3022:
22:
328:
is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal
266:
1233:
1112:
1066:
786:
727:
465:
956:
813:
754:
687:
660:
633:
606:
546:
408:
381:
1401:
574:
1030:
983:
929:
902:
875:
840:
519:
492:
2076:
39:
428:
354:
298:, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via â€
320:
has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (
1003:
159:
2159:
1300:
2473:
86:
2631:
1181:
1419:
58:
2486:
1809:
2071:
65:
2491:
2481:
2218:
1424:
1969:
1415:
2627:
105:
2724:
2468:
1293:
72:
2029:
1722:
1463:
1071:
2985:
2687:
2450:
2445:
2270:
1691:
1375:
43:
54:
2980:
2763:
2680:
2393:
2324:
2201:
1443:
1250:. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.
549:
2051:
2905:
2731:
2417:
1650:
2056:
2388:
2127:
1385:
1286:
287:
2783:
2778:
2712:
2302:
1696:
1664:
1355:
1429:
32:
3046:
3002:
2951:
2848:
2346:
2307:
1784:
1038:
2843:
1458:
3051:
2773:
2312:
2164:
2147:
1870:
1350:
316:, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its
79:
2675:
2652:
2613:
2499:
2440:
2086:
2006:
1850:
1794:
1407:
759:
700:
2965:
2692:
2670:
2637:
2530:
2376:
2361:
2334:
2285:
2169:
2104:
1929:
1895:
1890:
1764:
1595:
1572:
437:
2895:
2748:
2540:
2258:
1994:
1900:
1759:
1744:
1625:
1600:
295:
3021:
934:
791:
732:
665:
638:
611:
584:
524:
386:
359:
2868:
2830:
2707:
2511:
2351:
2275:
2253:
2081:
2039:
1938:
1905:
1769:
1557:
1468:
2997:
2888:
2873:
2853:
2810:
2697:
2647:
2573:
2518:
2455:
2248:
2243:
2191:
1959:
1948:
1620:
1520:
1448:
1439:
1435:
1370:
1365:
850:
of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and
552:
126:
3026:
2795:
2758:
2743:
2736:
2719:
2505:
2371:
2297:
2280:
2233:
2046:
1955:
1789:
1774:
1734:
1686:
1671:
1659:
1615:
1590:
1360:
1309:
1008:
961:
907:
880:
853:
818:
694:
497:
470:
272:
2523:
1979:
1005:(any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the
2961:
2768:
2578:
2568:
2460:
2341:
2176:
2152:
1933:
1917:
1822:
1799:
1676:
1645:
1610:
1505:
1340:
847:
413:
339:
2975:
2970:
2863:
2820:
2642:
2603:
2598:
2583:
2409:
2366:
2263:
2061:
2011:
1585:
1547:
120:
2956:
2946:
2900:
2883:
2838:
2800:
2702:
2622:
2429:
2356:
2329:
2317:
2223:
2137:
2111:
2066:
2034:
1835:
1637:
1580:
1530:
1495:
1453:
1161:
843:
690:
325:
313:
291:
261:{\displaystyle |U|=\mathrm {card} (U)=\inf\{\alpha \in \mathrm {ON} \ |\ \alpha =_{c}U\},}
142:
2941:
2920:
2878:
2858:
2753:
2608:
2206:
2196:
2186:
2181:
2115:
1989:
1865:
1754:
1749:
1727:
1328:
988:
321:
146:
130:
3040:
2915:
2593:
2100:
1885:
1875:
1845:
1830:
1500:
578:
2815:
2662:
2563:
2555:
2435:
2383:
2292:
2228:
2211:
2142:
2001:
1860:
1562:
1345:
1259:
431:
2925:
2805:
1984:
1974:
1921:
1605:
1525:
1510:
1390:
1335:
21:
1855:
1710:
1681:
1487:
317:
134:
3007:
2910:
1963:
1880:
1840:
1804:
1740:
1552:
1542:
1515:
286:
is well-orderable and that the class of ordinals is well-ordered, using the
2992:
2790:
2238:
1943:
1537:
282:
That such an ordinal exists and is unique is guaranteed by the fact that
153:, using the von Neumann definition of an ordinal number. More precisely:
1035:
Infinite initial ordinals are limit ordinals. Using ordinal arithmetic,
2588:
1380:
1228:{\displaystyle \varphi _{\alpha }(\omega _{\beta })=\omega _{\beta }\,}
1278:
2132:
1478:
1323:
1282:
324:) is initial, but most infinite ordinals are not initial. The
15:
904:
is the smallest ordinal whose cardinality is greater than
1107:{\displaystyle \alpha +\omega _{\beta }=\omega _{\beta }}
1184:
1074:
1041:
1011:
991:
964:
937:
910:
883:
856:
821:
794:
762:
735:
703:
668:
641:
614:
587:
555:
527:
500:
473:
440:
416:
389:
362:
342:
162:
846:
ordinal (to see that it exists, consider the set of
2934:
2829:
2661:
2554:
2406:
2099:
2022:
1916:
1820:
1709:
1636:
1571:
1486:
1477:
1399:
1316:
46:. Unsourced material may be challenged and removed.
