2173:
1509:
1966:
1999:
2718:
86:
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.
1788:
1715:
1382:
1842:
1609:
1247:
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
66:
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.
2225:
1166:
493:
2168:{\displaystyle ((\land _{\mu <\gamma }{(\lor _{\delta <\gamma }{A_{\mu ,\delta }})})\implies (\lor _{\epsilon <\gamma ^{\gamma }}{(\land _{\mu <\gamma }{A_{\mu ,\gamma _{\epsilon }(\mu )})}}))}
679:
225:
150:
836:
771:
1211:
107:
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
2784:
1837:
2624:
3132:
597:
430:
545:
2269:
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
2944:
2865:
1374:
2261:. 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.
896:
2984:
2901:
2825:
2550:
2509:
2413:
2372:
991:
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
2315:
1994:
1537:
1279:
1059:
349:
278:
2252:
1241:
922:
1720:
966:
944:
252:
3266:
2573:
2456:
398:
3024:
3004:
2596:
2479:
2433:
1634:
1342:
1322:
1105:, 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:
856:
617:
369:
301:
177:
105:
62:
sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses
1299:
706:
986:
1504:{\displaystyle ((\land _{\epsilon <\delta }{(A_{\delta }\implies A_{\epsilon })})\implies (A_{\delta }\implies \land _{\epsilon <\delta }{A_{\epsilon }}))}
1639:
1103:
1079:
1026:
432:, 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:
1961:{\displaystyle \forall g\in \gamma ^{\gamma }\exists \epsilon <\gamma :\{A_{\epsilon },\neg A_{\epsilon }\}\subseteq \{A_{\mu ,g(\mu )}:\mu <\gamma \}}
1542:
3162:
3377:
2727:
can only be expressed in a logic that allows infinitely many quantifiers in an individual statement. As a consequence many theories, including
3423:
3156:
3068:
2178:
2867:. The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size.
2731:, which cannot be properly axiomatised in finitary logic, can be in a suitable infinitary logic. Other examples include the theories of
3336:
1112:
439:
625:
3476:
182:
110:
776:
711:
1171:
2713:{\displaystyle \forall _{\gamma <\omega }{V_{\gamma }:}\neg \land _{\gamma <\omega }{V_{\gamma +}\in V_{\gamma }}.\,}
3438:
2745:
993:
310:
is assumed (as is often done when discussing infinitary logic) as this is necessary to have sensible distributivity laws.
1793:
3481:
3048:
2739:. These three theories can be defined without the use of infinite quantification; only infinite junctions are needed.
1006:
550:
403:
498:
862:
The language may also have function, relation, and predicate symbols of finite arity. Karp also defined languages
2512:
2946:
fails to be compact, but it is complete (under the axioms given above). Moreover, it satisfies a variant of the
3044:
2909:
2830:
1347:
2375:
865:
1085:
of statements that obeys the following conditions: Each statement is either a logical axiom, an element of
3310:
2956:
2873:
2797:
2522:
2488:
2385:
2351:
2732:
2287:
1973:
1516:
1304:
The logical axiom schemata specific to infinitary logic are presented below. Global schemata variables:
1251:
1031:
321:
257:
32:
2230:
1220:
2723:
Unlike the axiom of foundation, this statement admits no non-standard interpretations. The concept of
1783:{\displaystyle \forall \mu \forall \delta \exists \epsilon <\gamma :A_{\mu ,\delta }=A_{\epsilon }}
901:
75:
55:
3315:
2947:
2615:
153:
51:
949:
927:
3455:
3300:
2736:
230:
40:
2558:
2441:
3419:
3366:
3152:
3064:
1244:
377:
47:
36:
3009:
2989:
2581:
2464:
2418:
1619:
1327:
1307:
841:
602:
354:
286:
162:
90:
3447:
3411:
3345:
3275:
3257:
3144:
3136:
3103:
3084:
3056:
2728:
2724:
1613:
1284:
371:
684:
1710:{\displaystyle (\lor _{\mu <\gamma }{(\land _{\delta <\gamma }{A_{\mu ,\delta }})})}
971:
307:
227:. This is meant to represent an infinite sequence of quantifiers: a quantifier for each
3244:
3128:
1088:
1064:
1011:
63:
59:
3415:
3280:
3261:
3470:
2257:
The last two axiom schemata require the axiom of choice because certain sets must be
1604:{\displaystyle ((\land _{\epsilon <\delta }{A_{\epsilon }})\implies A_{\gamma })}
17:
3148:
3433:
3133:"The Continuum Hypothesis, the generic-multiverse of sets, and the Ω Conjecture"
3060:
157:
3350:
3331:
2794:
Two infinitary logics stand out in their completeness. These are the logics of
3403:
3141:
Set Theory, Arithmetic, and
Foundations of Mathematics: Theorems, Philosophies
3043:
Moore, Gregory H. (1997). "The prehistory of infinitary logic: 1885–1955". In
2611:
2258:
1061:
is a set of sentences in the logic. A proof in infinitary logic from a theory
71:
3107:
3247:". Stanford Encyclopedia of Philosophy, revised 2023. Accessed 26 July 2024.
