1111:, viewed as sets of positions where the expansion is one, almost give a one-to-one mapping from subsets of a countable set (the set of positions in the expansions) to real numbers, but it fails to be one-to-one for numbers with terminating binary expansions, which can also be represented by a non-terminating expansion that ends in a repeating sequence of 1s. This can be made into a one-to-one mapping by that adds one to the non-terminating repeating-1 expansions, mapping them into
2027:
1542:
1663:
1391:
3654:
1861:
1400:
2724:
3463:
1217:
388:, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the
3346:
147:
1555:
357:
2091:
3303:
554:
2867:
2210:
3737:
1737:
3571:
1260:
2262:
841:
2517:
4512:
4159:
4080:
3699:
845:
In practice, this means that there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. For more information on this topic, see
4193:
3807:
3771:
2438:
2368:
1271:
712:
3566:
95:
2315:
2952:
3557:
1007:
951:
3493:
1694:
4120:
3927:
3893:
3387:
460:
4019:
3253:
3031:
2891:
2769:
2656:
2554:
2392:
1065:
1041:
768:
674:
920:
4947:
4464:
4442:
4420:
4392:
4370:
4342:
4320:
4291:
4265:
4237:
4215:
4045:
3984:
3957:
3858:
3836:
3414:
3228:
3204:
3093:
3061:
2466:
1824:
1794:
976:
424:
266:
244:
218:
196:
171:
58:
3179:
2979:
2918:
2796:
2632:
2585:
1851:
799:
649:
614:
583:
294:
2022:{\displaystyle {\mathfrak {c}}\leq \aleph _{0}\cdot 10^{\aleph _{0}}\leq 2^{\aleph _{0}}\cdot {(2^{4})}^{\aleph _{0}}=2^{\aleph _{0}+4\cdot \aleph _{0}}=2^{\aleph _{0}}}
1537:{\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\aleph _{0}}^{\aleph _{0}}=n^{\aleph _{0}}={\mathfrak {c}}^{n}=\aleph _{0}{\mathfrak {c}}=n{\mathfrak {c}}={\mathfrak {c}}}
2131:
2163:
1141:
1109:
3522:
2822:
2605:
3129:
1796:. This makes it sensible to talk about, say, the first, the one-hundredth, or the millionth decimal place of Ï. Since the natural numbers have cardinality
1068:
1067:. This can be shown by providing one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, and applying the
4527:
2662:
1071:
according to which two sets with one-to-one mappings in both directions have the same cardinality. In one direction, reals can be equated with
846:
366:
3421:
1148:
3311:
1658:{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=(2^{\aleph _{0}})^{\mathfrak {c}}=2^{{\mathfrak {c}}\times \aleph _{0}}=2^{\mathfrak {c}}}
108:
4397:
2825:
302:
4956:
4936:
4918:
4729:
4694:
2036:
3261:
2729:
719:
471:
2824:
being the continuum hypothesis). The same is true for most other alephs, although in some cases, equality can be ruled out by
2835:
2170:
4890:
4608:
3712:
1703:
850:
370:
1224:
4681:. Elements of Logic via Numbers and Sets. Springer Undergraduate Mathematics Series. Springer London. pp. 113â130.
2219:
4603:
369:
of 1874, part of his groundbreaking study of different infinities. The inequality was later stated more simply in his
806:
4598:
1386:{\displaystyle {\mathfrak {c}}^{2}=(2^{\aleph _{0}})^{2}=2^{2\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}}.}
3649:{\displaystyle {\begin{aligned}f\colon \mathbb {R} ^{2}&\to \mathbb {C} \\(a,b)&\mapsto a+bi\end{aligned}}}
2477:
4475:
4129:
4050:
3669:
4989:
4167:
3781:
3745:
2982:
2403:
2320:
1771:
680:
3066:
74:
3183:
Furthermore, the real algebraic numbers and the real transcendental numbers are disjoint sets whose union is
2740:. That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero
2280:
2165:
and consider that decimal fractions containing only 3 or 7 are only a part of the real numbers, then we get
2923:
3142:
3070:
61:
4881:
3528:
981:
925:
4973:
4885:
3470:
1670:
4096:
3903:
3869:
3363:
436:
377:: two sets have the same cardinality if, and only if, there exists a bijective function between them.