1227:
1106:
1060:
1024:
997:
977:
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923:
896:
869:
834:
807:
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748:
721:
681:
654:
627:
600:
568:
540:
513:
486:
459:
422:
402:
375:
348:
260:
689:for writing ordinals. This is important because
205:
304:. This is a well-ordering of cardinal numbers.
275:of ordinals. This ordinal is also called the
1294:
8:
252:
208:
1061:{\displaystyle \alpha <\omega _{\beta }}
2120:
1715:
1483:
1301:
1287:
1279:
1224:
1218:
1202:
1189:
1183:
1098:
1085:
1073:
1052:
1040:
1016:
1010:
990:
969:
963:
942:
936:
915:
909:
888:
882:
861:
855:
826:
820:
799:
793:
772:
767:
761:
740:
734:
713:
708:
702:
673:
667:
646:
640:
619:
613:
592:
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478:
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394:
388:
367:
361:
341:
243:
228:
217:
179:
171:
163:
161:
106:Learn how and when to remove this message
356:-th infinite initial ordinal is written
781:{\displaystyle \omega _{\alpha }^{2}}
722:{\displaystyle \aleph _{\alpha }^{2}}
7:
44:adding citations to reliable sources
662:is used for writing cardinals, and
494:, which is also the cardinality of
460:{\displaystyle \omega _{0}=\omega }
434:). For example, the cardinality of
912:
737:
705:
643:
616:
475:
391:
221:
218:
189:
186:
183:
180:
145:to be the smallest ordinal number
14:
951:{\displaystyle \omega _{\omega }}
808:{\displaystyle \omega _{\alpha }}
749:{\displaystyle \aleph _{\alpha }}
682:{\displaystyle \omega _{\alpha }}
655:{\displaystyle \aleph _{\alpha }}
628:{\displaystyle \aleph _{\alpha }}
601:{\displaystyle \omega _{\alpha }}
541:{\displaystyle \omega ^{\omega }}
403:{\displaystyle \aleph _{\alpha }}
376:{\displaystyle \omega _{\alpha }}
55:"Von Neumann cardinal assignment"
3020:
877:is the order type of that set),
20:
312:Each ordinal has an associated
31:needs additional citations for
1208:
1195:
383:. Its cardinality is written
229:
199:
193:
172:
164:
1:
2981:History of mathematical logic
569:{\displaystyle \epsilon _{0}}
308:Initial ordinal of a cardinal
2906:Primitive recursive function
1275:(1994 Springer) p. 198
1025:{\displaystyle \omega _{n}}
978:{\displaystyle \omega _{n}}
924:{\displaystyle \aleph _{1}}
897:{\displaystyle \omega _{2}}
870:{\displaystyle \omega _{1}}
835:{\displaystyle \omega _{1}}
635:, except that the notation
581:ordinals). So we identify
514:{\displaystyle \omega ^{2}}
487:{\displaystyle \aleph _{0}}
296:every set is well-orderable