46:
Some infinitary logics may have different properties from those of standard
1082:
3088:
3459:
2345:. An infinitary logic can be complete without being strongly complete.
2220:{\displaystyle \{\gamma _{\epsilon }:\epsilon <\gamma ^{\gamma }\}}
946:
that allow for function and predicate symbols of infinite arity, with
3451:
838:
are formulae. (In each case the sequence of quantifiers has length
3305:
3296:
58:. Notions of compactness and completeness that are equivalent in
3006:
is strongly compact (because proofs in these logics cannot use
1161:{\displaystyle A=\{A_{\gamma }|\gamma <\delta <\alpha \}}
924:
an infinite cardinal and some more complicated restrictions on
488:{\displaystyle A=\{A_{\gamma }|\gamma <\delta <\alpha \}}
179:. The same notation may be applied to quantifiers, for example
1168:
that have occurred previously in the proof then the statement
674:{\displaystyle V=\{V_{\gamma }|\gamma <\delta <\beta \}}
2752:
220:{\displaystyle \forall _{\gamma <\delta }{V_{\gamma }:}}
2742:
Truth predicates for countable languages are definable in
145:{\displaystyle \bigvee _{\gamma <\delta }{A_{\gamma }}}
2986:
is strongly complete (under the axioms given above) then
2903:
is also strongly complete, compact and strongly compact.
831:{\displaystyle \exists V_{0}:\exists V_{1}\cdots (A_{0})}
766:{\displaystyle \forall V_{0}:\forall V_{1}\cdots (A_{0})}
1206:{\displaystyle \land _{\gamma <\delta }{A_{\gamma }}}
968:
controlling the maximum arity of a function symbol and
3262:"On the representation of α-complete Boolean algebras"
43:. The concept was introduced by Zermelo in the 1930s.
3055:. Springer-Science+Business Media. pp. 105–123.
3012:
2992:
2959:
2912:
2876:
2833:
2800:
2748:
2627:
2584:
2561:
2525:
2491:
2467:
2444:
2421:
2388:
2354:
2290:
2233:
2181:
2002:
1976:
1845:
1796:
1723:
1642:
1622:
1545:
1519:
1385:
1350:
1330:
1310:
1287:
1254:
1223:
1174:
1115:
1091:
1067:
1034:
1014:
974:
952:
930:
904:
868:
844:
779:
714:
687:
628:
605:
553:
501:
442:
406:
380:
357:
324:
289:
260:
233:
185:
165:
113:
93:
70:
Considering whether a certain infinitary logic named
2779:{\displaystyle {\mathcal {L}}_{\omega _{1},\omega }}
599:
are formulae. (In each case the sequence has length
1832:{\displaystyle A_{\mu ,\delta }=\neg A_{\epsilon }}
3018:
2998:
2978:
2938:
2895:
2859:
2819:
2778:
2712:
2590:
2567:
2544:
2503:
2473:
2450:
2427:
2407:
2366:
2309:
2265:Completeness, compactness, and strong completeness
2246:
2219:
2167:
1988:
1960:
1831:
1782:
1709:
1628:
1603:
1531:
1503:
1368:
1336:
1316:
1293:
1273:
1235:
1205:
1160:
1097:
1073:
1053:
1020:
980:
960:
938:
916:
890:
850:
830:
765:
700:
673:
611:
591:
539:
487:
424:
392:
363:
343:
295:
272:
246:
219:
171:
144:
99:
50:. In particular, infinitary logics may fail to be
3267:Transactions of the American Mathematical Society
3436:(1969). "Infinitary logic and admissible sets".
592:{\displaystyle (A_{0}\land A_{1}\land \cdots )}
3367:"Inexpressible longing for the intended model"
3143:. Cambridge University Press. pp. 13–42.
2273:a sentence is said to be valid for the theory
425:{\displaystyle \omega \leq \beta \leq \alpha }
3408:Languages with Expressions of Infinite Length
2321:valid in every model there exists a proof of
540:{\displaystyle (A_{0}\lor A_{1}\lor \cdots )}
303:are not part of formal infinitary languages.
