2896:
2966:
1014:
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.
772:
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.
955:
833:
905:
870:
701:
1275:
329:
295:
982:
766:
653:
595:
537:
502:
253:
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
197:
159:
626:
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356:
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1950:
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2267:
2198:
2075:
1317:
250:
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1925:
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359:
270:
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1930:
708:
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2001:
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2647:
2186:
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1224:
910:
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2549:
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1960:
1880:
1724:
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1281:
1092:
666:
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3182:
3002:
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2404:
2250:
2235:
2208:
2159:
2043:
1978:
1803:
1769:
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1638:
1469:
1446:
393:
3405:
3319:
3239:
3219:
3197:
2769:
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2414:
2132:
1868:
1774:
1633:
1618:
1499:
1474:
728:
50:
2895:
660:
3479:
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3303:
3234:
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3127:
3014:
2742:
2704:
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2385:
2225:
2149:
2127:
1955:
1913:
1812:
1779:
1643:
1431:
1342:
704:
300:
276:
211:, but the equivalence of the third and fourth cannot be proved without additional choice principles.
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116:
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1234:
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175:
137:
86:
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546:
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411:
334:
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1379:
1214:
1124:
1105:
1097:
1096:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
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31:
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2191:
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2011:
1985:
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58:
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1989:
1863:
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1601:
1202:
470:
258:
66:
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3522:
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3263:
2789:
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1974:
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54:
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2016:
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1024:
598:
505:
162:
62:
46:
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1209:
1116:
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1029:
246:
127:
38:
3507:
3375:
3278:
2941:
1729:
1584:
1555:
1361:
1143:
712:
366:
17:
3310:
3273:
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3122:
2881:
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1837:
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1714:
1678:
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1426:
1416:
1389:
385:
has dimension one). This is an example of the following fact: any subset of
90:
1123:, Springer Monographs in Mathematics (3rd millennium ed.), Springer,
469:
A more abstract example of an uncountable set is the set of all countable
2866:
2664:
2112:
1817:
1411:
436:
254:
104:
2462:
1254:
389:
of
Hausdorff dimension strictly greater than zero must be uncountable.
374:
203:
The first three of these characterizations can be proven equivalent in
111:, there exists at least one element of X not included in it. That is,
77:
There are many equivalent characterizations of uncountability. A set
3335:
3157:
1152:
776:
If the axiom of choice holds, the following conditions on a cardinal
262:
81:
is uncountable if and only if any of the following conditions hold:
3207:
2974:
2006:
1352:
1197:
1104:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
2914:
1156:
30:"Uncountable" redirects here. For the linguistic concept, see
2910:
61:: a set is uncountable if its cardinal number is larger than
392:
Another example of an uncountable set is the set of all
241:
The best known example of an uncountable set is the set
57:. The uncountability of a set is closely related to its
994:
963:
913:
880:
842:
805:
782:
747:
669:
634:
607:
576:
549:
518:
483:
445:
414:
337:
303:
279:
178:
140:
3488:
3451:
3363:
3253:
3141:
3082:
2973:
2948:
2808:
2703:
2535:
2428:
2280:
1973:
1896:
1790:
1694:
1583:
1510:
1445:
1360:
1351:
1273:
1190:
1003:
976:
949:
899:
864:
827:
788:
760:
695:
647:
620:
589:
562:
531:
496:
458:
427:
350:
323:
289:
265:of the set of natural numbers. The cardinality of
191:
153:
601:was the first to propose the question of whether
408:in the sense that the cardinality of this set is
2926:
1168:
8:
570:, the cardinality of the reals, is equal to
404:. This set is even "more uncountable" than
2933:
2919:
2911:
1994:
1589:
1357:
1175:
1161:
1153:
993:
968:
962:
950:{\displaystyle \aleph _{1}=|\omega _{1}|}
942:
936:
927:
918:
912:
891:
879:
853:
841:
828:{\displaystyle \kappa \nleq \aleph _{0};}
816:
804:
781:
752:
746:
687:
674:
668:
639:
633:
612:
606:
581:
575:
554:
548:
543:uncountable cardinal number. Thus either
523:
517:
488:
482:
450:
444:
419:
413:
342:
336:
313:
308:
302:
281:
280:
278:
183:
177:
145:
139:
707:, and is known to be independent of the
659:posed this question as the first of his
1051:
900:{\displaystyle \kappa \geq \aleph _{1}}
865:{\displaystyle \kappa >\aleph _{0};}
696:{\displaystyle \aleph _{1}=\beth _{1}}
172:has cardinality strictly greater than
3574:Basic concepts in infinite set theory
381:greater than zero but less than one (
7:
282:
915:
888:
850:
813:
749:
737:, there might exist cardinalities
671:
636:
578:
520:
485:
477:. The cardinality of Ω is denoted
310:
180:
142:
27:Infinite set that is not countable
25:
2964:
2894:
324:{\displaystyle 2^{\aleph _{0}}}
290:{\displaystyle {\mathfrak {c}}}
134:is neither finite nor equal to
943:
928:
768:(namely, the cardinalities of
508:). It can be shown, using the
97:to the set of natural numbers.