4825:
4758:
4297:
4000:
3234:
3012:
2872:
2750:
2637:
2535:
2527:
2373:
1046:
1012:
749:
743:
715:
655:
621:
872:
4994:
4347:
4268:
3813:
1263:
862:
463:
4447:
4425:
4403:
4375:
4353:
4325:
4303:
4274:
4248:
4220:
4198:
4028:
3967:
3940:
3841:
3819:
3397:
3211:
3187:
3076:
3044:
2449:
1799:
1777:
959:
407:
249:
227:
201:
179:
154:
41:
4644:
4090:
3863:
3157:
2986:
2957:
2896:
2774:
2610:
2563:
1829:
777:
627:
592:
561:
374:
272:
3150:
2098:
4999:
4952:
4932:
4914:
4913:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
4861:
4843:
4794:
4776:
4725:
4690:
3136:
2998:
1748:
32:
2136:
4851:
4833:
4784:
4766:
4717:
4682:
4636:
4242:
3146:
2994:
1076:
4656:
1114:
1082:
4652:
3498:
3358:
1856:
Since each real number can be broken into an integer part and a decimal fraction, we get:
771:
742:
to compare the sizes of infinite sets. He famously showed that the set of real numbers is
431:
174:
68:
2801:
4829:
4762:
4546:
3664:
3392:
2990:
2741:
2590:
954:
4856:
4813:
4789:
4746:
3307:
A similar result follows for complex transcendental numbers, once we have proved that
3102:
4983:
4942:
3934:
3096:
2733:
1767:
1763:(This is true even in the case the expansion repeats, as in the first two examples.)
1079:. In the other direction, the binary expansions of numbers in the half-open interval
389:
4965:
4640:
2557:
1072:
735:
617:
586:
401:
362:
269:
65:
36:
865:, which states that the cardinality of any set is strictly less than that of its
4924:
4906:
4686:
4515:
3039:
2272:
739:
28:
4574:
4969:
4123:
3353:
2829:
2737:
20:
4847:
4780:
2719:{\displaystyle \nexists A\quad :\quad \aleph _{0}<|A|<{\mathfrak {c}}.}
624:, which asserts that there are no sets whose cardinality is strictly between
4086:
3960:
3705:
3660:
2469:
866:
221:
4865:
4798:
4771:
4711:
4838:
4721:
4161: â the indicator function chooses elements of each subset to include)
3896:
4468:
the set of all automorphisms of the (discrete) field of complex numbers.
4648:
3458:{\displaystyle \left\vert \mathbb {R} ^{2}\right\vert ={\mathfrak {c}}}
2587:. In other words, the continuum hypothesis states that there is no set
98:
1212:{\displaystyle {\mathfrak {c}}=|\wp (\mathbb {N} )|=2^{\aleph _{0}}.}
3341:{\displaystyle \left\vert \mathbb {C} \right\vert ={\mathfrak {c}}}
3009:
A great many sets studied in mathematics have cardinality equal to
1395:
By using the rules of cardinal arithmetic, one can also show that
142:{\displaystyle {\mathbf {|}}{\mathbf {\mathbb {R} }}{\mathbf {|}}}
2728:
This statement is now known to be independent of the axioms of
861:
A variation of Cantor's diagonal argument can be used to prove
352:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}>\aleph _{0}.}
4929:
Set Theory: The Third
Millennium Edition, Revised and Expanded
714:. The truth or falsity of this hypothesis is undecidable and
4135:
4056:
3675:
2086:{\displaystyle \aleph _{0}+4\cdot \aleph _{0}=\aleph _{0}\,}
3298:{\displaystyle {\mathfrak {c}}-\aleph _{0}={\mathfrak {c}}}
426:, as well as with several other infinite sets, such as any
3417:
Per Cantor's proof of the cardinality of
Euclidean space,