8:
3378:Uniwersytet im. Adama Mickiewicza w Poznaniu
3297:"Four departures in Mathematics and Physics"
2325:. It is strongly complete if for any theory
2214:
2182:
1955:
1915:
1909:
1880:
1155:
1122:
1001:Definition of Hilbert-type infinitary logics
668:
635:
482:
449:
74:is complete promises to throw light on the
2074:
2070:
1587:
1583:
1466:
1462:
1448:
1444:
1426:
1422:
1281:forbidding formulas from having more than
82:A word on notation and the axiom of choice
3349:
3314:
3304:
3279:
3011:
2991:
2964:
2958:
2922:
2917:
2911:
2881:
2875:
2843:
2838:
2832:
2805:
2799:
2762:
2757:
2751:
2750:
2747:
2709:
2699:
2683:
2678:
2666:
2649:
2644:
2632:
2626:
2583:
2560:
2530:
2524:
2490:
2466:
2443:
2420:
2393:
2387:
2353:
2295:
2289:
2238:
2232:
2208:
2189:
2180:
2137:
2126:
2121:
2109:
2101:
2093:
2082:
2050:
2045:
2033:
2025:
2013:
2001:
1975:
1922:
1903:
1887:
1859:
1844:
1823:
1801:
1795:
1774:
1755:
1722:
1687:
1682:
1670:
1662:
1650:
1641:
1621:
1592:
1573:
1568:
1556:
1544:
1518:
1488:
1483:
1471:
1456:
1431:
1416:
1408:
1396:
1384:
1349:
1329:
1309:
1286:
1259:
1253:
1222:
1196:
1191:
1179:
1173:
1135:
1129:
1114:
1090:
1066:
1039:
1033:
1013:
973:
953:
951:
931:
929:
903:
873:
867:
843:
819:
803:
787:
778:
754:
738:
722:
713:
692:
686:
648:
642:
627:
604:
574:
561:
552:
522:
509:
500:
462:
456:
441:
405:
379:
356:
329:
323:
288:
259:
238:
232:
207:
202:
190:
184:
164:
135:
130:
118:
112:
92:
2606:Concepts expressible in infinitary logic
2552:, without restriction on size, if every
3035:
2939:{\displaystyle L_{\omega _{1},\omega }}
2860:{\displaystyle L_{\omega _{1},\omega }}
1369:{\displaystyle 0<\delta <\alpha }
891:{\displaystyle L_{\alpha \beta o\pi }}
7:
3410:. North-Holland Publishing Company.
3231:
3219:
3207:
3195:
3183:
2979:{\displaystyle L_{\alpha ,\alpha }}
2896:{\displaystyle L_{\omega ,\omega }}
2820:{\displaystyle L_{\omega ,\omega }}
2545:{\displaystyle L_{\kappa ,\kappa }}
2504:{\displaystyle \kappa \neq \omega }
2408:{\displaystyle L_{\kappa ,\kappa }}
2367:{\displaystyle \kappa \neq \omega }
3365:Pogonowski, Jerzy (10 June 2010).
3337:Notre Dame Journal of Formal Logic
2659:
2629:
2614:the following statement expresses
2317:is complete if for every sentence
2310:{\displaystyle L_{\alpha ,\beta }}
1989:{\displaystyle \gamma <\alpha }
1896:
1865:
1846:
1816:
1736:
1730:
1724:
1532:{\displaystyle \gamma <\delta }
1274:{\displaystyle L_{\alpha ,\beta }}
1054:{\displaystyle L_{\alpha ,\beta }}
954:
932:
796:
780:
731:
715:
344:{\displaystyle L_{\alpha ,\beta }}
318:A first-order infinitary language
273:{\displaystyle \gamma <\delta }
187:
25:
3281:10.1090/S0002-9947-1957-0086792-1
3047:; Doets, Kees; Mundici, Daniele;
2247:{\displaystyle \gamma ^{\gamma }}
1616:'s distributivity laws (for each
1236:{\displaystyle \beta <\alpha }
152:are used to indicate an infinite
917:{\displaystyle \pi \leq \alpha }
3053:Structures and Norms in Science
2277:if it is true in all models of
988:controlling predicate symbols.
3096:The Bulletin of Symbolic Logic
3026:or more of the given axioms).
2162:
2159:
2154:
2149:
2143:
2102:
2075:
2071:
2067:
2063:
2026:
2006:
2003:
1938:
1932:
1704:
1700:
1663:
1643:
1598:
1584:
1580:
1549:
1546:
1498:
1495:
1463:
1449:
1445:
1441:
1437:
1423:
1409:
1389:
1386:
1136:
825:
812:
760:
747:
649:
586:
554:
534:
502:
463:
1:
3439:The Journal of Symbolic Logic
3416:10.1016/S0049-237X(08)70423-3
3295:Rosinger, Elemer E. (2010).
3149:10.1017/CBO9780511910616.003
961:{\displaystyle \mathrm {o} }
939:{\displaystyle \mathrm {o} }
35:that allows infinitely long
3061:10.1007/978-94-017-0538-7_7
247:{\displaystyle V_{\gamma }}
3498:
3330:Bennett, David W. (1980).
2790:Complete infinitary logics
2568:{\displaystyle \subseteq }
2451:{\displaystyle \subseteq }
1109:Given a set of statements
283:All usage of suffixes and
156:over a set of formulae of
3045:Dalla Chiara, Maria Luisa
2578:of cardinality less than
2461:of cardinality less than
1081:is a (possibly infinite)
622:Given a set of variables
3374:Zakład Logiki Stosowanej
3351:10.1305/ndjfl/1093882943
3139:; Kossak, Roman (eds.).
