2885:
2955:
1003:
However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.
761:
infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.
944:
822:
894:
859:
690:
318:
284:
971:
755:
642:
584:
526:
491:
242:
shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite
186:
148:
615:
557:
453:
422:
345:
998:
783:
1264:
3412:
1939:
3562:
2022:
1163:
2336:
2494:
1098:
1282:
3101:
2921:
2349:
1672:
3429:
2354:
2344:
2081:
1934:
1287:
1278:
2490:
1117:
1090:
1832:
2587:
2331:
1156:
3407:
1892:
1585:
193:
3287:
1326:
2848:
2550:
2313:
2308:
2133:
1554:
1238:
899:
791:
3181:
3060:
2843:
2626:
2543:
2256:
2187:
2064:
1306:
866:
828:
239:
655:
3424:
2768:
2594:
2280:
1914:
1513:
348:
259:
3417:
3055:
3018:
2646:
2641:
2251:
1990:
1919:
1248:
1149:
697:
2575:
2165:
1559:
1527:
1218:
3072:
3572:
3106:
2998:
2986:
2981:
2865:
2814:
2711:
2209:
2170:
1647:
1292:
1023:
289:
265:
200:, but the equivalence of the third and fourth cannot be proved without additional choice principles.
1321:
2914:
2706:
2636:
2175:
2027:
2010:
1733:
1213:
3526:
3444:
3319:
3271:
3085:
3008:
2538:
2515:
2476:
2362:
2303:
1949:
1869:
1713:
1657:
1270:
1081:
949:
733:
620:
562:
504:
469:
164:
126:
593:
535:
431:
400:
323:
3478:
3359:
3171:
2991:
2828:
2555:
2533:
2500:
2393:
2239:
2224:
2197:
2148:
2032:
1967:
1792:
1758:
1753:
1627:
1458:
1435:
382:
1085:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
980:
3394:
3364:
3308:
3228:
3208:
3186:
2758:
2611:
2403:
2121:
1857:
1763:
1622:
1607:
1488:
1463:
768:
717:
649:
39:
3468:
3458:
3292:
3223:
3176:
3116:
3003:
2731:
2693:
2570:
2374:
2214:
2138:
2116:
1944:
1902:
1801:
1768:
1632:
1420:
1331:
693:
8:
3463:
3374:
3282:
3277:
3091:
3033:
2971:
2907:
2860:
2751:
2736:
2716:
2673:
2560:
2510:
2436:
2381:
2318:
2111:
2106:
2054:
1822:
1811:
1483:
1383:
1311:
1302:
1298:
1233:
1228:
1049:
367:
105:
3386:
3381:
3166:
3121:
3028:
2889:
2658:
2621:
2606:
2599:
2582:
2386:
2368:
2234:
2160:
2143:
2096:
1909:
1818:
1652:
1637:
1597:
1549:
1534:
1522:
1478:
1453:
1223:
1172:
1028:
75:
1842:
3567:
3243:
3080:
3043:
3013:
2944:
2884:
2824:
2631:
2441:
2431:
2323:
2204:
2039:
2015:
1796:
1780:
1685:
1662:
1539:
1508:
1473:
1368:
1203:
1113:
1094:
1086:
3531:
3521:
3506:
3501:
3369:
3023:
2838:
2833:
2726:
2683:
2505:
2466:
2461:
2446:
2272:
2229:
2126:
1924:
1874:
1448:
1410:
20:
3400:
3156:
2976:
2819:
2809:
2763:
2746:
2701:
2663:
2565:
2485:
2292:
2219:
2192:
2180:
2086:
2000:
1974:
1929:
1897:
1500:
1443:
1393:
1358:
1316:
974:
758:
723:
705:
498:
197:
47:
3536:
3333:
3314:
3218:
3203:
3160:
3096:
3038:
2804:
2783:
2741:
2721:
2616:
2471:
2069:
2059:
2049:
2044:
1978:
1852:
1728:
1617:
1612:
1590:
1191:
459:
247:
55:
1132:
3556:
3541:
3343:
3257:
3252:
2778:
2456:
1963:
1748:
1738:
1708:
1693:
1363:
727:
645:
43:
3511:
374:
has dimension one). This is an example of the following fact: any subset of
3491:
3486:
3304:
3233:
3191:
3050:
2954:
2678:
2525:
2426:
2418:
2298:
2246:
2155:
2091:
2074:
2005:
1864:
1723:
1425:
1208:
1013:
587:
494:
151:
51:
35:
1112:, Springer Monographs in Mathematics (3rd millennium ed.), Springer,
458:
A more abstract example of an uncountable set is the set of all countable
3516:
3151:
2788:
2668:
1847:
1837:
1784:
1468:
1388:
1373:
1253:
1198:
1105:
1076:
1018:
235:
116:
27:
3496:
3267:
2930:
1718:
1573:
1544:
1350:
701:
378:
of
Hausdorff dimension strictly greater than zero must be uncountable.