549:{\displaystyle |(a,b)|=|\mathbb {R} |=|\mathbb {R} ^{n}|.}
3153:.) So the cardinality of the real algebraic numbers is
3257:
the cardinality of the real transcendental numbers is
4478:
4450:
4428:
4406:
4378:
4356:
4328:
4306:
4277:
4251:
4223:
4201:
4170:
4132:
4099:
4053:
4031:
4003:
3970:
3943:
3906:
3872:
3844:
3822:
3784:
3748:
3715:
3672:
3569:
3531:
3501:
3473:
3424:
3400:
3366:
3314:
3264:
3237:
3214:
3190:
3160:
3105:
3079:
3047:
3015:
2960:
2926:
2899:
2875:
2862:{\displaystyle {\mathfrak {c}}\neq \aleph _{\omega }}
2838:
2804:
2777:
2753:
2665:
2640:
2613:
2593:
2566:
2538:
2480:
2452:
2406:
2376:
2323:
2283:
2222:
2205:{\displaystyle 2^{\aleph _{0}}\leq {\mathfrak {c}}\,}
2173:
2139:
2101:
2039:
1864:
1832:
1802:
1780:
1706:
1673:
1558:
1403:
1274:
1227:
1151:
1117:
1085:
1049:
1015:
984:
962:
928:
875:
809:
780:
752:
683:
658:
630:
595:
564:
474:
439:
410:
305:
275:
252:
230:
204:
182:
157:
111:
77:
44:
3732:{\displaystyle \mathbb {N} \rightarrow \mathbb {Z} }
1732:{\displaystyle 2^{\mathfrak {c}}>{\mathfrak {c}}}
2277:The sequence of beth numbers is defined by setting
1766:In any given case, the number of decimal places is
1255:{\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}}}
4948:Set Theory: An Introduction to Independence Proofs
4814:"The Independence of the Continuum Hypothesis, Ii"
4506:
4458:
4436:
4414:
4386:
4364:
4336:
4314:
4285:
4259:
4231:
4209:
4187:
4153:
4114:
4074:
4039:
4013:
3978:
3951:
3921:
3887:
3852:
3830:
3801:
3765:
3731:
3693:
3648:
3551:
3516:
3487:
3457:
3408:
3381:
3340:
3297:
3247:
3222:
3198:
3173:
3149:is countably infinite (assign to each formula its
3123:
3087:
3055:
3025:
2973:
2946:
2912:
2885:
2861:
2816:
2790:
2763:
2718:
2650:
2626:
2599:
2579:
2548:
2511:
2460:
2432:
2386:
2362:
2309:
2257:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}\,.}
2256:
2204:
2157:
2125:
2085:
2021:
1845:
1818:
1788:
1731:
1688:
1657:
1536:
1385:
1254:
1211:
1135:
1103:
1059:
1035:
1001:
970:
945:
914:
835:
793:
762:
706:
668:
643:
608:
577:
548:
454:
418:
351:
288:
260:
238:
212:
190:
165:
141:
89:
52:
4627:Stillwell, John (2002). "The continuum problem".
133:
115:
4974:Creative Commons Attribution/Share-Alike License
770:is strictly greater than the cardinality of the
373:in 1891. Cantor defined cardinality in terms of
4818:Proceedings of the National Academy of Sciences
4751:Proceedings of the National Academy of Sciences
4713:Consistency of the Continuum Hypothesis. (AM-3)
3701:(the set of all subsets of the natural numbers)
836:{\displaystyle \aleph _{0}<{\mathfrak {c}}.}
4747:"The Independence of the Continuum Hypothesis"
2512:{\displaystyle 2^{\mathfrak {c}}=\beth _{2}.}
978:is uncountable). In fact, the cardinality of
8:
4507:{\displaystyle 2^{\mathfrak {c}}=\beth _{2}}
4154:{\displaystyle {\mathcal {P}}(\mathbb {R} )}
4075:{\displaystyle {\mathcal {P}}(\mathbb {R} )}
3694:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
2732:with the axiom of choice (ZFC), as shown by
2152:
2140:
2120:
2108:
1747:Every real number has at least one infinite
4677:Johnson, D. L. (1998). "Cardinal Numbers".