3089:"Zermelo and set theory"
2485:has a model. A cardinal
2435:many formulas, if every
2284:A logic in the language
436:Given a set of formulae
393:{\displaystyle \beta =0}
3477:Systems of formal logic
3019:{\displaystyle \alpha }
2999:{\displaystyle \alpha }
2591:{\displaystyle \kappa }
2474:{\displaystyle \kappa }
2428:{\displaystyle \kappa }
1629:{\displaystyle \gamma }
1337:{\displaystyle \gamma }
1317:{\displaystyle \delta }
1028:in infinitary language
851:{\displaystyle \delta }
612:{\displaystyle \delta }
364:{\displaystyle \alpha }
296:{\displaystyle \cdots }
172:{\displaystyle \delta }
100:{\displaystyle \cdots }
39:and/or infinitely long
3108:10.2178/bsl/1102083759
3020:
3000:
2980:
2940:
2897:
2861:
2821:
2780:
2733:non-archimedean fields
2714:
2592:
2569:
2546:
2515:when for every theory
2505:
2475:
2452:
2429:
2409:
2378:when for every theory
2368:
2311:
2248:
2227:is a well ordering of
2221:
2169:
1990:
1962:
1833:
1784:
1711:
1630:
1605:
1533:
1505:
1370:
1338:
1318:
1295:
1294:{\displaystyle \beta }
1275:
1237:
1207:
1162:
1099:
1075:
1055:
1022:
982:
962:
940:
918:
892:
852:
832:
767:
702:
675:
613:
593:
541:
489:
426:
394:
365:
345:
297:
274:
248:
221:
173:
146:
101:
3021:
3001:
2981:
2941:
2898:
2862:
2822:
2781:
2715:
2593:
2570:
2547:
2506:
2476:
2453:
2430:
2410:
2369:
2312:
2249:
2222:
2170:
1991:
1963:
1834:
1785:
1712:
1631:
1606:
1534:
1506:
1371:
1339:
1319:
1296:
1276:
1238:
1208:
1163:
1100:
1076:
1056:
1023:
983:
963:
941:
919:
893:
853:
833:
768:
703:
701:{\displaystyle A_{0}}
676:
614:
594:
542:
490:
427:
395:
366:
346:
298:
275:
249:
222:
174:
147:
102:
3010:
2990:
2957:
2910:
2874:
2831:
2798:
2746:
2625:
2582:
2559:
2523:
2489:
2465:
2442:
2419:
2386:
2352:
2337:there is a proof of
2288:
2231:
2179:
2000:
1974:
1843:
1794:
1721:
1640:
1620:
1543:
1517:
1383:
1348:
1328:
1308:
1285:
1252:
1221:
1172:
1113:
1089:
1065:
1032:
1012:
981:{\displaystyle \pi }
972:
950:
928:
902:
866:
842:
777:
712:
685:
626:
603:
551:
499:
440:
404:
378:
355:
322:
287:
258:
231:
183:
163:
111:
91:
76:continuum hypothesis
3482:Non-classical logic
3210:, pp. 101–102.
2948:Craig interpolation
2737:torsion-free groups
2610:In the language of
2415:containing at most
2329:for every sentence
18:Infinitary language
3049:van Benthem, Johan
3016:
2996:
2976:
2936:
2893:
2857:
2817:
2776:
2710:
2598:has a model, then
2588:
2565:
2542:
2501:
2481:has a model, then
2471:
2448:
2425:
2405:
2364:
2307:
2244:
2217:
2165:
1986:
1958:
1829:
1780:
1707:
1626:
1601:
1529:
1501:
1366:
1334:
1314:
1291:
1271:
1245:universal closures
1233:
1203:
1158:
1095:
1071:
1051:
1018:
978:
958:
936:
914:
888:
848:
828:
763:
698:
671:
609:
589:
537:
485:
422:
390:
361:
341:
293:
270:
244:
217:
169:
142:
129:
97:
3425:978-0-444-53401-9
3222:, pp. 39–54.
3158:978-0-511-91061-6
3137:Kennedy, Juliette
3085:Kanamori, Akihiro
3070:978-94-017-0538-7
1098:{\displaystyle T}
1074:{\displaystyle T}
1021:{\displaystyle T}
114:
48:first-order logic
16:(Redirected from
3489:
3463:
3429:
3390:
3389:
3387:
3385:
3371:
3362:
3356:
3355:
3353:
3327:
3321:
3320:
3318:
3308:
3292:
3286:
3285:
3283:
3254:
3248:
3245:Infinitary Logic
3241:
3235:
3229:
3223:
3217:
3211:
3205:
3199:
3193:
3187:
3181:
3175:
3174:
3172:
3170:
3161:. Archived from
3125:
3119:
3118:
3116:
3114:
3093:
3081:
3075:
3074:
3040:
3025:
3023:
3022:
3017:
3005:
3003:
3002:
2997:
2985:
2983:
2982:
2977:
2975:
2974:
2953:If the logic of
2945:
2943:
2942:
2937:
2935:
2934:
2927:
2926:
2902:
2900:
2899:
2894:
2892:
2891:
2866:
2864:
2863:
2858:
2856:
2855:
2848:
2847:
2826:
2824:
2823:
2818:
2816:
2815:
2785:
2783:
2782:
2777:
2775:
2774:
2767:
2766:
2756:
2755:
2729:Peano arithmetic
2725:well-foundedness
2719:
2717:
2716:
2711:
2705:
2704:
2703:
2691:
2690:
2677:
2676:
2658:
2654:
2653:
2643:
2642:
2597:
2595:
2594:
2589:
2574:
2572:
2571:
2566:
2551:
2549:
2548:
2543:
2541:
2540:
2513:strongly compact
2510:
2508:
2507:
2502:
2480:
2478:
2477:
2472:
2457:
2455:
2454:
2449:
2434:
2432:
2431:
2426:
2414:
2412:
2411:
2406:
2404:
2403:
2373:
2371:
2370:
2365:
2316:
2314:
2313:
2308:
2306:
2305:
2253:
2251:
2250:
2245:
2243:
2242:
2226:
2224:
2223:
2218:
2213:
2212:
2194:
2193:
2174:
2172:
2171:
2166:
2158:
2157:
2153:
2152:
2142:
2141:
2120:
2119:
2100:
2099:
2098:
2097:
2066:
2062:
2061:
2060:
2044:
2043:
2024:
2023:
1995:
1993:
1992:
1987:
1967:
1965:
1964:
1959:
1942:
1941:
1908:
1907:
1892:
1891:
1864:
1863:
1838:
1836:
1835:
1830:
1828:
1827:
1812:
1811:
1789:
1787:
1786:
1781:
1779:
1778:
1766:
1765:
1716:
1714:
1713:
1708:
1703:
1699:
1698:
1697:
1681:
1680:
1661:
1660:
1635:
1633:
1632:
1627:
1610:
1608:
1607:
1602:
1597:
1596:
1579:
1578:
1577:
1567:
1566:
1538:
1536:
1535:
1530:
1510:
1508:
1507:
1502:
1494:
1493:
1492:
1482:
1481:
1461:
1460:
1440:
1436:
1435:
1421:
1420:
1407:
1406:
1375:
1373:
1372:
1367:
1343:
1341:
1340:
1335:
1323:
1321:
1320:
1315:
1301:free variables.
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1270:
1269:
1242:
1240:
1239:
1234:
1213:can be inferred.
1212:
1210:
1209:
1204:
1202:
1201:
1200:
1190:
1189:
1167:
1165:
1164:
1159:
1139:
1134:
1133:
1104:
1102:
1101:
1096:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1050:
1049:
1027:
1025:
1024:
1019:
987:
985:
984:
979:
967:
965:
964:
959:
957:
945:
943:
942:
937:
935:
923:
921:
920:
915:
897:
895:
894:
889:
887:
886:
857:
855:
854:
849:
837:
835:
834:
829:
824:
823:
808:
807:
792:
791:
772:
770:
769:
764:
759:
758:
743:
742:
727:
726:
707:
705:
704:
699:
697:
696:
680:
678:
677:
672:
652:
647:
646:
618:
616:
615:
610:
598:
596:
595:
590:
579:
578:
566:
565:
546:
544:
543:
538:
527:
526:
514:
513:
494:
492:
491:
486:
466:
461:
460:
431:
429:
428:
423:
399:
397:
396:
391:
370:
368:
367:
362:
350:
348:
347:
342:
340:
339:
314:Formal languages
302:
300:
299:
294:
279:
277:
276:
271:
253:
251:
250:
245:
243:
242:
226:
224:
223:
218:
216:
212:
211:
201:
200:
178:
176:
175:
170:
151:
149:
148:
143:
141:
140:
139:
128:
106:
104:
103:
98:
29:infinitary logic
21:
3497:
3496:
3492:
3491:
3490:
3488:
3487:
3486:
3467:
3466:
3452:10.2307/2271099
3432:
3426:
3402:
3399:
3394:
3393:
3383:
3381:
3369:
3364:
3363:
3359:
3329:
3328:
3324:
3316:10.1.1.760.6726
3294:
3293:
3289:
3256:
3255:
3251:
3242:
3238:
3230:
3226:
3218:
3214:
3206:
3202:
3194:
3190:
3186:, pp. 1–2.
3182:
3178:
3168:
3166:
3165:on 1 March 2024
3159:
3129:Woodin, W. Hugh
3127:
3126:
3122:
3112:
3110:
3091:
3083:
3082:
3078:
3071:
3042:
3041:
3037:
3032:
3008:
3007:
2988:
2987:
2960:
2955:
2954:
2918:
2913:
2908:
2907:
2877:
2872:
2871:
2839:
2834:
2829:
2828:
2801:
2796:
2795:
2792:
2758:
2749:
2744:
2743:
2695:
2679:
2662:
2645:
2628:
2623:
2622:
2608:
2580:
2579:
2557:
2556:
2526:
2521:
2520:
2487:
2486:
2463:
2462:
2440:
2439:
2417:
2416:
2389:
2384:
2383:
2350:
2349:
2291:
2286:
2285:
2267:
2234:
2229:
2228:
2204:
2185:
2177:
2176:
2133:
2122:
2105:
2089:
2078:
2046:
2029:
2009:
1998:
1997:
1972:
1971:
1918:
1899:
1883:
1855:
1841:
1840:
1819:
1797:
1792:
1791:
1770:
1751:
1719:
1718:
1683:
1666:
1646:
1638:
1637:
1618:
1617:
1588:
1569:
1552:
1541:
1540:
1515:
1514:
1484:
1467:
1452:
1427:
1412:
1392:
1381:
1380:
1346:
1345:
1326:
1325:
1306:
1305:
1283:
1282:
1255:
1250:
1249:
1219:
1218:
1192:
1175:
1170:
1169:
1125:
1111:
1110:
1087:
1086:
1063:
1062:
1035:
1030:
1029:
1010:
1009:
1003:
970:
969:
948:
947:
926:
925:
900:
899:
869:
864:
863:
840:
839:
815:
799:
783:
775:
774:
750:
734:
718:
710:
709:
688:
683:
682:
638:
624:
623:
601:
600:
570:
557:
549:
548:
518:
505:
497:
496:
452:
438:
437:
402:
401:
376:
375:
353:
352:
325:
320:
319:
316:
308:axiom of choice
285:
284:
256:
255:
234:
229:
228:
203:
186:
181:
180:
161:
160:
131:
109:
108:
89:
88:
84:
23:
22:
15:
12:
11:
5:
3495:
3493:
3485:
3484:
3479:
3469:
3468:
3465:
3464:
3446:(2): 226–252.