355:
192:
The first three of these characterizations can be proven equivalent in
100:, there exists at least one element of X not included in it. That is,
66:
There are many equivalent characterizations of uncountability. A set
3299:
3262:
3213:
3111:
2870:
2773:
1826:
1743:
1703:
1667:
1603:
1415:
1405:
1378:
1141:
765:
If the axiom of choice holds, the following conditions on a cardinal
79:
70:
is uncountable if and only if any of the following conditions hold:
2855:
2653:
2101:
1806:
1400:
425:
243:
93:
1093:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
2451:
1243:
363:
3324:
3146:
251:
19:"Uncountable" redirects here. For the linguistic concept, see
3196:
2963:
2899:
1995:
1341:
1186:
50:: a set is uncountable if its cardinal number is larger than
381:
Another example of an uncountable set is the set of all
230:
The best known example of an uncountable set is the set
46:. The uncountability of a set is closely related to its
983:
952:
902:
869:
831:
794:
771:
736:
658:
623:
596:
565:
538:
507:
472:
434:
403:
326:
292:
268:
167:
129:
992:
965:
938:
888:
853:
816:
777:
749:
684:
636:
609:
578:
551:
520:
485:
447:
416:
339:
312:
278:
254:of the set of natural numbers. The cardinality of
180:
142:
590:was the first to propose the question of whether
397:in the sense that the cardinality of this set is
3554:
711:
2915:
1157:
559:, the cardinality of the reals, is equal to
393:. This set is even "more uncountable" than
2922:
2908:
1349:
1164:
1150:
939:{\displaystyle \aleph _{1}=|\omega _{1}|}
817:{\displaystyle \kappa \nleq \aleph _{0};}
532:uncountable cardinal number. Thus either
696:, and is known to be independent of the
648:posed this question as the first of his
889:{\displaystyle \kappa \geq \aleph _{1}}
854:{\displaystyle \kappa >\aleph _{0};}
3555:
1171:
685:{\displaystyle \aleph _{1}=\beth _{1}}
161:has cardinality strictly greater than
3563:Basic concepts in infinite set theory
2903:
1145:
1047:
370:greater than zero but less than one (
1104:
61:
271:
13:
904:
877:
839:
802:
738:
726:, there might exist cardinalities
660:
625:
567:
509:
474:
466:. The cardinality of Ω is denoted
299:
169:
131:
16:Infinite set that is not countable
14:
3584:
1126:
1041:
2953:
2883:
1070:
313:{\displaystyle 2^{\aleph _{0}}}
279:{\displaystyle {\mathfrak {c}}}
123:is neither finite nor equal to
2929:
932:
917:
757:(namely, the cardinalities of
497:). It can be shown, using the
86:to the set of natural numbers.
1:
2844:History of mathematical logic
1034:
203:
2769:Primitive recursive function
358:is an uncountable subset of
260:cardinality of the continuum
108:from the natural numbers to
104:is nonempty and there is no
92:is nonempty and for every Ï-
7:
1007:
966:{\displaystyle \omega _{1}}
750:{\displaystyle \aleph _{0}}
712:Without the axiom of choice
637:{\displaystyle \aleph _{1}}
579:{\displaystyle \aleph _{1}}
521:{\displaystyle \aleph _{1}}
486:{\displaystyle \aleph _{1}}
225:
194:ZermeloâFraenkel set theory
181:{\displaystyle \aleph _{0}}
143:{\displaystyle \aleph _{0}}
10:
3589:
3413:von NeumannâBernaysâGödel
1833:SchröderâBernstein theorem
1560:Monadic predicate calculus
1219:Foundations of mathematics
715:
610:{\displaystyle \beth _{1}}
586:or it is strictly larger.
552:{\displaystyle \beth _{1}}
448:{\displaystyle \beth _{1}}
417:{\displaystyle \beth _{2}}
340:{\displaystyle \beth _{1}}
240:Cantor's diagonal argument
18:
3477:
3440:
3352:
3242:
3214:One-to-one correspondence
3130:
3071:
2962:
2951:
2937:
2879:
2866:Philosophy of mathematics
2815:Automated theorem proving
2797:
2692:
2524:
2417:
2269:
1986:
1962:
1940:Von NeumannâBernaysâGödel
1885:
1779:
1683:
1581:
1572:
1499:
1434:
1340:
1262:
1179:
1024:First uncountable ordinal
54:, the cardinality of the
993:{\displaystyle \omega .}
428:), which is larger than
2516:Self-verifying theories
2337:Tarski's axiomatization
1288:Tarski's undefinability
1283:incompleteness theorems
778:{\displaystyle \kappa }
698:ZermeloâFraenkel axioms
38:that contains too many
3172:Constructible universe
2999:Constructibility (V=L)
2890:Mathematics portal
2501:Proof of impossibility
2149:propositional variable
1459:Propositional calculus
1050:"Uncountably Infinite"
994:
967:
940:
890:
855:
818:
779:
751:
686:
638:
611:
580:
553:
522:
487:
449:
418:
362:. The Cantor set is a
341:
314:
280:
209:If an uncountable set
182:
144:
3395:Principia Mathematica
3229:Transfinite induction
3088:(i.e. set difference)
2759:Kolmogorov complexity
2712:Computably enumerable
2612:Model complete theory
2404:Principia Mathematica
1464:Propositional formula
1293:BanachâTarski paradox
1054:mathworld.wolfram.com
995:
968:
941:
891:
856:
819:
780:
752:
718:Dedekind-infinite set
687:
652:. The statement that
639:
612:
581:
554:
523:
488:
450:
419:
342:
315:
281:
183:
145:
3469:Burali-Forti paradox
3224:Set-builder notation
3177:Continuum hypothesis
3117:Symmetric difference
2707:ChurchâTuring thesis
2694:Computability theory
1903:continuum hypothesis
1421:Square of opposition
1279:Gödel's completeness
1101:(Paperback edition).