1:
2855:History of mathematical logic
2780:Primitive recursive function
369:is an uncountable subset of
271:cardinality of the continuum
119:from the natural numbers to
115:is nonempty and there is no
103:is nonempty and for every Ï-
977:{\displaystyle \omega _{1}}
761:{\displaystyle \aleph _{0}}
723:Without the axiom of choice
648:{\displaystyle \aleph _{1}}
590:{\displaystyle \aleph _{1}}
532:{\displaystyle \aleph _{1}}
497:{\displaystyle \aleph _{1}}
205:ZermeloâFraenkel set theory
192:{\displaystyle \aleph _{0}}
154:{\displaystyle \aleph _{0}}
3600:
3424:von NeumannâBernaysâGödel
1844:SchröderâBernstein theorem
1571:Monadic predicate calculus
1230:Foundations of mathematics
726:
621:{\displaystyle \beth _{1}}
597:or it is strictly larger.
563:{\displaystyle \beth _{1}}
459:{\displaystyle \beth _{1}}
428:{\displaystyle \beth _{2}}
351:{\displaystyle \beth _{1}}
251:Cantor's diagonal argument
29:
3225:One-to-one correspondence
2962:
2890:
2877:Philosophy of mathematics
2826:Automated theorem proving
1997:
1951:Von NeumannâBernaysâGödel
1592:
1035:First uncountable ordinal
65:, the cardinality of the
1004:{\displaystyle \omega .}
439:), which is larger than
2527:Self-verifying theories
2348:Tarski's axiomatization
1299:Tarski's undefinability
1294:incompleteness theorems
789:{\displaystyle \kappa }
709:ZermeloâFraenkel axioms
49:that contains too many
3183:Constructible universe
3010:Constructibility (V=L)
2901:Mathematics portal
2512:Proof of impossibility
2160:propositional variable
1470:Propositional calculus
1061:"Uncountably Infinite"
1005:
978:
951:
901:
866:
829:
790:
762:
697:
649:
622:
591:
564:
533:
498:
460:
429:
373:. The Cantor set is a
352:
325:
291:
220:If an uncountable set
193:
155:
3406:Principia Mathematica
3240:Transfinite induction
3099:(i.e. set difference)
2770:Kolmogorov complexity
2723:Computably enumerable
2623:Model complete theory
2415:Principia Mathematica
1475:Propositional formula
1304:BanachâTarski paradox
1065:mathworld.wolfram.com
1006:
979:
952:
902:
867:
830:
791:
763:
729:Dedekind-infinite set
698:
663:. The statement that
650:
623:
592:
565:
534:
499:
461:
430:
353:
326:
292:
194:
156:
3480:Burali-Forti paradox
3235:Set-builder notation
3188:Continuum hypothesis
3128:Symmetric difference
2718:ChurchâTuring thesis
2705:Computability theory
1914:continuum hypothesis
1432:Square of opposition
1290:Gödel's completeness
1112:(Paperback edition).
992:
961:
911:
878:
840:
803:
780:
745:
705:continuum hypothesis
667:
632:
605:
574:
547:
516:
481:
443:
412:
335:
301:
277:
269:is often called the
176:
138:
45:, informally, is an
3441:TarskiâGrothendieck
2872:Mathematical object
2763:P versus NP problem
2728:Computable function
2522:Reverse mathematics
2448:Logical consequence
2325:primitive recursive
2320:elementary function
2093:Free/bound variable
1946:TarskiâGrothendieck
1465:Logical connectives
1395:Logical equivalence
1245:Logical consequence
1059:Weisstein, Eric W.