3068:
1970:SchröderâBernstein theorem
1697:Monadic predicate calculus
1356:Foundations of mathematics
3016:
3003:Philosophy of mathematics
2952:Automated theorem proving
2123:
2077:Von NeumannâBernaysâGödel
1718:
2653:Self-verifying theories
2474:Tarski's axiomatization
1425:Tarski's undefinability
1420:incompleteness theorems
691:arithmetic on cardinals
423:{\displaystyle \alpha }
349:{\displaystyle \alpha }
3027:Mathematics portal
2638:Proof of impossibility
2286:propositional variable
1596:Propositional calculus
1229:
1108:
1062:
1026:
999:
979:
952:
925:
898:
871:
836:
809:
782:
750:
723:
695:arithmetic on ordinals
683:
656:
629:
602:
570:
542:
515:
488:
461:
424:
404:
377:
350:
262:
2896:Kolmogorov complexity
2849:Computably enumerable
2749:Model complete theory
2541:Principia Mathematica
1601:Propositional formula
1430:BanachâTarski paradox
1230:
1109:
1063:
1027:
1000:
980:
953:
926:
899:
872:
837:
810:
783:
751:
724:
684:
657:
630:
603:
571:
543:
516:
489:
462:
425:
405:
378:
351:
263:
2844:ChurchâTuring thesis
2831:Computability theory
2040:continuum hypothesis
1558:Square of opposition
1416:Gödel's completeness
1182:
1072:
1039:
1009:
989:
985:for natural numbers
962:
935:
908:
881:
854:
819:
792:
760:
733:
701:
666:
639:
612:
585:
553:
525:
498:
471:
438:
414:
387:
360:
340:
288:axiom of replacement
160:
40:improve this article
2998:Mathematical object
2889:P versus NP problem
2854:Computable function
2648:Reverse mathematics
2574:Logical consequence
2451:primitive recursive
2446:elementary function
2219:Free/bound variable
2072:TarskiâGrothendieck
1591:Logical connectives
1521:Logical equivalence
1371:Logical consequence
1273:Notes on Set Theory
848:equivalence classes
777:
718:
127:cardinal assignment
123:cardinal assignment
2796:Transfer principle
2759:Semantics of logic
2744:Categorical theory
2720:Non-standard model
2234:Logical connective
1361:Information theory
1310:Mathematical logic
1225:
1104:
1058:
1022:
995:
975:
948:
921:
894:
867:
832:
805:
778:
763:
746:
719:
704:
693:is different from
679:
652:
625:
598:
566:
538:
511:
484:
457:
420:
400:
373:
346:
258:
3034:
3033:
2966:Abstract category
2769:Theories of truth
2579:Rule of inference
2569:Natural deduction
2550:
2549:
2095:
2094:
1800:Cartesian product
1705:
1704:
1611:Many-valued logic
1586:Boolean functions
1469:Russell's paradox
1444:diagonal argument
1341:First-order logic
1271:Y.N. Moschovakis
998:{\displaystyle n}
931:, and so on, and
279:of the cardinal.