981:
950:
900:
867:
829:
792:
769:
734:
694:continuum hypothesis
656:
621:
594:
563:
536:
505:
470:
432:
401:
324:
290:
266:
258:is often called the
165:
127:
34:, informally, is an
3430:TarskiâGrothendieck
2861:Mathematical object
2752:P versus NP problem
2717:Computable function
2511:Reverse mathematics
2437:Logical consequence
2314:primitive recursive
2309:elementary function
2082:Free/bound variable
1935:TarskiâGrothendieck
1454:Logical connectives
1384:Logical equivalence
1234:Logical consequence
1048:Weisstein, Eric W.
462:, denoted by Ω or Ï
368:Hausdorff dimension
250:and the set of all
213:is a subset of set
106:surjective function
3019:Limitation of size
2659:Transfer principle
2622:Semantics of logic
2607:Categorical theory
2583:Non-standard model
2097:Logical connective
1224:Information theory
1173:Mathematical logic
1029:Injective function
990:
963:
936:
886:
851:
814:
775:
747:
692:is now called the
682:
634:
607:
576:
549:
518:
483:
445:
414:
337:
310:
276:
178:
140:
76:injective function
3550:
3549:
3459:Russell's paradox
3408:ZermeloâFraenkel
3309:Dedekind-infinite
3182:Diagonal argument
3081:Cartesian product
2945:Set (mathematics)
2897:
2896:
2829:Abstract category
2632:Theories of truth
2442:Rule of inference
2432:Natural deduction
2413:
2412:
1958:
1957:
1663:Cartesian product
1568:
1567:
1474:Many-valued logic
1449:Boolean functions
1332:Russell's paradox
1307:diagonal argument
1204:First-order logic
1099:978-1-61427-131-4
262:, and denoted by
62:Characterizations
3580:
3573:Cardinal numbers
3532:Bertrand Russell
3522:John von Neumann
3507:Abraham Fraenkel
3502:Richard Dedekind
3464:Suslin's problem
3375:Cantor's theorem
3092:De Morgan's laws
2957:
2924:
2917:
2910:
2901:
2900:
2888:
2887:
2839:History of logic
2834:Category of sets
2727:Decision problem
2506:Ordinal analysis
2447:Sequent calculus
2345:Boolean algebras
2285:
2284:
2259:
2230:logical/constant
1984:
1983:
1970:
1893:ZermeloâFraenkel
1644:Set operations:
1579:
1578:
1516:
1347:
1346:
1327:LöwenheimâSkolem
1214:Formal semantics
1166:
1159:
1152:
1143:
1142:
1122:
1082:Naive Set Theory
1064:
1063:
1061:
1060:
1045:
999:
997:
996:
991:
972:
970:
969:
964:
962:
961:
945:
943:
942:
937:
935:
930:
929:
920:
912:
911:
895:
893:
892:
887:
885:
884:
860:
858:
857:
852:
847:
846:
823:
821:
820:
815:
810:
809:
785:are equivalent:
784:
782:
781:
776:
756:
754:
753:
748:
746:
745:
691:
689:
688:
683:
681:
680:
668:
667:
643:
641:
640:
635:
633:
632:
616:
614:
613:
608:
606:
605:
585:
583:
582:
577:
575:
574:
558:
556:
555:
550:
548:
547:
527:
525:
524:
519:
517:
516:
492:
490:
489:
484:
482:
481:
454:
452:
451:
446:
444:
443:
423:
421:
420:
415:
413:
412:
346:
344:
343:
338:
336:
335:
319:
317:
316:
311:
309:
308:
307:
306:
285:
283:
282:
277:
275:
274:
187:
185:
184:
179:
177:
176:
149:
147:
146:
141:
139:
138:
21:Uncountable noun
3588:
3587:
3583:
3582:
3581:
3579:
3578:
3577:
3553:
3552:
3551:
3546:
3473:
3452:
3436:
3401:New Foundations
3348:
3238:
3157:Cardinal number
3140:
3126:
3067:
2958:
2949:
2933:
2928:
2898:
2893:
2882:
2875:
2820:Category theory
2810:Algebraic logic
2793:
2764:Lambda calculus
2702:Church encoding
2688:
2664:Truth predicate
2520:
2486:Complete theory
2409:
2278:
2274:
2270:
2265:
2257:
1977: and
1973:
1968:
1954:
1930:New Foundations
1898:axiom of choice
1881:
1843:Gödel numbering
1783: and
1775:
1679:
1564:
1514:
1495:
1444:Boolean algebra
1430:
1394:Equiconsistency
1359:Classical logic
1336:
1317:Halting problem