473:, denoted by Ω or Ï
379:Hausdorff dimension
261:and the set of all
224:is a subset of set
117:surjective function
3030:Limitation of size
2670:Transfer principle
2633:Semantics of logic
2618:Categorical theory
2594:Non-standard model
2108:Logical connective
1235:Information theory
1184:Mathematical logic
1040:Injective function
1001:
974:
947:
897:
862:
825:
786:
758:
703:is now called the
693:
645:
618:
587:
560:
529:
494:
456:
425:
348:
321:
287:
189:
151:
87:injective function
3561:
3560:
3470:Russell's paradox
3419:ZermeloâFraenkel
3320:Dedekind-infinite
3193:Diagonal argument
3092:Cartesian product
2956:Set (mathematics)
2908:
2907:
2840:Abstract category
2643:Theories of truth
2453:Rule of inference
2443:Natural deduction
2424:
2423:
1969:
1968:
1674:Cartesian product
1579:
1578:
1485:Many-valued logic
1460:Boolean functions
1343:Russell's paradox
1318:diagonal argument
1215:First-order logic
1110:978-1-61427-131-4
273:, and denoted by
73:Characterizations
16:(Redirected from
3591:
3584:Cardinal numbers
3543:Bertrand Russell
3533:John von Neumann
3518:Abraham Fraenkel
3513:Richard Dedekind
3475:Suslin's problem
3386:Cantor's theorem
3103:De Morgan's laws
2968:
2935:
2928:
2921:
2912:
2899:
2898:
2850:History of logic
2845:Category of sets
2738:Decision problem
2517:Ordinal analysis
2458:Sequent calculus
2356:Boolean algebras
2296:
2295:
2270:
2241:logical/constant
1995:
1981:
1904:ZermeloâFraenkel
1655:Set operations:
1590:
1527:
1358:
1338:LöwenheimâSkolem
1225:Formal semantics
1177:
1170:
1163:
1154:
1133:
1093:Naive Set Theory
1075:
1074:
1072:
1071:
1056:
1010:
1008:
1007:
1002:
983:
981:
980:
975:
973:
972:
956:
954:
953:
948:
946:
941:
940:
931:
923:
922:
906:
904:
903:
898:
896:
895:
871:
869:
868:
863:
858:
857:
834:
832:
831:
826:
821:
820:
796:are equivalent:
795:
793:
792:
787:
767:
765:
764:
759:
757:
756:
702:
700:
699:
694:
692:
691:
679:
678:
654:
652:
651:
646:
644:
643:
627:
625:
624:
619:
617:
616:
596:
594:
593:
588:
586:
585:
569:
567:
566:
561:
559:
558:
538:
536:
535:
530:
528:
527:
503:
501:
500:
495:
493:
492:
465:
463:
462:
457:
455:
454:
434:
432:
431:
426:
424:
423:
357:
355:
354:
349:
347:
346:
330:
328:
327:
322:
320:
319:
318:
317:
296:
294:
293:
288:
286:
285:
198:
196:
195:
190:
188:
187:
160:
158:
157:
152:
150:
149:
32:Uncountable noun
21:
3599:
3598:
3594:
3593:
3592:
3590:
3589:
3588:
3564:
3563:
3562:
3557:
3484:
3463:
3447:
3412:New Foundations
3359:
3249:
3168:Cardinal number
3151:
3137:
3078:
2969:
2960:
2944:
2939:
2909:
2904:
2893:
2886:
2831:Category theory
2821:Algebraic logic
2804:
2775:Lambda calculus
2713:Church encoding
2699:
2675:Truth predicate
2531:
2497:Complete theory
2420:
2289:
2285:
2281:
2276:
2268:
1988: and
1984:
1979:
1965:
1941:New Foundations
1909:axiom of choice
1892:
1854:Gödel numbering
1794: and
1786:
1690:
1575:
1525:
1506:
1455:Boolean algebra
1441:
1405:Equiconsistency
1370:Classical logic
1347:
1328:Halting problem
1316: and
1292: and
1280: and
1279:
1274:Theorems (
1269:
1186:
1181:
1140:
1131:
1115:
1084:
1079:
1078:
1069:
1067:
1058:
1057:
1053:
1048:
1021:
990:
989:
986:initial ordinal
964:
959:
958:
932:
914:
909:
908:
887:
876:
875:
849:
838:
837:
812:
801:
800:
778:
777:
770:Dedekind-finite
748:
743:
742:
735:axiom of choice
731:
725:
717:axiom of choice
715:(including the
683:
670:
665:
664:
635:
630:
629:
608:
603:
602:
577:
572:
571:
550:
545:
544:
519:
514:
513:
510:axiom of choice
484:
479:
478:
476:
471:ordinal numbers
446:
441:
440:
415:
410:
409:
338:
333:
332:
309:
304:
299:
298:
275:
274:
259:natural numbers
239:
232:is uncountable.