235:
227:
116:
115:
108:
90:
3059:
3047:Cardinal numbers
3025:
3024:
2976:History of logic
2971:Category of sets
2864:Decision problem
2643:Ordinal analysis
2584:Sequent calculus
2482:Boolean algebras
2422:
2421:
2396:
2367:logical/constant
2121:
2107:
2030:ZermeloâFraenkel
1781:Set operations:
1716:
1653:
1484:
1464:LöwenheimâSkolem
1351:Formal semantics
1303:
1296:
1289:
1280:
1234:
1232:
1231:
1226:
1223:
1222:
1207:
1206:
1194:
1193:
1162:Veblen hierarchy
1128: · Ï
1113:
1111:
1110:
1105:
1103:
1102:
1090:
1089:
1067:
1065:
1064:
1059:
1057:
1056:
1031:
1029:
1028:
1023:
1021:
1020:
1004:
1002:
1001:
996:
984:
982:
981:
976:
974:
973:
958:is the limit of
957:
955:
954:
949:
947:
946:
930:
928:
927:
922:
920:
919:
903:
901:
900:
895:
893:
892:
876:
874:
873:
868:
866:
865:
842:is the smallest
841:
839:
838:
833:
831:
830:
814:
812:
811:
806:
804:
803:
788: >
787:
785:
784:
779:
776:
771:
755:
753:
752:
747:
745:
744:
728:
726:
725:
720:
717:
712:
688:
686:
685:
680:
678:
677:
661:
659:
658:
653:
651:
650:
634:
632:
631:
626:
624:
623:
607:
605:
604:
599:
597:
596:
575:
573:
572:
567:
565:
564:
547:
545:
544:
539:
537:
536:
520:
518:
517:
512:
510:
509:
493:
491:
490:
485:
483:
482:
466:
464:
463:
458:
450:
449:
429:
427:
426:
421:
409:
407:
406:
401:
399:
398:
382:
380:
379:
374:
372:
371:
355:
353:
352:
347:
290:. With the full
271:where ON is the
267:
265:
264:
259:
248:
247:
233:
232:
225:
224:
192:
175:
167:
141:, we define its
111:
104:
100:
97:
91:
89:
48:
24:
16:
3067:
3066:
3062:
3061:
3060:
3058:
3057:
3056:
3052:Ordinal numbers
3037:
3036:
3035:
3030:
3019:
3012:
2957:Category theory
2947:Algebraic logic
2930:
2901:Lambda calculus
2839:Church encoding
2825:
2801:Truth predicate
2657:
2623:Complete theory
2546:
2415:
2411:
2407:
2402:
2394:
2114: and
2110:
2105:
2091:
2067:New Foundations
2035:axiom of choice
2018:
1980:Gödel numbering
1920: and
1912:
1816:
1701:
1651:
1632:
1581:Boolean algebra
1567:
1531:Equiconsistency
1496:Classical logic
1473:
1454:Halting problem
1442: and
1418: and
1406: and
1405:
1400:Theorems (
1395:
1312:
1307:
1268:
1256:
1249:
1243:
1242:
1214:
1198:
1185:
1180:
1179:
1177:
1159:
1149:
1139:
1133:
1123:
1094:
1081:
1070:
1069:
1048:
1037:
1036:
1012:
1007:
1006:
987:
986:
965:
960:
959:
938:
933:
932:
911:
906:
905:
884:
879:
878:
857:
852:
851:
822:
817:
816:
795:
790:
789:
758:
757:
736:
731:
730:
699:
698:
669:
664:
663:
642:
637:
636:
615:
610:
609:
588:
583:
582:
556:
551:
550:
528:
523:
522:
501:
496:
495:
474:
469:
468:
441:
436:
435:
412:
411:
390:
385:
384:
363:
358:
357:
338:
337:
326:axiom of choice
310:
303:
292:axiom of choice
277:initial ordinal
239:
158:
157:
143:cardinal number
131:ordinal numbers
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
3065:
3063:
3055:
3054:
3049:
3039:
3038:
3032:
3031:
3017:
3014:
3013:
3011:
3010:
3005:
3000:
2995:
2990:
2989:
2988:
2978:
2973:
2968:
2959:
2954:
2949:
2944:
2942:Abstract logic
2938:
2936:
2932:
2931:
2929:
2928:
2923:
2921:Turing machine
2918:
2913:
2908:
2903:
2898:
2893:
2892:
2891:
2886:
2881:
2876:
2871:
2861:
2859:Computable set
2856:
2851:
2846:
2841:
2835:
2833:
2827:
2826:
2824:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2787:
2786:
2781:
2776:
2766:
2761:
2756:
2754:Satisfiability
2751:
2746:
2741:
2740:
2739:
2729:
2728:
2727:
2717:
2716:
2715:
2710:
2705:
2700:
2695:
2685:
2684:
2683:
2678:
2671:Interpretation
2667:
2665:
2659:
2658:
2656:
2655:
2650:
2645:
2640:
2635:
2625:
2620:
2619:
2618:
2617:
2616:
2606:
2601:
2591:
2586:
2581:
2576:
2571:
2566:
2560:
2558:
2552:
2551:
2548:
2547:
2545:
2544:
2536:
2535:
2534:
2533:
2528:
2527:
2526:
2521:
2516:
2496:
2495:
2494:
2492:minimal axioms
2489:
2478:
2477:
2476:
2465:
2464:
2463:
2458:
2453:
2448:
2443:
2438:
2425:
2423:
2404:
2403:
2401:
2400:
2399:
2398:
2386:
2381:
2380:
2379:
2374:
2369:
2364:
2354:
2349:
2344:
2339:
2338:
2337:
2332:
2322:
2321:
2320:
2315:
2310:
2305:
2295:
2290:
2289:
2288:
2283:
2278:
2268:
2267:
2266:
2261:
2256:
2251:
2246:
2241:
2231:
2226:
2221:
2216:
2215:
2214:
2209:
2204:
2199:
2189:
2184:
2182:Formation rule
2179:
2174:
2173:
2172:
2167:
2157:
2156:
2155:
2145:
2140:
2135:
2130:
2124:
2118:
2101:Formal systems
2097:
2096:
2093:
2092:
2090:
2089:
2084:
2079:
2074:
2069:
2064:
2059:
2054:
2049:
2044:
2043:
2042:
2037:
2026:
2024:
2020:
2019:
2017:
2016:
2015:
2014:
2004:
1999:
1998:
1997:
1990:Large cardinal
1987:
1982:
1977:
1972:
1967:
1953:
1952:
1951:
1946:
1941:
1926:
1924:
1914:
1913:
1911:
1910:
1909:
1908:
1903:
1898:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1853:
1848:
1843:
1838:
1833:
1827:
1825:
1818:
1817:
1815:
1814:
1813:
1812:
1807:
1802:
1797:
1792:
1787:
1779:
1778:
1777:
1772:
1762:
1757:
1755:Extensionality
1752:
1750:Ordinal number
1747:
1737:
1732:
1731:
1730:
1719:
1713:
1707:
1706:
1703:
1702:
1700:
1699:
1694:
1689:
1684:
1679:
1674:
1669:
1668:
1667:
1657:
1656:
1655:
1642:
1640:
1634:
1633:
1631:
1630:
1629:
1628:
1623:
1618:
1608:
1603:
1598:
1593:
1588:
1583:
1577:
1575:
1569:
1568:
1566:
1565:
1560:
1555:
1550:
1545:
1540:
1535:
1534:
1533:
1523:
1518:
1513:
1508:
1503:
1498:
1492:
1490:
1481:
1475:
1474:
1472:
1471:
1466:
1461:
1456:
1451:
1446:
1434:Cantor's
1432:
1427:
1422:
1412:
1410:
1397:
1396:
1394:
1393:
1388:
1383:
1378:
1373:
1368:
1363:
1358:
1353:
1348:
1343:
1338:
1333:
1332:
1331:
1320:
1318:
1314:
1313:
1308:
1306:
1305:
1298:
1291:
1283:
1277:
1276:
1267:
1264:
1263:
1262:
1255:
1252:
1245:
1238:
1236:
1221:
1217:
1213:
1210:
1205:
1201:
1197:
1192:
1188:
1173:
1155:
1145:
1135:
1129:
1119:
1101:
1097:
1093:
1088:
1084:
1080:
1077:
1055:
1051:
1047:
1044:
1019:
1015:
994:
972:
968:
945:
941:
918:
914:
891:
887:
864:
860:
829:
825:
802:
798:
775:
770:
766:
743:
739:
716:
711:
707:
697:, for example
676:
672:
649:
645:
622:
618:
595:
591:
563:
559:
535:
531:
508:
504:
481:
477:
456:
453:
448:
444:
419:
397:
393:
370:
366:
345:
322:natural number
309:
306:
299:
269:
268:
257:
254:
251:
246:
242:
238:
231:
223:
220:
216:
213:
210:
207:
204:
201:
198:
195:
191:
188:
185:
182:
178:
174:
170:
166:
135:well-orderable
114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
3064:
3053:
3050:
3048:
3045:
3044:
3042:
3029:
3028:
3023:
3015:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2987:
2984:
2983:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2939:
2937:
2933:
2927:
2924:
2922:
2919:
2917:
2916:Recursive set
2914:
2912:
2909:
2907:
2904:
2902:
2899:
2897:
2894:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2866:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2836:
2834:
2832:
2828:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2785:
2782:
2780:
2777:
2775:
2772:
2771:
2770:
2767:
2765:
2762:
2760:
2757:
2755:
2752:
2750:
2747:
2745:
2742:
2738:
2735:
2734:
2733:
2730:
2726:
2725:of arithmetic
2723:
2722:
2721:
2718:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2690:
2689:
2686:
2682:
2679:
2677:
2674:
2673:
2672:
2669:
2668:
2666:
2664:
2660:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2633:
2632:from ZFC
2629:
2626:
2624:
2621:
2615:
2612:
2611:
2610:
2607:
2605:
2602:
2600:
2597:
2596:
2595:
2592:
2590:
2587:
2585:
2582:
2580:
2577:
2575:
2572:
2570:
2567:
2565:
2562:
2561:
2559:
2557:
2553:
2543:
2542:
2538:
2537:
2532:
2531:non-Euclidean
2529:
2525:
2522:
2520:
2517:
2515:
2514:
2510:
2509:
2507:
2504:
2503:
2501:
2497:
2493:
2490:
2488:
2485:
2484:
2483:
2479:
2475:
2472:
2471:
2470:
2466:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2433:
2431:
2427:
2426:
2424:
2419:
2413:
2408:Example
2405:
2397:
2392:
2391:
2390:
2387:
2385:
2382:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2359:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2336:
2333:
2331:
2328:
2327:
2326:
2323:
2319:
2316:
2314:
2311:
2309:
2306:
2304:
2301:
2300:
2299:
2296:
2294:
2291:
2287:
2284:
2282:
2279:
2277:
2274:
2273:
2272:
2269:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2236:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2213:
2210:
2208:
2205:
2203:
2200:
2198:
2195:
2194:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2171:
2168:
2166:
2165:by definition
2163:
2162:
2161:
2158:
2154:
2151:
2150:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2125:
2122:
2119:
2117:
2113:
2108:
2102:
2098:
2088:
2085:
2083:
2080:
2078:
2075:
2073:
2070:
2068:
2065:
2063:
2060:
2058:
2055:
2053:
2052:KripkeâPlatek
2050:
2048:
2045:
2041:
2038:
2036:
2033:
2032:
2031:
2028:
2027:
2025:
2021:
2013:
2010:
2009:
2008:
2005:
2003:
2000:
1996:
1993:
1992:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1971:
1968:
1965:
1961:
1957:
1954:
1950:
1947:
1945:
1942:
1940:
1937:
1936:
1935:
1931:
1928:
1927:
1925:
1923:
1919:
1915:
1907:
1904:
1902:
1899:
1897:
1896:constructible
1894:
1893:
1892:
1889:
1887:
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1852:
1849:
1847:
1844:
1842:
1839:
1837:
1834:
1832:
1829:
1828:
1826:
1824:
1819:
1811:
1808:
1806:
1803:
1801:
1798:
1796:
1793:
1791:
1788:
1786:
1783:
1782:
1780:
1776:
1773:
1771:
1768:
1767:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1742:
1738:
1736:
1733:
1729:
1726:
1725:
1724:
1721:
1720:
1717:
1714:
1712:
1708:
1698:
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1670:
1666:
1663:
1662:
1661:
1658:
1654:
1649:
1648:
1647:
1644:
1643:
1641:
1639:
1635:
1627:
1624:
1622:
1619:
1617:
1614:
1613:
1612:
1609:
1607:
1604:
1602:
1599:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1578:
1576:
1574:
1573:Propositional
1570:
1564:
1561:
1559:
1556:
1554:
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1532:
1529:
1528:
1527:
1524:
1522:
1519:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1501:Logical truth
1499:
1497:
1494:
1493:
1491:
1489:
1485:
1482:
1480:
1476:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1441:
1437:
1433:
1431:
1428:
1426:
1423:
1421:
1417:
1414:
1413:
1411:
1409:
1403:
1398:
1392:
1389:
1387:
1384:
1382:
1379:
1377:
1374:
1372:
1369:
1367:
1364:
1362:
1359:
1357:
1354:
1352:
1349:
1347:
1344:
1342:
1339:
1337:
1334:
1330:
1327:
1326:
1325:
1322:
1321:
1319:
1315:
1311:
1304:
1299:
1297:
1292:
1290:
1285:
1284:
1281:
1274:
1270:
1269:
1265:
1261:
1258:
1257:
1253:
1251:
1248:
1241:
1219:
1215:
1211:
1203:
1199:
1190:
1186:
1176:
1171:
1167:
1163:
1158:
1153:
1148:
1143:
1138:
1132:
1127:
1122:
1117:
1099:
1095:
1091:
1086:
1082:
1078:
1075:
1053:
1049:
1045:
1042:
1033:
1017:
1013:
992:
970:
966:
943:
939:
916:
889:
885:
862:
858:
849:
845:
827:
823:
800:
796:
773:
768:
764:
741:
729: =
714:
709:
696:
692:
674:
670:
647:
620:
593:
589:
580:
576:
561:
557:
533:
529:
506:
502:
479:
454:
451:
446:
442:
433:
417:
395:
368:
364:
343:
334:
332:
327:
323:
319:
315:
307:
305:
302:
297:
293:
289:
285:
280:
278:
274:
255:
249:
244:
240:
236:
214:
211:
202:
196:
176:
168:
156:
155:
154:
152:
148:
144:
140:
136:
132:
128:
124:
122:
110:
107:
99:
96:February 2010
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: â
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
3018:
2816:Ultraproduct
2663:Model theory
2628:Independence
2564:Formal proof
2556:Proof theory
2539:
2512:
2469:real numbers
2441:second-order
2352:Substitution
2229:Metalanguage
2170:conservative
2143:Axiom schema
2087:Constructive
2057:MorseâKelley
2023:Set theories
2002:Aleph number
1995:inaccessible
1901:Grothendieck
1785:intersection
1672:Higher-order
1660:Second-order
1606:Truth tables
1563:Venn diagram
1346:Formal proof
1272:
1260:Aleph number
1246:
1239:
1174:
1169:
1165:
1160:. Using the
1156:
1151:
1146:
1141:
1136:
1130:
1125:
1120:
1115:
1034:
432:aleph number
335:
333:a cardinal.
330:
311:
300:
283:
281:
276:
270:
150:
147:equinumerous
138:
119:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2926:Type theory
2874:undecidable
2806:Truth value
2693:equivalence
2372:non-logical
1985:Enumeration
1975:Isomorphism
1922:cardinality
1906:Von Neumann
1871:Ultrafilter
1836:Uncountable
1770:equivalence
1687:Quantifiers
1677:Fixed-point
1646:First-order
1526:Consistency
1511:Proposition
1488:Traditional
1459:Lindström's
1449:Compactness
1391:Type theory
1336:Cardinality
844:uncountable
121:von Neumann
3041:Categories
2737:elementary
2430:arithmetic
2298:Quantifier
2276:functional
2148:Expression
1866:Transitive
1810:identities
1795:complement
1728:hereditary
1711:Set theory
1266:References
1235:and Γ
1140:, and 2 â€
1114:, and 1 â€
318:order type
129:that uses
66:newspapers
3008:Supertask
2911:Recursion
2869:decidable
2703:saturated