1305: and
1281: and
1269: and
1268:
1263:Theorems (
1258:
1175:
1170:
1129:
1120:
1073:
1068:
1067:
1058:
1056:
1046:
1042:
1037:
1010:
982:
979:
978:
975:initial ordinal
957:
953:
951:
948:
947:
931:
925:
921:
916:
907:
903:
901:
898:
897:
880:
876:
868:
865:
864:
842:
838:
830:
827:
826:
805:
801:
793:
790:
789:
770:
767:
766:
759:Dedekind-finite
741:
737:
735:
732:
731:
724:axiom of choice
720:
714:
706:axiom of choice
704:(including the
676:
672:
663:
659:
657:
654:
653:
628:
624:
622:
619:
618:
601:
597:
595:
592:
591:
570:
566:
564:
561:
560:
543:
539:
537:
534:
533:
512:
508:
506:
503:
502:
499:axiom of choice
477:
473:
471:
468:
467:
465:
460:ordinal numbers
439:
435:
433:
430:
429:
408:
404:
402:
399:
398:
331:
327:
325:
322:
321:
302:
298:
297:
293:
291:
288:
287:
270:
269:
267:
264:
263:
248:natural numbers
228:
221:is uncountable.
206:
198:axiom of choice
172:
168:
166:
163:
162:
134:
130:
128:
125:
124:
96:of elements of
64:
56:natural numbers
48:cardinal number
32:uncountable set
24:
17:
12:
11:
5:
3586:
3576:
3575:
3570:
3565:
3548:
3547:
3545:
3544:
3539:
3537:Thoralf Skolem
3534:
3529:
3524:
3519:
3514:
3509:
3504:
3499:
3494:
3489:
3483:
3481:
3475:
3474:
3472:
3471:
3466:
3461:
3455:
3453:
3451:
3450:
3447:
3441:
3438:
3437:
3435:
3434:
3433:
3432:
3427:
3422:
3421:
3420:
3405:
3404:
3403:
3391:
3390:
3389:
3378:
3377:
3372:
3367:
3362:
3356:
3354:
3350:
3349:
3347:
3346:
3341:
3336:
3331:
3322:
3317:
3312:
3302:
3297:
3296:
3295:
3290:
3285:
3275:
3265:
3260:
3255:
3249:
3247:
3240:
3239:
3237:
3236:
3231:
3226:
3221:
3219:Ordinal number
3216:
3211:
3206:
3201:
3200:
3199:
3194:
3184:
3179:
3174:
3169:
3164:
3154:
3149:
3143:
3141:
3139:
3138:
3135:
3131:
3128:
3127:
3125:
3124:
3119:
3114:
3109:
3104:
3099:
3097:Disjoint union
3094:
3089:
3083:
3077:
3075:
3069:
3068:
3066:
3065:
3064:
3063:
3058:
3047:
3046:
3044:Martin's axiom
3041:
3036:
3031:
3026:
3021:
3016:
3011:
3009:Extensionality
3006:
3001:
2996:
2995:
2994:
2989:
2984:
2974:
2968:
2966:
2960:
2959:
2952:
2950:
2948:
2947:
2941:
2939:
2935:
2934:
2927:
2926:
2919:
2912:
2904:
2895:
2894:
2880:
2877:
2876:
2874:
2873:
2868:
2863:
2858:
2853:
2852:
2851:
2841:
2836:
2831:
2822:
2817:
2812:
2807:
2805:Abstract logic
2801:
2799:
2795:
2794:
2792:
2791:
2786:
2784:Turing machine
2781:
2776:
2771:
2766:
2761:
2756:
2755:
2754:
2749:
2744:
2739:
2734:
2724:
2722:Computable set
2719:
2714:
2709:
2704:
2698:
2696:
2690:
2689:
2687:
2686:
2681:
2676:
2671:
2666:
2661:
2656:
2651:
2650:
2649:
2644:
2639:
2629:
2624:
2619:
2617:Satisfiability
2614:
2609:
2604:
2603:
2602:
2592:
2591:
2590:
2580:
2579:
2578:
2573:
2568:
2563:
2558:
2548:
2547:
2546:
2541:
2534:Interpretation
2530:
2528:
2522:
2521:
2519:
2518:
2513:
2508:
2503:
2498:
2488:
2483:
2482:
2481:
2480:
2479:
2469:
2464:
2454:
2449:
2444:
2439:
2434:
2429:
2423:
2421:
2415:
2414:
2411:
2410:
2408:
2407:
2399:
2398:
2397:
2396:
2391:
2390:
2389:
2384:
2379:
2359:
2358:
2357:
2355:minimal axioms
2352:
2341:
2340:
2339:
2328:
2327:
2326:
2321:
2316:
2311:
2306:
2301:
2288:
2286:
2267:
2266:
2264:
2263:
2262:
2261:
2249:
2244:
2243:
2242:
2237:
2232:
2227:
2217:
2212:
2207:
2202:
2201:
2200:
2195:
2185:
2184:
2183:
2178:
2173:
2168:
2158:
2153:
2152:
2151:
2146:
2141:
2131:
2130:
2129:
2124:
2119:
2114:
2109:
2104:
2094:
2089:
2084:
2079:
2078:
2077:
2072:
2067:
2062:
2052:
2047:
2045:Formation rule
2042:
2037:
2036:
2035:
2030:
2020:
2019:
2018:
2008:
2003:
1998:
1993:
1987:
1981:
1964:Formal systems
1960:
1959:
1956:
1955:
1953:
1952:
1947:
1942:
1937:
1932:
1927:
1922:
1917:
1912:
1907:
1906:
1905:
1900:
1889:
1887:
1883:
1882:
1880:
1879:
1878:
1877:
1867:
1862:
1861:
1860:
1853:Large cardinal
1850:
1845:
1840:
1835:
1830:
1816:
1815:
1814:
1809:
1804:
1789:
1787:
1777:
1776:
1774:
1773:
1772:
1771:
1766:
1761:
1751:
1746:
1741:
1736:
1731:
1726:
1721:
1716:
1711:
1706:
1701:
1696:
1690:
1688:
1681:
1680:
1678:
1677:
1676:
1675:
1670:
1665:
1660:
1655:
1650:
1642:
1641:
1640:
1635:
1625:
1620:
1618:Extensionality
1615:
1613:Ordinal number
1610:
1600:
1595:
1594:
1593:
1582:
1576:
1570:
1569:
1566:
1565:
1563:
1562:
1557:
1552:
1547:
1542:
1537:
1532:
1531:
1530:
1520:
1519:
1518:
1505:
1503:
1497:
1496:
1494:
1493:
1492:
1491:
1486:
1481:
1471:
1466:
1461:
1456:
1451:
1446:
1440:
1438:
1432:
1431:
1429:
1428:
1423:
1418:
1413:
1408:
1403:
1398:
1397:
1396:
1386:
1381:
1376:
1371:
1366:
1361:
1355:
1353:
1344:
1338:
1337:
1335:
1334:
1329:
1324:
1319:
1314:
1309:
1297:Cantor's
1295:
1290:
1285:
1275:
1273:
1260:
1259:
1257:
1256:
1251:
1246:
1241:
1236:
1231:
1226:
1221:
1216:
1211:
1206:
1201:
1196:
1195:
1194:
1183:
1181:
1177:
1176:
1169:
1168:
1161:
1154:
1146:
1140:
1139:
1137:is uncountable
1128:
1127:External links
1125:
1124:
1123:
1118:
1102:
1072:
1069:
1066:
1065:
1039:
1038:
1036:
1033:
1032:
1031:
1026:
1021:
1016:
1009:
1006:
1001:
1000:
989:
986:
960:
956:
934:
928:
924:
919:
915:
910:
906:
883:
879:
875:
872:
862:
850:
845:
841:
837:
834:
824:
813:
808:
804:
800:
797:
774:
744:
740:
716:Main article:
713:
710:
679:
675:
671:
666:
662:
631:
627:
604:
600:
573:
569:
546:
542:
515:
511:
480:
476:
463:
442:
438:
411:
407:
334:
330:
305:
301:
296:
273:
227:
224:
223:
222:
205:
202:
190:
189:
175:
171:
155:
137:
133:
113:
87:
63:
60:
15:
9:
6:
4:
3:
2:
3585:
3574:
3571:
3569:
3566:
3564:
3561:
3560:
3558:
3543:
3542:Ernst Zermelo
3540:
3538:
3535:
3533:
3530:
3528:
3527:Willard Quine
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3505:
3503:
3500:
3498:
3495:
3493:
3490:
3488:
3485:
3484:
3482:
3480:
3479:Set theorists
3476:
3470:
3467:
3465:
3462:
3460:
3457:
3456:
3454:
3448:
3446:
3443:
3442:
3439:
3431:
3428:
3426:
3425:KripkeâPlatek
3423:
3419:
3416:
3415:
3414:
3411:
3410:
3409:
3406:
3402:
3399:
3398:
3397:
3396:
3392:
3388:
3385:
3384:
3383:
3380:
3379:
3376:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3357:
3355:
3351:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3328:
3323:
3321:
3318:
3316:
3313:
3310:
3306:
3303:
3301:
3298:
3294:
3291:
3289:
3286:
3284:
3281:
3280:
3279:
3276:
3273:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3250:
3248:
3245:
3241:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3205:
3202:
3198:
3195:
3193:
3190:
3189:
3188:
3185:
3183:
3180:
3178:
3175:
3173:
3170:
3168:
3165:
3162:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3142:
3136:
3133:
3132:
3129:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3087:
3084:
3082:
3079:
3078:
3076:
3074:
3070:
3062:
3061:specification
3059:
3057:
3054:
3053:
3052:
3049:
3048:
3045:
3042:
3040:
3037:
3035:
3032:
3030:
3027:
3025:
3022:
3020:
3017:
3015:
3012:
3010:
3007:
3005:
3002:
3000:
2997:
2993:
2990:
2988:
2985:
2983:
2980:
2979:
2978:
2975:
2973:
2970:
2969:
2967:
2965:
2961:
2956:
2946:
2943:
2942:
2940:
2936:
2932:
2925:
2920:
2918:
2913:
2911:
2906:
2905:
2902:
2892:
2891:
2886:
2878:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2850:
2847:
2846:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2826:
2823:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2802:
2800:
2796:
2790:
2787:
2785:
2782:
2780:
2779:Recursive set
2777:
2775:
2772:
2770:
2767:
2765:
2762:
2760:
2757:
2753:
2750:
2748:
2745:
2743:
2740:
2738:
2735:
2733:
2730:
2729:
2728:
2725:
2723:
2720:
2718:
2715:
2713:
2710:
2708:
2705:
2703:
2700:
2699:
2697:
2695:
2691:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2648:
2645:
2643:
2640:
2638:
2635:
2634:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2601:
2598:
2597:
2596:
2593:
2589:
2588:of arithmetic
2586:
2585:
2584:
2581:
2577:
2574:
2572:
2569:
2567:
2564:
2562:
2559:
2557:
2554:
2553:
2552:
2549:
2545:
2542:
2540:
2537:
2536:
2535:
2532:
2531:
2529:
2527:
2523:
2517:
2514:
2512:
2509:
2507:
2504:
2502:
2499:
2496:
2495:from ZFC
2492:
2489:
2487:
2484:
2478:
2475:
2474:
2473:
2470:
2468:
2465:
2463:
2460:
2459:
2458:
2455:
2453:
2450:
2448:
2445:
2443:
2440:
2438:
2435:
2433:
2430:
2428:
2425:
2424:
2422:
2420:
2416:
2406:
2405:
2401:
2400:
2395:
2394:non-Euclidean
2392:
2388:
2385:
2383:
2380:
2378:
2377:
2373:
2372:
2370:
2367:
2366:
2364:
2360:
2356:
2353:
2351:
2348:
2347:
2346:
2342:
2338:
2335:
2334:
2333:
2329:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2296:
2294:
2290:
2289:
2287:
2282:
2276:
2271:Example
2268:
2260:
2255:
2254:
2253:
2250:
2248:
2245:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2222:
2221:
2218:
2216:
2213:
2211:
2208:
2206:
2203:
2199:
2196:
2194:
2191:
2190:
2189:
2186:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2163:
2162:
2159:
2157:
2154:
2150:
2147:
2145:
2142:
2140:
2137:
2136:
2135:
2132:
2128:
2125:
2123:
2120:
2118:
2115:
2113:
2110:
2108:
2105:
2103:
2100:
2099:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2076:
2073:
2071:
2068:
2066:
2063:
2061:
2058:
2057:
2056:
2053:
2051:
2048:
2046:
2043:
2041:
2038:
2034:
2031:
2029:
2028:by definition
2026:
2025:
2024:
2021:
2017:
2014:
2013:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1988:
1985:
1982:
1980:
1976:
1971:
1965:
1961:
1951:
1948:
1946:
1943:
1941:
1938:
1936:
1933:
1931:
1928:
1926:
1923:
1921:
1918:
1916:
1915:KripkeâPlatek
1913:
1911:
1908:
1904:
1901:
1899:
1896:
1895:
1894:
1891:
1890:
1888:
1884:
1876:
1873:
1872:
1871:
1868:
1866:
1863:
1859:
1856:
1855:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1828:
1824:
1820:
1817:
1813:
1810:
1808:
1805:
1803:
1800:
1799:
1798:
1794:
1791:
1790:
1788:
1786:
1782:
1778:
1770:
1767:
1765:
1762:
1760:
1759:constructible
1757:
1756:
1755:
1752:
1750:
1747:
1745:
1742:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1697:
1695:
1692:
1691:
1689:
1687:
1682:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1643:
1639:
1636:
1634:
1631:
1630:
1629:
1626:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1605:
1601:
1599:
1596:
1592:
1589:
1588:
1587:
1584:
1583:
1580:
1577:
1575:
1571:
1561:
1558:
1556:
1553:
1551:
1548:
1546:
1543:
1541:
1538:
1536:
1533:
1529:
1526:
1525:
1524:
1521:
1517:
1512:
1511:
1510:
1507:
1506:
1504:
1502:
1498:
1490:
1487:
1485:
1482:
1480:
1477:
1476:
1475:
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1452:
1450:
1447:
1445:
1442:
1441:
1439:
1437:
1436:Propositional
1433:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1402:
1399:
1395:
1392:
1391:
1390:
1387:
1385:
1382:
1380:
1377:
1375:
1372:
1370:
1367:
1365:
1364:Logical truth
1362:
1360:
1357:
1356:
1354:
1352:
1348:
1345:
1343:
1339:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1308:
1304:
1300:
1296:
1294:
1291:
1289:
1286:
1284:
1280:
1277:
1276:
1274:
1272:
1266:
1261:
1255:
1252:
1250:
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1193:
1190:
1189:
1188:
1185:
1184:
1182:
1178:
1174:
1167:
1162:
1160:
1155:
1153:
1148:
1147:
1144:
1138:
1136:
1131:
1130:
1121:
1119:3-540-44085-2
1115:
1111:
1107:
1103:
1100:
1096:
1092:
1091:0-387-90092-6
1088:
1084:
1083:
1078:
1075:
1074:
1055:
1051:
1044:
1040:
1030:
1027:
1025:
1022:
1020:
1017:
1015:
1012:
1011:
1005:
987:
984:
977:greater than
976:
973:is the least
958:
954:
926:
922:
913:
908:
881:
873:
870:
863:
848:
843:
835:
832:
825:
811:
806:
798:
795:
788:
787:
786:
772:
763:
760:
742:
729:
725:
719:
709:
707:
703:
699:
695:
677:
673:
669:
664:
651:
647:
646:David Hilbert
629:
602:
598:
589:
571:
544:
540:
531:
513:
500:
496:
478:
461:
456:
440:
436:
427:
409:
405:
396:
392:
388:
384:
379:
377:
373:
369:
365:
361:
357:
352:
350:
332:
328:
303:
294:
261:
257:
253:
249:
245:
241:
237:
233:
220:
216:
212:
208:
207:
201:
199:
195:
173:
160:
156:
153:
135:
122:
118:
114:
111:
107:
103:
99:
95:
91:
88:
85:
81:
77:
73:
72:
71:
69:
59:
57:
53:
49:
45:
41:
37:
33:
29:
22:
3492:Georg Cantor
3487:Paul Bernays
3418:MorseâKelley
3393:
3338:
3326:
3325:Subset
3272:hereditarily
3234:Venn diagram
3192:ordered pair
3107:Intersection
3051:Axiom schema
2881:
2679:Ultraproduct
2526:Model theory
2491:Independence
2427:Formal proof
2419:Proof theory
2402:
2375:
2332:real numbers
2304:second-order
2215:Substitution
2092:Metalanguage
2033:conservative
2006:Axiom schema
1950:Constructive
1920:MorseâKelley
1886:Set theories
1865:Aleph number
1858:inaccessible
1764:Grothendieck
1698:
1648:intersection
1535:Higher-order
1523:Second-order
1469:Truth tables
1426:Venn diagram
1209:Formal proof
1134:
1109:
1106:Jech, Thomas
1080:
1077:Halmos, Paul
1071:Bibliography
1057:. Retrieved
1053:
1043:
1014:Aleph number
1002:
764:
728:incomparable
722:Without the
721:
644:. In 1900,
617:is equal to
588:Georg Cantor
529:
457:
394:
390:
386:
380:
375:
371:
359:
353:
255:
236:real numbers
231:
229:
218:
214:
210:
196:without the
191:
158:
120:
109:
101:
97:
89:
83:
74:There is no
67:
65:
36:infinite set
31:
25:
3517:Thomas Jech
3360:Alternative
3339:Uncountable
3293:Ultrafilter
3152:Cardinality
3056:replacement
3004:Determinacy
2789:Type theory
2737:undecidable
2669:Truth value
2556:equivalence
2235:non-logical
1848:Enumeration
1838:Isomorphism
1785:cardinality
1769:Von Neumann
1734:Ultrafilter
1699:Uncountable
1633:equivalence
1550:Quantifiers
1540:Fixed-point
1509:First-order
1389:Consistency
1374:Proposition
1351:Traditional
1322:Lindström's
1312:Compactness
1254:Type theory
1199:Cardinality
1133:Proof that
1019:Beth number
650:23 problems
117:cardinality
28:mathematics
3557:Categories
3512:Kurt Gödel
3497:Paul Cohen
3334:Transitive
3102:Identities
3086:Complement
3073:Operations
3034:Regularity
2972:Adjunction
2931:Set theory
2600:elementary
2293:arithmetic
2161:Quantifier
2139:functional
2011:Expression
1729:Transitive
1673:identities
1658:complement
1591:hereditary
1574:Set theory
1110:Set Theory
1059:2020-09-05
1035:References
702:set theory
356:Cantor set
204:Properties
152:aleph-null
78:(hence no
52:aleph-null
3445:Paradoxes
3365:Axiomatic
3344:Universal
3320:Singleton
3315:Recursive
3258:Countable
3253:Amorphous
3112:Power set
3029:Power set
2987:dependent
2982:countable
2871:Supertask
2774:Recursion
2732:decidable
