:
3068:
3041:
3027:
2941:
2894:
2858:
2782:
2730:
2711:
2598:
2588:
2564:
2542:
2505:
2471:
2447:
2425:
2368:
2336:
2331:
2327:
2323:
2319:
2315:
2311:
2307:
2272:
2267:
2263:
2259:
2257:
2245:
1292:
1205:
993:
981:
979:
850:
306:
294:
271:
74:
58:
53:Hilbert-type
34:
17:
15:
3369:. p. 4
3321:"Junctions"
2337:A cardinal
147:cardinality
143:disjunction
3460:Categories
3019:References
2939:property.
2605:foundation
2601:set theory
1333:such that
1232:, forming
26:statements
3300:CiteSeerX
3295:1003.0360
3221:Karp 1964
3209:Karp 1964
3197:Karp 1964
3185:Karp 1964
3173:Karp 1964
3102:22 August
3003:α
2983:α
2961:α
2955:α
2921:ω
2909:ω
2878:ω
2872:ω
2842:ω
2830:ω
2802:ω
2796:ω
2761:ω
2749:ω
2690:γ
2682:∈
2674:γ
2663:ω
2657:γ
2653:∧
2649:¬
2640:γ
2629:ω
2623:γ
2619:∀
2575:κ
2552:⊆
2527:κ
2521:κ
2488:ω
2485:≠
2482:κ
2458:κ
2435:⊆
2412:κ
2390:κ
2384:κ
2351:ω
2348:≠
2345:κ
2322:valid in
2292:β
2286:α
2229:γ
2225:γ
2199:γ
2195:γ
2188:ϵ
2180:ϵ
2176:γ
2136:μ
2128:ϵ
2124:γ
2117:μ
2106:γ
2100:μ
2096:∧
2084:γ
2080:γ
2073:ϵ
2069:∨
2061:⟹
2047:δ
2041:μ
2030:γ
2024:δ
2020:∨
2010:γ
2004:μ
2000:∧
1973:α
1967:γ
1942:γ
1936:μ
1925:μ
1913:μ
1902:⊆
1894:ϵ
1886:¬
1878:ϵ
1864:γ
1858:ϵ
1855:∃
1850:γ
1846:γ
1842:∈
1836:∀
1814:ϵ
1806:¬
1798:δ
1792:μ
1765:ϵ
1752:δ
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1729:ϵ
1726:∃
1723:δ
1720:∀
1717:μ
1714:∀
1684:δ
1678:μ
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1661:δ
1657:∧
1647:γ
1641:μ
1637:∨
1613:γ
1583:γ
1574:⟹
1564:ϵ
1553:δ
1547:ϵ
1543:∧
1516:δ
1510:γ
1502:For each
1479:ϵ
1468:δ
1462:ϵ
1458:∧
1453:⟹
1447:δ
1435:⟹
1422:ϵ
1413:⟹
1407:δ
1393:δ
1387:ϵ
1383:∧
1353:α
1347:δ
1321:γ
1301:δ
1278:β
1256:β
1250:α
1220:α
1214:β
1187:γ
1176:δ
1170:γ
1166:∧
1142:α
1136:δ
1130:γ
1120:γ
1036:β
1030:α
965:π
901:α
898:≤
895:π
873:π
867:β
864:α
835:δ
799:⋯
786:∃
770:∃
734:⋯
721:∀
705:∀
655:β
649:δ
643:γ
633:γ
596:δ
573:⋯
570:∧
557:∧
521:⋯
518:∨
505:∨
469:α
463:δ
457:γ
447:γ
409:α
406:≤
403:β
400:≤
397:ω
371:β
348:α
326:β
320:α
280:⋯
257:δ
251:γ
229:γ
198:γ
187:δ
181:γ
177:∀
156:δ
126:γ
115:δ
109:γ
105:⋁
84:⋯
3395:(1964).
3249:(1957).
3120:(2011).
3076:(2004).
3040:(eds.).
2164:, where
1706:, where
1072:sequence
983:sentence
45:complete
3449:2271099
3386:Sources
3373:1 March
3158:1 March
361:regular
61:Ω-logic
41:compact
3447:
3411:
3302:
3144:
3056:
1828:, and
996:theory
243:where
30:proofs
3445:JSTOR
3359:(PDF)
3290:arXiv
3124:. In
3081:(PDF)
2330:from
1603:Chang
887:with
697:then
484:then
22:logic
20:is a
3409:ISBN
3375:2024
3160:2024
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3054:ISBN
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