217:
209:axiom of choice
179:
174:
173:
141:
136:
135:
107:of elements of
75:
67:natural numbers
59:cardinal number
43:uncountable set
35:
28:
23:
22:
15:
12:
11:
5:
3597:
3595:
3587:
3586:
3581:
3576:
3566:
3565:
3559:
3558:
3556:
3555:
3550:
3548:Thoralf Skolem
3545:
3540:
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3494:
3492:
3486:
3485:
3483:
3482:
3477:
3472:
3466:
3464:
3462:
3461:
3458:
3452:
3449:
3448:
3446:
3445:
3444:
3443:
3438:
3433:
3432:
3431:
3416:
3415:
3414:
3402:
3401:
3400:
3389:
3388:
3383:
3378:
3373:
3367:
3365:
3361:
3360:
3358:
3357:
3352:
3347:
3342:
3333:
3328:
3323:
3313:
3308:
3307:
3306:
3301:
3296:
3286:
3276:
3271:
3266:
3260:
3258:
3251:
3250:
3248:
3247:
3242:
3237:
3232:
3230:Ordinal number
3227:
3222:
3217:
3212:
3211:
3210:
3205:
3195:
3190:
3185:
3180:
3175:
3165:
3160:
3154:
3152:
3150:
3149:
3146:
3142:
3139:
3138:
3136:
3135:
3130:
3125:
3120:
3115:
3110:
3108:Disjoint union
3105:
3100:
3094:
3088:
3086:
3080:
3079:
3077:
3076:
3075:
3074:
3069:
3058:
3057:
3055:Martin's axiom
3052:
3047:
3042:
3037:
3032:
3027:
3022:
3020:Extensionality
3017:
3012:
3007:
3006:
3005:
3000:
2995:
2985:
2979:
2977:
2971:
2970:
2963:
2961:
2959:
2958:
2952:
2950:
2946:
2945:
2940:
2938:
2937:
2930:
2923:
2915:
2906:
2905:
2891:
2888:
2887:
2885:
2884:
2879:
2874:
2869:
2864:
2863:
2862:
2852:
2847:
2842:
2833:
2828:
2823:
2818:
2816:Abstract logic
2812:
2810:
2806:
2805:
2803:
2802:
2797:
2795:Turing machine
2792:
2787:
2782:
2777:
2772:
2767:
2766:
2765:
2760:
2755:
2750:
2745:
2735:
2733:Computable set
2730:
2725:
2720:
2715:
2709:
2707:
2701:
2700:
2698:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2661:
2660:
2655:
2650:
2640:
2635:
2630:
2628:Satisfiability
2625:
2620:
2615:
2614:
2613:
2603:
2602:
2601:
2591:
2590:
2589:
2584:
2579:
2574:
2569:
2559:
2558:
2557:
2552:
2545:Interpretation
2541:
2539:
2533:
2532:
2530:
2529:
2524:
2519:
2514:
2509:
2499:
2494:
2493:
2492:
2491:
2490:
2480:
2475:
2465:
2460:
2455:
2450:
2445:
2440:
2434:
2432:
2426:
2425:
2422:
2421:
2419:
2418:
2410:
2409:
2408:
2407:
2402:
2401:
2400:
2395:
2390:
2370:
2369:
2368:
2366:minimal axioms
2363:
2352:
2351:
2350:
2339:
2338:
2337:
2332:
2327:
2322:
2317:
2312:
2299:
2297:
2278:
2277:
2275:
2274:
2273:
2272:
2260:
2255:
2254:
2253:
2248:
2243:
2238:
2228:
2223:
2218:
2213:
2212:
2211:
2206:
2196:
2195:
2194:
2189:
2184:
2179:
2169:
2164:
2163:
2162:
2157:
2152:
2142:
2141:
2140:
2135:
2130:
2125:
2120:
2115:
2105:
2100:
2095:
2090:
2089:
2088:
2083:
2078:
2073:
2063:
2058:
2056:Formation rule
2053:
2048:
2047:
2046:
2041:
2031:
2030:
2029:
2019:
2014:
2009:
2004:
1998:
1992:
1975:Formal systems
1971:
1970:
1967:
1966:
1964:
1963:
1958:
1953:
1948:
1943:
1938:
1933:
1928:
1923:
1918:
1917:
1916:
1911:
1900:
1898:
1894:
1893:
1891:
1890:
1889:
1888:
1878:
1873:
1872:
1871:
1864:Large cardinal
1861:
1856:
1851:
1846:
1841:
1827:
1826:
1825:
1820:
1815:
1800:
1798:
1788:
1787:
1785:
1784:
1783:
1782:
1777:
1772:
1762:
1757:
1752:
1747:
1742:
1737:
1732:
1727:
1722:
1717:
1712:
1707:
1701:
1699:
1692:
1691:
1689:
1688:
1687:
1686:
1681:
1676:
1671:
1666:
1661:
1653:
1652:
1651:
1646:
1636:
1631:
1629:Extensionality
1626:
1624:Ordinal number
1621:
1611:
1606:
1605:
1604:
1593:
1587:
1581:
1580:
1577:
1576:
1574:
1573:
1568:
1563:
1558:
1553:
1548:
1543:
1542:
1541:
1531:
1530:
1529:
1516:
1514:
1508:
1507:
1505:
1504:
1503:
1502:
1497:
1492:
1482:
1477:
1472:
1467:
1462:
1457:
1451:
1449:
1443:
1442:
1440:
1439:
1434:
1429:
1424:
1419:
1414:
1409:
1408:
1407:
1397:
1392:
1387:
1382:
1377:
1372:
1366:
1364:
1355:
1349:
1348:
1346:
1345:
1340:
1335:
1330:
1325:
1320:
1308:Cantor's
1306:
1301:
1296:
1286:
1284:
1271:
1270:
1268:
1267:
1262:
1257:
1252:
1247:
1242:
1237:
1232:
1227:
1222:
1217:
1212:
1207:
1206:
1205:
1194:
1192:
1188:
1187:
1182:
1180:
1179:
1172:
1165:
1157:
1151:
1150:
1148:is uncountable
1139:
1138:External links
1136:
1135:
1134:
1129:
1113:
1083:
1080:
1077:
1076:
1050:
1049:
1047:
1044:
1043:
1042:
1037:
1032:
1027:
1020:
1017:
1012:
1011:
1000:
997:
971:
967:
945:
939:
935:
930:
926:
921:
917:
894:
890:
886:
883:
873:
861:
856:
852:
848:
845:
835:
824:
819:
815:
811:
808:
785:
755:
751:
727:Main article:
724:
721:
690:
686:
682:
677:
673:
642:
638:
615:
611:
584:
580:
557:
553:
526:
522:
491:
487:
474:
453:
449:
422:
418:
345:
341:
316:
312:
307:
284:
238:
235:
234:
233:
216:
213:
201:
200:
186:
182:
166:
148:
144:
124:
98:
74:
71:
26:
24:
18:Uncountability
14:
13:
10:
9:
6:
4:
3:
2:
3596:
3585:
3582:
3580:
3577:
3575:
3572:
3571:
3569:
3554:
3553:Ernst Zermelo
3551:
3549:
3546:
3544:
3541:
3539:
3538:Willard Quine
3536:
3534:
3531:
3529:
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3495:
3493:
3491:
3490:Set theorists
3487:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3459:
3457:
3454:
3453:
3450:
3442:
3439:
3437:
3436:KripkeâPlatek
3434:
3430:
3427:
3426:
3425:
3422:
3421:
3420:
3417:
3413:
3410:
3409:
3408:
3407:
3403:
3399:
3396:
3395:
3394:
3391:
3390:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3368:
3366:
3362:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3339:
3334:
3332:
3329:
3327:
3324:
3321:
3317:
3314:
3312:
3309:
3305:
3302:
3300:
3297:
3295:
3292:
3291:
3290:
3287:
3284:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3259:
3256:
3252:
3246:
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3209:
3206:
3204:
3201:
3200:
3199:
3196:
3194:
3191:
3189:
3186:
3184:
3181:
3179:
3176:
3173:
3169:
3166:
3164:
3161:
3159:
3156:
3155:
3153:
3147:
3144:
3143:
3140:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3098:
3095:
3093:
3090:
3089:
3087:
3085:
3081:
3073:
3072:specification
3070:
3068:
3065:
3064:
3063:
3060:
3059:
3056:
3053:
3051:
3048:
3046:
3043:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3004:
3001:
2999:
2996:
2994:
2991:
2990:
2989:
2986:
2984:
2981:
2980:
2978:
2976:
2972:
2967:
2957:
2954:
2953:
2951:
2947:
2943:
2936:
2931:
2929:
2924:
2922:
2917:
2916:
2913:
2903:
2902:
2897:
2889:
2883:
2880:
2878:
2875:
2873:
2870:
2868:
2865:
2861:
2858:
2857:
2856:
2853:
2851:
2848:
2846:
2843:
2841:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2813:
2811:
2807:
2801:
2798:
2796:
2793:
2791:
2790:Recursive set
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2740:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2710:
2708:
2706:
2702:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2659:
2656:
2654:
2651:
2649:
2646:
2645:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2616:
2612:
2609:
2608:
2607:
2604:
2600:
2599:of arithmetic
2597:
2596:
2595:
2592:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2564:
2563:
2560:
2556:
2553:
2551:
2548:
2547:
2546:
2543:
2542:
2540:
2538:
2534:
2528:
2525:
2523:
2520:
2518:
2515:
2513:
2510:
2507:
2506:from ZFC
2503:
2500:
2498:
2495:
2489:
2486:
2485:
2484:
2481:
2479:
2476:
2474:
2471:
2470:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2435:
2433:
2431:
2427:
2417:
2416:
2412:
2411:
2406:
2405:non-Euclidean
2403:
2399:
2396:
2394:
2391:
2389:
2388:
2384:
2383:
2381:
2378:
2377:
2375:
2371:
2367:
2364:
2362:
2359:
2358:
2357:
2353:
2349:
2346:
2345:
2344:
2340:
2336:
2333:
2331:
2328:
2326:
2323:
2321:
2318:
2316:
2313:
2311:
2308:
2307:
2305:
2301:
2300:
2298:
2293:
2287:
2282:Example
2279:
2271:
2266:
2265:
2264:
2261:
2259:
2256:
2252:
2249:
2247:
2244:
2242:
2239:
2237:
2234:
2233:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2210:
2207:
2205:
2202:
2201:
2200:
2197:
2193:
2190:
2188:
2185:
2183:
2180:
2178:
2175:
2174:
2173:
2170:
2168:
2165:
2161:
2158:
2156:
2153:
2151:
2148:
2147:
2146:
2143:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2110:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2068:
2067:
2064:
2062:
2059:
2057:
2054:
2052:
2049:
2045:
2042:
2040:
2039:by definition
2037:
2036:
2035:
2032:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1999:
1996:
1993:
1991:
1987:
1982:
1976:
1972:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1932:
1929:
1927:
1926:KripkeâPlatek
1924:
1922:
1919:
1915:
1912:
1910:
1907:
1906:
1905:
1902:
1901:
1899:
1895:
1887:
1884:
1883:
1882:
1879:
1877:
1874:
1870:
1867:
1866:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1839:
1835:
1831:
1828:
1824:
1821:
1819:
1816:
1814:
1811:
1810:
1809:
1805:
1802:
1801:
1799:
1797:
1793:
1789:
1781:
1778:
1776:
1773:
1771:
1770:constructible
1768:
1767:
1766:
1763:
1761:
1758:
1756:
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1736:
1733:
1731:
1728:
1726:
1723:
1721:
1718:
1716:
1713:
1711:
1708:
1706:
1703:
1702:
1700:
1698:
1693:
1685:
1682:
1680:
1677:
1675:
1672:
1670:
1667:
1665:
1662:
1660:
1657:
1656:
1654:
1650:
1647:
1645:
1642:
1641:
1640:
1637:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1616:
1612:
1610:
1607:
1603:
1600:
1599:
1598:
1595:
1594:
1591:
1588:
1586:
1582:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1552:
1549:
1547:
1544:
1540:
1537:
1536:
1535:
1532:
1528:
1523:
1522:
1521:
1518:
1517:
1515:
1513:
1509:
1501:
1498:
1496:
1493:
1491:
1488:
1487:
1486:
1483:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1463:
1461:
1458:
1456:
1453:
1452:
1450:
1448:
1447:Propositional
1444:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1406:
1403:
1402:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1375:Logical truth
1373:
1371:
1368:
1367:
1365:
1363:
1359:
1356:
1354:
1350:
1344:
1341:
1339:
1336:
1334:
1331:
1329:
1326:
1324:
1321:
1319:
1315:
1311:
1307:
1305:
1302:
1300:
1297:
1295:
1291:
1288:
1287:
1285:
1283:
1277:
1272:
1266:
1263:
1261:
1258:
1256:
1253:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1231:
1228:
1226:
1223:
1221:
1218:
1216:
1213:
1211:
1208:
1204:
1201:
1200:
1199:
1196:
1195:
1193:
1189:
1185:
1178:
1173:
1171:
1166:
1164:
1159:
1158:
1155:
1149:
1147:
1142:
1141:
1137:
1132:
1130:3-540-44085-2
1126:
1122:
1118:
1114:
1111:
1107:
1103:
1102:0-387-90092-6
1099:
1095:
1094:
1089:
1086:
1085:
1081:
1066:
1062:
1055:
1052:
1045:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1022:
1018:
1016:
998:
995:
988:greater than
987:
984:is the least
969:
965:
937:
933:
924:
919:
892:
884:
881:
874:
859:
854:
846:
843:
836:
822:
817:
809:
806:
799:
798:
797:
783:
774:
771:
753:
740:
736:
730:
722:
720:
718:
714:
710:
706:
688:
684:
680:
675:
662:
658:
657:David Hilbert
640:
613:
609:
600:
582:
555:
551:
542:
524:
511:
507:
489:
472:
467:
451:
447:
438:
420:
416:
407:
403:
399:
395:
390:
388:
384:
380:
376:
372:
368:
363:
361:
343:
339:
314:
305:
272:
268:
264:
260:
256:
252:
248:
244:
236:
231:
227:
223:
219:
218:
214:
212:
210:
206:
184:
171:
167:
164:
146:
133:
129:
125:
122:
118:
114:
110:
106:
102:
99:
96:
92:
88:
84:
83:
82:
80:
72:
70:
68:
64:
60:
56:
52:
48:
44:
40:
33:
19:
3503:Georg Cantor
3498:Paul Bernays
3429:MorseâKelley
3404:
3349:
3337:
3336:Subset
3283:hereditarily
3245:Venn diagram
3203:ordered pair
3118:Intersection
3062:Axiom schema
2892:
2690:Ultraproduct
2537:Model theory
2502:Independence
2438:Formal proof
2430:Proof theory
2413:
2386:
2343:real numbers
2315:second-order
2226:Substitution
2103:Metalanguage
2044:conservative
2017:Axiom schema
1961:Constructive
1931:MorseâKelley
1897:Set theories
1876:Aleph number
1869:inaccessible
1775:Grothendieck
1709:
1659:intersection
1546:Higher-order
1534:Second-order
1480:Truth tables
1437:Venn diagram
1220:Formal proof
1145:
1120:
1117:Jech, Thomas
1091:
1088:Halmos, Paul
1082:Bibliography
1068:. Retrieved
1064:
1054:
1025:Aleph number
1013:
775:
739:incomparable
733:Without the
732:
655:. In 1900,
628:is equal to
599:Georg Cantor
540:
468:
405:
401:
397:
391:
386:
382:
370:
364:
266:
247:real numbers
242:
240:
229:
225:
221:
207:without the
202:
169:
131:
120:
112:
108:
100:
94:
85:There is no
78:
76:
47:infinite set
42:
36:
3528:Thomas Jech
3371:Alternative
3350:Uncountable
3304:Ultrafilter
3163:Cardinality
3067:replacement
3015:Determinacy
2800:Type theory
2748:undecidable
2680:Truth value
2567:equivalence
2246:non-logical
1859:Enumeration
1849:Isomorphism
1796:cardinality
1780:Von Neumann
1745:Ultrafilter
1710:Uncountable
1644:equivalence
1561:Quantifiers
1551:Fixed-point
1520:First-order
1400:Consistency
1385:Proposition
1362:Traditional
1333:Lindström's
1323:Compactness
1265:Type theory
1210:Cardinality
1144:Proof that
1030:Beth number
661:23 problems
128:cardinality
39:mathematics
3568:Categories
3523:Kurt Gödel
3508:Paul Cohen
3345:Transitive
3113:Identities
3097:Complement
3084:Operations
3045:Regularity
2983:Adjunction
2942:Set theory
2611:elementary
2304:arithmetic
2172:Quantifier
2150:functional
2022:Expression
1740:Transitive
1684:identities
1669:complement
1602:hereditary
1585:Set theory
1121:Set Theory
1070:2020-09-05
1046:References
713:set theory
367:Cantor set
215:Properties
163:aleph-null
89:(hence no
63:aleph-null
3456:Paradoxes
3376:Axiomatic
3355:Universal
3331:Singleton
3326:Recursive
3269:Countable
3264:Amorphous
3123:Power set
3040:Power set
2998:dependent
2993:countable
2882:Supertask
2785:Recursion
2743:decidable
2577:saturated
2555:of models
2478:deductive
2473:axiomatic
2393:Hilbert's
2380:Euclidean
2361:canonical
2284:axiomatic
2216:Signature
2145:Predicate
2034:Extension
1956:Ackermann
1881:Operation
1760:Universal
1750:Recursive
1725:Singleton
1720:Inhabited
1705:Countable
1695:Types of
1679:power set
1649:partition
1566:Predicate
1512:Predicate
1427:Syllogism
1417:Soundness
1390:Inference
1380:Tautology
1282:paradoxes
996:ω
966:ω
934:ω
916:ℵ
889:ℵ
885:≥
882:κ
851:ℵ
844:κ
814:ℵ
810:≰
807:κ
784:κ
750:ℵ
685:ℶ
672:ℵ
637:ℵ
610:ℶ
579:ℵ
552:ℶ
521:ℵ
506:aleph-one
486:ℵ
448:ℶ
417:ℶ
394:functions
340:ℶ
311:ℵ
255:sequences