3430:
3424:
3404:Karp, Carol R.
3398:
3395:
3392:
3391:
3357:
3344:(1): 111–118.
3322:
3287:
3274:(1): 208–218.
3249:
3236:
3234:, p. 127.
3224:
3212:
3200:
3188:
3176:
3157:
3120:
3102:(4): 487–553.
3076:
3069:
3034:
3033:
3031:
3028:
3015:
2995:
2973:
2970:
2967:
2963:
2933:
2930:
2925:
2921:
2916:
2890:
2887:
2884:
2880:
2854:
2851:
2846:
2842:
2837:
2814:
2811:
2808:
2804:
2791:
2788:
2773:
2770:
2765:
2761:
2754:
2721:
2720:
2708:
2702:
2698:
2694:
2689:
2686:
2682:
2675:
2672:
2669:
2665:
2661:
2657:
2652:
2648:
2641:
2638:
2635:
2631:
2607:
2604:
2587:
2564:
2539:
2536:
2533:
2529:
2500:
2497:
2494:
2470:
2447:
2424:
2402:
2399:
2396:
2392:
2376:weakly compact
2363:
2360:
2357:
2304:
2301:
2298:
2294:
2266:
2263:
2259:well orderable
2255:
2254:
2241:
2237:
2216:
2211:
2207:
2203:
2200:
2197:
2192:
2188:
2184:
2164:
2161:
2156:
2151:
2148:
2145:
2140:
2136:
2132:
2129:
2125:
2118:
2115:
2112:
2108:
2104:
2096:
2092:
2088:
2085:
2081:
2077:
2073:
2069:
2065:
2059:
2056:
2053:
2049:
2042:
2039:
2036:
2032:
2028:
2022:
2019:
2016:
2012:
2008:
2005:
1985:
1982:
1979:
1968:
1957:
1954:
1951:
1948:
1945:
1940:
1937:
1934:
1931:
1928:
1925:
1921:
1917:
1914:
1911:
1906:
1902:
1898:
1895:
1890:
1886:
1882:
1879:
1876:
1873:
1870:
1867:
1862:
1858:
1854:
1851:
1848:
1826:
1822:
1818:
1815:
1810:
1807:
1804:
1800:
1777:
1773:
1769:
1764:
1761:
1758:
1754:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1706:
1702:
1696:
1693:
1690:
1686:
1679:
1676:
1673:
1669:
1665:
1659:
1656:
1653:
1649:
1645:
1625:
1611:
1600:
1595:
1591:
1586:
1582:
1576:
1572:
1565:
1562:
1559:
1555:
1551:
1548:
1528:
1525:
1522:
1511:
1500:
1497:
1491:
1487:
1480:
1477:
1474:
1470:
1465:
1459:
1455:
1451:
1447:
1443:
1439:
1434:
1430:
1425:
1419:
1415:
1411:
1405:
1402:
1399:
1395:
1391:
1388:
1365:
1362:
1359:
1356:
1353:
1333:
1313:
1290:
1268:
1265:
1262:
1258:
1232:
1229:
1226:
1215:
1214:
1199:
1195:
1188:
1185:
1182:
1178:
1157:
1154:
1151:
1148:
1145:
1142:
1138:
1132:
1128:
1124:
1121:
1118:
1094:
1070:
1048:
1045:
1042:
1038:
1017:
1002:
999:
977:
956:
934:
913:
910:
907:
885:
882:
879:
876:
872:
860:
859:
847:
827:
822:
818:
814:
811:
806:
802:
798:
795:
790:
786:
782:
762:
757:
753:
749:
746:
741:
737:
733:
730:
725:
721:
717:
695:
691:
681:and a formula
670:
667:
664:
661:
658:
655:
651:
645:
641:
637:
634:
631:
620:
608:
588:
585:
582:
577:
573:
569:
564:
560:
556:
536:
533:
530:
525:
521:
517:
512:
508:
504:
484:
481:
478:
475:
472:
469:
465:
459:
455:
451:
448:
445:
421:
418:
415:
412:
409:
389:
386:
383:
360:
338:
335:
332:
328:
315:
312:
292:
269:
266:
263:
241:
237:
215:
210:
206:
199:
196:
193:
189:
168:
138:
134:
127:
124:
121:
117:
96:
83:
80:
60:finitary logic
24:
14:
13:
10:
9:
6:
4:
3:
2:
3494:
3483:
3480:
3478:
3475:
3474:
3472:
3461:
3457:
3453:
3449:
3445:
3441:
3440:
3435:
3431:
3427:
3421:
3417:
3413:
3409:
3405:
3401:
3400:
3396:
3379:
3375:
3368:
3361:
3358:
3352:
3347:
3343:
3339:
3338:
3333:
3326:
3323:
3317:
3312:
3307:
3302:
3298:
3291:
3288:
3282:
3277:
3273:
3269:
3268:
3263:
3259:
3253:
3250:
3246:
3243:J. L. Bell, "
3240:
3237:
3233:
3228:
3225:
3221:
3216:
3213:
3209:
3204:
3201:
3197:
3192:
3189:
3185:
3180:
3177:
3164:
3160:
3154:
3150:
3146:
3142:
3138:
3134:
3130:
3124:
3121:
3109:
3105:
3101:
3097:
3090:
3086:
3080:
3077:
3072:
3066:
3062:
3058:
3054:
3050:
3046:
3039:
3036:
3029:
3027:
3013:
2993:
2971:
2968:
2965:
2961:
2951:
2949:
2931:
2928:
2923:
2919:
2914:
2906:The logic of
2904:
2888:
2885:
2882:
2878:
2870:The logic of
2868:
2852:
2849:
2844:
2840:
2835:
2812:
2809:
2806:
2802:
2789:
2787:
2771:
2768:
2763:
2759:
2740:
2738:
2734:
2730:
2726:
2706:
2700:
2696:
2692:
2687:
2684:
2680:
2673:
2670:
2667:
2663:
2655:
2650:
2646:
2639:
2636:
2633:
2621:
2620:
2619:
2617:
2613:
2605:
2603:
2602:has a model.
2601:
2585:
2577:
2562:
2555:
2537:
2534:
2531:
2527:
2518:
2514:
2498:
2495:
2492:
2484:
2468:
2460:
2445:
2438:
2422:
2400:
2397:
2394:
2390:
2381:
2377:
2361:
2358:
2355:
2346:
2344:
2340:
2336:
2332:
2328:
2324:
2320:
2302:
2299:
2296:
2292:
2282:
2280:
2276:
2272:
2264:
2262:
2260:
2239:
2235:
2209:
2205:
2201:
2198:
2195:
2190:
2186:
2146:
2138:
2134:
2130:
2127:
2123:
2116:
2113:
2110:
2106:
2094:
2090:
2086:
2083:
2079:
2057:
2054:
2051:
2047:
2040:
2037:
2034:
2030:
2020:
2017:
2014:
2010:
1983:
1980:
1977:
1969:
1952:
1949:
1946:
1943:
1935:
1929:
1926:
1923:
1919:
1912:
1904:
1900:
1893:
1888:
1884:
1877:
1874:
1871:
1868:
1860:
1856:
1852:
1849:
1824:
1820:
1813:
1808:
1805:
1802:
1798:
1775:
1771:
1767:
1762:
1759:
1756:
1752:
1748:
1745:
1742:
1739:
1733:
1727:
1694:
1691:
1688:
1684:
1677:
1674:
1671:
1667:
1657:
1654:
1651:
1647:
1623:
1615:
1612:
1593:
1589:
1574:
1570:
1563:
1560:
1557:
1553:
1526:
1523:
1520:
1512:
1489:
1485:
1478:
1475:
1472:
1468:
1457:
1453:
1432:
1428:
1417:
1413:
1403:
1400:
1397:
1393:
1379:
1378:
1377:
1363:
1360:
1357:
1354:
1351:
1331:
1311:
1302:
1288:
1266:
1263:
1260:
1256:
1246:
1230:
1227:
1224:
1197:
1193:
1186:
1183:
1180:
1176:
1152:
1149:
1146:
1143:
1140:
1130:
1126:
1119:
1116:
1108:
1107:
1106:
1092:
1084:
1068:
1046:
1043:
1040:
1036:
1015:
1008:
1000:
998:
996:
995:
989:
975:
911:
908:
905:
883:
880:
877:
874:
870:
845:
820:
816:
809:
804:
800:
793:
788:
784:
755:
751:
744:
739:
735:
728:
723:
719:
693:
689:
665:
662:
659:
656:
653:
643:
639:
632:
629:
621:
606:
583:
580:
575:
571:
567:
562:
558:
531:
528:
523:
519:
515:
510:
506:
479:
476:
473:
470:
467:
457:
453:
446:
443:
435:
434:
433:
419:
416:
413:
410:
407:
387:
384:
381:
373:
358:
336:
333:
330:
326:
313:
311:
309:
304:
290:
281:
267:
264:
261:
239:
235:
213:
208:
204:
197:
194:
191:
166:
159:
155:
136:
132:
125:
122:
119:
115:
94:
81:
79:
77:
73:
68:
65:
61:
57:
53:
49:
44:
42:
38:
34:
30:
19:
3443:
3437:
3434:Barwise, Jon
3407:
3382:. Retrieved
3373:
3360:
3341:
3335:
3325:
3290:
3271:
3265:
3258:Chang, C. C.