2566:saturated
2544:of models
2467:deductive
2462:axiomatic
2382:Hilbert's
2369:Euclidean
2350:canonical
2273:axiomatic
2205:Signature
2134:Predicate
2023:Extension
1945:Ackermann
1870:Operation
1749:Universal
1739:Recursive
1714:Singleton
1709:Inhabited
1694:Countable
1684:Types of
1668:power set
1638:partition
1555:Predicate
1501:Predicate
1416:Syllogism
1406:Soundness
1379:Inference
1369:Tautology
1271:paradoxes
985:ω
955:ω
923:ω
905:ℵ
878:ℵ
874:≥
871:κ
840:ℵ
833:κ
803:ℵ
799:≰
796:κ
773:κ
739:ℵ
674:ℶ
661:ℵ
626:ℵ
599:ℶ
568:ℵ
541:ℶ
510:ℵ
495:aleph-one
475:ℵ
437:ℶ
406:ℶ
383:functions
329:ℶ
300:ℵ
244:sequences
170:ℵ
132:ℵ
80:bijection
44:countable
3568:Infinity
3449:Problems
3353:Theories
3329:Superset
3305:Infinite
3134:Concepts
3014:Infinity
2938:Overview
2856:Logicism
2849:timeline
2825:Concrete
2684:Validity
2654:T-schema
2647:Kripke's
2642:Tarski's
2637:semantic
2627:Strength
2576:submodel
2571:spectrum
2539:function
2387:Tarski's
2376:Elements
2363:geometry
2319:Robinson
2240:variable
2225:function
2198:spectrum
2188:Sentence
2144:variable
2087:Language
2040:Relation
2001:Automata
1991:Alphabet
1975:language
1829:-jection
1807:codomain
1793:Function
1754:Universe
1724:Infinite
1628:Relation
1411:Validity
1401:Argument
1299:theorem,
1108:(2002),
1008:See also
896:, where
530:smallest
426:beth-two
366:and has
349:beth-one
226:Examples
157:The set
94:sequence
40:elements
3387:General
3382:Zermelo
3288:subbase
3270: (
3209:Forcing
3187:Element
3159: (
3137:Methods
3024:Pairing
2798:Related
2595:Diagram
2493: (
2472:Hilbert
2457:Systems
2452:Theorem
2330:of the
2275:systems
2055:Formula
2050:Grammar
1966: (
1910:General
1623:Forcing
1608:Element
1528:Monadic
1303:paradox
1244:Theorem
1180:General
528:is the
501:, that
364:fractal
252:subsets
234:of all
217:, then
82:) from
3278:Filter
3268:Finite
3204:Family
3147:Almost
2992:global
2977:Choice
2964:Axioms
2561:finite
2324:Skolem
2277:
2252:Theory
2220:Symbol
2210:String
2193:atomic
2070:ground
2065:closed
2060:atomic
2016:ground
1979:syntax
1875:binary
1802:domain
1719:Finite
1484:finite
1342:Logics
1301:
1249:Theory
1116:
1097:
1089:
42:to be
3370:Naive
3300:Fuzzy
3263:Empty
3246:types
3197:tuple
3167:Class
3161:large
3122:Union
3039:Union
2551:Model
2299:Peano
2156:Proof
1996:Arity
1925:Naive
1812:image
1744:Fuzzy
1704:Empty
1653:union
1598:Class
1239:Model
1229:Lemma
1187:Axiom
385:from
320:, or
286:, or
30:, an
3283:base
2674:Type
2477:list
2281:list
2258:list
2247:Term
2181:rank
2075:open
1969:list
1781:Maps
1686:sets
1545:Free
1515:list
1265:list
1192:list
1114:ISBN
1095:ISBN
1087:ISBN
946:and
836:>
700:for
354:The
115:The
3244:Set
2361:of
2343:of
2291:of
1823:Sur
1797:Map
1604:Ur-
1586:Set
861:and
730:to
708:).
389:to
351:).
246:of
119:of
26:In
3559::
2747:NP
2371::
2365::
2295::
1972:),
1827:Bi
1819:In
1079:,
1052:.
455:.
238:;
154:).
58:.
3327:·
3311:)
3307:(
3274:)
3163:)
2923:e
2916:t
2909:v
2827:/
2742:P
2497:)
2283:)
2279:(
2176:â
2171:!
2166:â
2127:=
2122:â
2117:â
2112:â§
2107:âš
2102:ÂŹ
1825:/
1821:/
1795:/
1606:)
1602:(
1489:â
1479:3
1267:)
1165:e
1158:t
1151:v
1135:R
1062:.
988:.
959:1
933:|
927:1
918:|
914:=
909:1
882:1
849:;
844:0
812:;
807:0
743:0
678:1
670:=
665:1
630:1
603:1
572:1
545:1
514:1
493:(
479:1
464:1
441:1
424:(
410:2
395:R
391:R
387:R
376:R
372:R
360:R
347:(
333:1
304:0
295:2
272:c
256:R
232:R
219:Y
215:Y
211:X
188:.
174:0
159:X
150:(
136:0
121:X
112:.
110:X
102:X
98:X
90:X
84:X
68:X
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.