181:ℵ
143:ℵ
91:bijection
55:countable
3579:Infinity
3460:Problems
3364:Theories
3340:Superset
3316:Infinite
3145:Concepts
3025:Infinity
2949:Overview
2867:Logicism
2860:timeline
2836:Concrete
2695:Validity
2665:T-schema
2658:Kripke's
2653:Tarski's
2648:semantic
2638:Strength
2587:submodel
2582:spectrum
2550:function
2398:Tarski's
2387:Elements
2374:geometry
2330:Robinson
2251:variable
2236:function
2209:spectrum
2199:Sentence
2155:variable
2098:Language
2051:Relation
2012:Automata
2002:Alphabet
1986:language
1840:-jection
1818:codomain
1804:Function
1765:Universe
1735:Infinite
1639:Relation
1422:Validity
1412:Argument
1310:theorem,
1119:(2002),
1019:See also
907:, where
541:smallest
437:beth-two
377:and has
360:beth-one
237:Examples
168:The set
105:sequence
51:elements
3398:General
3393:Zermelo
3299:subbase
3281: (
3220:Forcing
3198:Element
3170: (
3148:Methods
3035:Pairing
2809:Related
2606:Diagram
2504: (
2483:Hilbert
2468:Systems
2463:Theorem
2341:of the
2286:systems
2066:Formula
2061:Grammar
1977: (
1921:General
1634:Forcing
1619:Element
1539:Monadic
1314:paradox
1255:Theorem
1191:General
539:is the
512:, that
375:fractal
263:subsets
245:of all
228:, then
93:) from
3289:Filter
3279:Finite
3215:Family
3158:Almost
3003:global
2988:Choice
2975:Axioms
2572:finite
2335:Skolem
2288:
2263:Theory
2231:Symbol
2221:String
2204:atomic
2081:ground
2076:closed
2071:atomic
2027:ground
1990:syntax
1886:binary
1813:domain
1730:Finite
1495:finite
1353:Logics
1312:
1260:Theory
1127:
1108:
1100:
53:to be
3381:Naive
3311:Fuzzy
3274:Empty
3257:types
3208:tuple
3178:Class
3172:large
3133:Union
3050:Union
2562:Model
2310:Peano
2167:Proof
2007:Arity
1936:Naive
1823:image
1755:Fuzzy
1715:Empty
1664:union
1609:Class
1250:Model
1240:Lemma
1198:Axiom
396:from
331:, or
297:, or
41:, an
3294:base
2685:Type
2488:list
2292:list
2269:list
2258:Term
2192:rank
2086:open
1980:list
1792:Maps
1697:sets
1556:Free
1526:list
1276:list
1203:list
1125:ISBN
1106:ISBN
1098:ISBN
957:and
847:>
711:for
365:The
126:The
3255:Set
2372:of
2354:of
2302:of
1834:Sur
1808:Map
1615:Ur-
1597:Set
872:and
741:to
719:).
400:to
362:).
257:of
130:of
37:In
3570::
2758:NP
2382::
2376::
2306::
1983:),
1838:Bi
1830:In
1090:,
1063:.
466:.
249:;
165:).
69:.
3338:·
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2177:â
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2118:âš
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1073:.
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970:1
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938:1
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925:=
920:1
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860:;
855:0
823:;
818:0
754:0
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681:=
676:1
641:1
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490:1
475:1
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435:(
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406:R
402:R
398:R
387:R
383:R
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358:(
344:1
315:0
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283:c
267:R
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230:Y
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123:.
121:X
113:X
109:X
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95:X
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34:.
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