3252:
3239:
3227:
3215:
3203:
3198:, p. 1.
3191:
3179:
3167:. Retrieved
3163:the original
3140:
3123:
3111:. Retrieved
3099:
3095:
3079:
3052:
3038:
2952:
2905:
2869:
2793:
2741:
2722:
2609:
2599:
2575:
2553:
2516:
2482:
2458:
2436:
2379:
2347:
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2283:
2278:
2274:
2270:
2268:
2256:
1303:
1216:
1004:
992:
990:
861:
317:
305:
282:
85:
69:
64:Hilbert-type
45:
28:
26:
3380:. p. 4
3332:"Junctions"
2348:A cardinal
158:cardinality
154:disjunction
3471:Categories
3030:References
2950:property.
2616:foundation
2612:set theory
1344:such that
1243:, forming
37:statements
3311:CiteSeerX
3306:1003.0360
3232:Karp 1964
3220:Karp 1964
3208:Karp 1964
3196:Karp 1964
3184:Karp 1964
3113:22 August
3014:α
2994:α
2972:α
2966:α
2932:ω
2920:ω
2889:ω
2883:ω
2853:ω
2841:ω
2813:ω
2807:ω
2772:ω
2760:ω
2701:γ
2693:∈
2685:γ
2674:ω
2668:γ
2664:∧
2660:¬
2651:γ
2640:ω
2634:γ
2630:∀
2586:κ
2563:⊆
2538:κ
2532:κ
2499:ω
2496:≠
2493:κ
2469:κ
2446:⊆
2423:κ
2401:κ
2395:κ
2362:ω
2359:≠
2356:κ
2333:valid in
2303:β
2297:α
2240:γ
2236:γ
2210:γ
2206:γ
2199:ϵ
2191:ϵ
2187:γ
2147:μ
2139:ϵ
2135:γ
2128:μ
2117:γ
2111:μ
2107:∧
2095:γ
2091:γ
2084:ϵ
2080:∨
2072:⟹
2058:δ
2052:μ
2041:γ
2035:δ
2031:∨
2021:γ
2015:μ
2011:∧
1984:α
1978:γ
1953:γ
1947:μ
1936:μ
1924:μ
1913:⊆
1905:ϵ
1897:¬
1889:ϵ
1875:γ
1869:ϵ
1866:∃
1861:γ
1857:γ
1853:∈
1847:∀
1825:ϵ
1817:¬
1809:δ
1803:μ
1776:ϵ
1763:δ
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1740:ϵ
1737:∃
1734:δ
1731:∀
1728:μ
1725:∀
1695:δ
1689:μ
1678:γ
1672:δ
1668:∧
1658:γ
1652:μ
1648:∨
1624:γ
1594:γ
1585:⟹
1575:ϵ
1564:δ
1558:ϵ
1554:∧
1527:δ
1521:γ
1513:For each
1490:ϵ
1479:δ
1473:ϵ
1469:∧
1464:⟹
1458:δ
1446:⟹
1433:ϵ
1424:⟹
1418:δ
1404:δ
1398:ϵ
1394:∧
1364:α
1358:δ
1332:γ
1312:δ
1289:β
1267:β
1261:α
1231:α
1225:β
1198:γ
1187:δ
1181:γ
1177:∧
1153:α
1147:δ
1141:γ
1131:γ
1047:β
1041:α
976:π
912:α
909:≤
906:π
884:π
878:β
875:α
846:δ
810:⋯
797:∃
781:∃
745:⋯
732:∀
716:∀
666:β
660:δ
654:γ
644:γ
607:δ
584:⋯
581:∧
568:∧
532:⋯
529:∨
516:∨
480:α
474:δ
468:γ
458:γ
420:α
417:≤
414:β
411:≤
408:ω
382:β
359:α
337:β
331:α
291:⋯
268:δ
262:γ
240:γ
209:γ
198:δ
192:γ
188:∀
167:δ
137:γ
126:δ
120:γ
116:⋁
95:⋯
3406:(1964).
3260:(1957).
3131:(2011).
3087:(2004).
3051:(eds.).
2175:, where
1717:, where
1083:sequence
994:sentence
56:complete
3460:2271099
3397:Sources
3384:1 March
3169:1 March
372:regular
72:Ω-logic
52:compact
3458:
3422:
3313:
3155:
3067:
1839:, and
1007:theory
254:where
41:proofs
3456:JSTOR
3370:(PDF)
3301:arXiv
3135:. In
3092:(PDF)
2341:from
1614:Chang
898:with
708:then
495:then
33:logic
31:is a
3420:ISBN
3386:2024
3171:2024
3153:ISBN
3115:2023
3065:ISBN
2827:and
2735:and
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