4622:
4620:
4618:
4188:{\displaystyle \mathbb {R} ^{\mathbb {R} }}
3802:{\displaystyle \mathbb {R} ^{\mathbb {N} }}
3766:{\displaystyle \mathbb {Z} ^{\mathbb {N} }}
2433:{\displaystyle {\mathfrak {c}}=\beth _{1}.}
2363:{\displaystyle \beth _{k+1}=2^{\beth _{k}}}
707:{\displaystyle {\mathfrak {c}}=\aleph _{1}}
3033:. Some common examples are the following:
1075:, sets of rational numbers, or with their
4855:
4837:
4788:
4770:
4672:
4670:
4668:
4666:
4498:
4484:
4483:
4477:
4452:
4451:
4449:
4430:
4429:
4427:
4408:
4407:
4405:
4380:
4379:
4377:
4358:
4357:
4355:
4330:
4329:
4327:
4308:
4307:
4305:
4279:
4278:
4276:
4253:
4252:
4250:
4225:
4224:
4222:
4203:
4202:
4200:
4179:
4178:
4177:
4173:
4172:
4169:
4144:
4143:
4134:
4133:
4131:
4106:
4105:
4104:
4098:
4093:defined on subsets of the reals (the set
4065:
4064:
4055:
4054:
4052:
4033:
4032:
4030:
4005:
4004:
4002:
3972:
3971:
3969:
3945:
3944:
3942:
3913:
3909:
3908:
3905:
3879:
3875:
3874:
3871:
3846:
3845:
3843:
3824:
3823:
3821:
3793:
3792:
3791:
3787:
3786:
3783:
3757:
3756:
3755:
3751:
3750:
3747:
3725:
3724:
3717:
3716:
3714:
3684:
3683:
3674:
3673:
3671:
3600:
3599:
3586:
3582:
3581:
3570:
3568:
3545:
3544:
3530:
3500:
3481:
3480:
3472:
3449:
3448:
3435:
3431:
3430:
3423:
3402:
3401:
3399:
3373:
3369:
3368:
3365:
3332:
3331:
3320:
3319:
3313:
3289:
3288:
3279:
3266:
3265:
3263:
3239:
3238:
3236:
3216:
3215:
3213:
3192:
3191:
3189:
3165:
3159:
3104:
3081:
3080:
3078:
3049:
3048:
3046:
3017:
3016:
3014:
2965:
2959:
2936:
2931:
2925:
2904:
2898:
2877:
2876:
2874:
2853:
2840:
2839:
2837:
2803:
2782:
2776:
2755:
2754:
2752:
2707:
2706:
2698:
2690:
2681:
2664:
2642:
2641:
2639:
2618:
2612:
2592:
2571:
2565:
2540:
2539:
2537:
2500:
2486:
2485:
2479:
2454:
2453:
2451:
2446:, is the cardinality of the power set of
2421:
2408:
2407:
2405:
2378:
2377:
2375:
2352:
2347:
2328:
2322:
2301:
2288:
2282:
2250:
2242:
2237:
2224:
2223:
2221:
2201:
2195:
2194:
2183:
2178:
2172:
2138:
2100:
2082:
2076:
2063:
2044:
2038:
2011:
2006:
1991:
1972:
1967:
1952:
1947:
1937:
1929:
1917:
1912:
1897:
1892:
1879:
1866:
1865:
1863:
1837:
1831:
1807:
1801:
1782:
1781:
1779:
1743:Alternative explanation for 𝔠 = 2
1723:
1722:
1712:
1711:
1705:
1679:
1678:
1672:
1648:
1647:
1632:
1619:
1618:
1617:
1603:
1602:
1590:
1585:
1568:
1567:
1561:
1560:
1557:
1528:
1527:
1518:
1517:
1505:
1504:
1498:
1485:
1479:
1478:
1466:
1461:
1446:
1441:
1434:
1429:
1417:
1412:
1406:
1405:
1402:
1374:
1373:
1362:
1357:
1341:
1336:
1329:
1316:
1304:
1299:
1283:
1277:
1276:
1273:
1246:
1245:
1236:
1230:
1229:
1226:
1198:
1193:
1181:
1174:
1173:
1162:
1153:
1152:
1150:
1116:
1084:
1051:
1050:
1048:
1025:
1020:
1014:
992:
991:
983:
964:
963:
961:
936:
935:
927:
905:
897:
896:
884:
876:
874:
824:
823:
814:
808:
785:
779:
754:
753:
751:
698:
685:
684:
682:
660:
659:
657:
635:
629:
600:
594:
569:
563:
558:The smallest infinite cardinal number is
538:
532:
528:
527:
521:
513:
509:
508:
503:
495:
475:
473:
446:
442:
441:
438:
412:
411:
409:
340:
325:
320:
307:
306:
304:
280:
274:
254:
253:
251:
232:
231:
229:
206:
205:
203:
184:
183:
181:
159:
158:
156:
132:
131:
130:
124:
123:
122:
121:
114:
113:
112:
110:
90:{\displaystyle {\mathbf {\mathfrak {c}}}}
80:
79:
78:
76:
46:
45:
43:
4964:This article incorporates material from
4877:
4875:
4528:Cardinal characteristic of the continuum
2607:whose cardinality lies strictly between
4538:
1696:is the cardinality of the power set of
220:has the same number of elements as the
3777:the set of sequences of real numbers,
3005:Sets with cardinality of the continuum
2532:The continuum hypothesis asserts that
2310:{\displaystyle \beth _{0}=\aleph _{0}}
297:, the cardinality of the continuum is
246:. Symbolically, if the cardinality of
16:Cardinality of the set of real numbers
2947:{\displaystyle \aleph _{\omega _{1}}}
7:
4568:
4566:
2468:(i.e. the set of all subsets of the
4485:
4006:
3997:Sets with cardinality greater than
3552:{\displaystyle a,b\in \mathbb {R} }
3450:
3333:
3290:
3267:
3240:
3018:
2878:
2841:
2756:
2708:
2643:
2541:
2487:
2409:
2379:
2225:
2196:
1867:
1724:
1713:
1680:
1649:
1620:
1604:
1569:
1562:
1529:
1519:
1506:
1480:
1407:
1375:
1278:
1247:
1231:
1154:
1052:
847:Cantor's first uncountability proof
825:
755:
686:
661:
308:
81:
4547:"Transfinite number | mathematics"
3561:We therefore define the bijection
3276:
3162:
2928:
2901:
2850:
2779:
2678:
2615:
2568:
2298:
2239:
2180:
2073:
2060:
2041:
2008:
1988:
1969:
1949:
1914:
1894:
1876:
1834:
1804:
1629:
1587:
1495:
1463:
1443:
1431:
1414:
1359:
1338:
1301:
1195:
1167:
1069:CantorâBernsteinâSchroeder theorem
1022:
1002:{\displaystyle \wp (\mathbb {N} )}
985:
946:{\displaystyle \wp (\mathbb {N} )}
929:
811:
782:
695:
632:
597:
566:
337:
322:
277:
14:
3488:{\displaystyle c\in \mathbb {C} }
1689:{\displaystyle 2^{\mathfrak {c}}}
4745:Cohen, Paul J. (December 1963).
4115:{\displaystyle 2^{\mathbb {R} }}
3922:{\displaystyle \mathbb {R} ^{n}}
3888:{\displaystyle \mathbb {R} ^{n}}
3708:of integers (i.e. all functions
3382:{\displaystyle \mathbb {R} ^{n}}
1774:with the set of natural numbers
455:{\displaystyle \mathbb {R} ^{n}}
4812:Cohen, Paul J. (January 1964).