2681:of models
2604:deductive
2599:axiomatic
2519:Hilbert's
2506:Euclidean
2487:canonical
2410:axiomatic
2342:Signature
2271:Predicate
2160:Extension
2082:Ackermann
2007:Operation
1886:Universal
1876:Recursive
1851:Singleton
1846:Inhabited
1831:Countable
1821:Types of
1805:power set
1775:partition
1692:Predicate
1638:Predicate
1553:Syllogism
1543:Soundness
1516:Inference
1506:Tautology
1408:paradoxes
1220:β
1216:ω
1204:β
1200:ω
1191:α
1187:φ
1100:β
1096:ω
1087:β
1083:ω
1076:α
1054:β
1050:ω
1043:α
1014:ω
967:ω
944:ω
940:ω
913:ℵ
886:ω
859:ω
824:ω
801:α
797:ω
769:α
765:ω
742:α
738:ℵ
710:α
706:ℵ
675:α
671:ω
648:α
644:ℵ
621:α
617:ℵ
594:α
590:ω
579:countable
577:(all are
558:ϵ
534:ω
530:ω
503:ω
476:ℵ
455:ω
443:ω
418:α
396:α
392:ℵ
369:α
365:ω
344:α
237:α
215:∈
212:α
2993:Logicism
2986:timeline
2962:Concrete
2821:Validity
2791:T-schema
2784:Kripke's
2779:Tarski's
2774:semantic
2764:Strength
2713:submodel
2708:spectrum
2676:function
2524:Tarski's
2513:Elements
2500:geometry
2456:Robinson
2377:variable
2362:function
2335:spectrum
2325:Sentence
2281:variable
2224:Language
2177:Relation
2138:Automata
2128:Alphabet
2112:language
1966:-jection
1944:codomain
1930:Function
1891:Universe
1861:Infinite
1765:Relation
1548:Validity
1538:Argument
1436:theorem,
1254:See also
1168:â 0 and
1150:implies
1124:implies
1068:implies
815:. Also,
756:whereas
314:cardinal
133:. For a
2935:Related
2732:Diagram
2630: (
2609:Hilbert
2594:Systems
2589:Theorem
2467:of the
2412:systems
2192:Formula
2187:Grammar
2103: (
2047:General
1760:Forcing
1745:Element
1665:Monadic
1440:paradox
1381:Theorem
1317:General
80:scholar
2698:finite
2461:Skolem
2414:
2389:Theory
2357:Symbol
2347:String
2330:atomic
2207:ground
2202:closed
2197:atomic
2153:ground
2116:syntax
2012:binary
1939:domain
1856:Finite
1621:finite
1479:Logics
1438:
1386:Theory
1178:imply
1172:< Ï
1144:< Ï
1118:< Ï
548:, and
234:
226:
82:
75:
68:
61:
53:
2688:Model
2436:Peano
2293:Proof
2133:Arity
2062:Naive
1949:image
1881:Fuzzy
1841:Empty
1790:union
1735:Class
1376:Model
1366:Lemma
1324:Axiom
608:with
410:(the
273:class
125:is a
87:JSTOR
73:books
2811:Type
2614:list
2418:list
2395:list
2384:Term
2318:rank
2212:open
2106:list
1918:Maps
1823:sets
1682:Free
1652:list
1402:list
1329:list
1046:<
430:-th
336:The
137:set
118:The
59:news
2498:of
2480:of
2428:of
1960:Sur
1934:Map
1741:Ur-
1723:Set
1244:= Ï
1154:= Ï
1134:= Ï
1032:).
467:is
206:inf
149:to
42:by
3043::
2884:NP
2508::
2502::
2432::
2109:),
1964:Bi
1956:In
1164:,
521:,
331:is
294:,
2964:/
2879:P
2634:)
2420:)
2416:(
2313:â
2308:!
2303:â
2264:=
2259:â
2254:â
2249:â§
2244:âš
2239:ÂŹ
1962:/
1958:/
1932:/
1743:)
1739:(
1626:â
1616:3
1404:)
1302:e
1295:t
1288:v
1247:ÎČ
1240:ÎČ
1237:Ï
1212:=
1209:)
1196:(
1175:ÎČ
1170:α
1166:ÎČ
1157:ÎČ
1152:α
1147:ÎČ
1142:α
1137:ÎČ
1131:ÎČ
1126:α
1121:ÎČ
1116:α
1092:=
1079:+
1018:n
993:n
971:n
917:1
890:2
863:1
828:1
774:2
715:2
562:0
507:2
480:0
452:=
447:0
301:c
284:U
256:,
253:}
250:U
245:c
241:=
230:|
222:N
219:O
209:{
203:=
200:)
197:U
194:(
190:d
187:r
184:a
181:c
177:=
173:|
169:U
165:|
151:U
139:U
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