4014:{\displaystyle {\mathfrak {c}}}
3248:{\displaystyle {\mathfrak {c}}}
3208:Thus, since the cardinality of
3026:{\displaystyle {\mathfrak {c}}}
2886:{\displaystyle {\mathfrak {c}}}
2764:{\displaystyle {\mathfrak {c}}}
2676:
2672:
2651:{\displaystyle {\mathfrak {c}}}
2549:{\displaystyle {\mathfrak {c}}}
2387:{\displaystyle {\mathfrak {c}}}
1550:is any finite cardinal â„ 2 and
1060:{\displaystyle {\mathfrak {c}}}
1036:{\displaystyle 2^{\aleph _{0}}}
763:{\displaystyle {\mathfrak {c}}}
669:{\displaystyle {\mathfrak {c}}}
4972:, which is licensed under the
4641:10.1080/00029890.2002.11919865
4148:
4140:
4069:
4061:
3721:
3688:
3680:
3627:
3620:
3608:
3596:
3118:
3106:
2699:
2691:
1943:
1930:
1599:
1578:
1313:
1292:
1182:
1178:
1170:
1163:
1130:
1118:
1098:
1086:
996:
988:
940:
932:
915:{\displaystyle |A|<2^{|A|}}
906:
898:
885:
877:
539:
522:
514:
504:
496:
492:
480:
476:
1:
4891:American Mathematical Monthly
4629:American Mathematical Monthly
3993:Sets with greater cardinality
3495:can be uniquely expressed as
2095:On the other hand, if we map
1770:since they can be put into a
380:Between any two real numbers
4966:cardinality of the continuum
4459:{\displaystyle \mathbb {R} }
4437:{\displaystyle \mathbb {Q} }
4415:{\displaystyle \mathbb {N} }
4398:StoneâÄech compactifications
4387:{\displaystyle \mathbb {R} }
4365:{\displaystyle \mathbb {R} }
4337:{\displaystyle \mathbb {R} }
4315:{\displaystyle \mathbb {R} }
4286:{\displaystyle \mathbb {R} }
4260:{\displaystyle \mathbb {R} }
4232:{\displaystyle \mathbb {R} }
4210:{\displaystyle \mathbb {R} }
4040:{\displaystyle \mathbb {R} }
3979:{\displaystyle \mathbb {R} }
3952:{\displaystyle \mathbb {R} }
3853:{\displaystyle \mathbb {R} }
3831:{\displaystyle \mathbb {R} }
3409:{\displaystyle \mathbb {C} }
3223:{\displaystyle \mathbb {R} }
3199:{\displaystyle \mathbb {R} }
3088:{\displaystyle \mathbb {R} }
3056:{\displaystyle \mathbb {R} }
2798:is independent of ZFC (case
2461:{\displaystyle \mathbb {R} }
2031:where we used the fact that
1819:{\displaystyle \aleph _{0},}
1789:{\displaystyle \mathbb {N} }
971:{\displaystyle \mathbb {N} }
722:with axiom of choice (ZFC).
419:{\displaystyle \mathbb {R} }
261:{\displaystyle \mathbb {N} }
239:{\displaystyle \mathbb {N} }
213:{\displaystyle \mathbb {R} }
191:{\displaystyle \mathbb {N} }
166:{\displaystyle \mathbb {R} }
53:{\displaystyle \mathbb {R} }
25:cardinality of the continuum
4687:10.1007/978-1-4471-0603-6_6
4679:Chapter 6: Cardinal numbers
4604:Encyclopedia of Mathematics
4472:These all have cardinality
3174:{\displaystyle \aleph _{0}}
2974:{\displaystyle \omega _{1}}
2913:{\displaystyle \aleph _{1}}
2791:{\displaystyle \aleph _{n}}
2730:ZermeloâFraenkel set theory
2627:{\displaystyle \aleph _{0}}
2580:{\displaystyle \aleph _{1}}
2394:is the second beth number,
1846:{\displaystyle \aleph _{0}}
922:(and so that the power set
794:{\displaystyle \aleph _{0}}
720:ZermeloâFraenkel set theory
644:{\displaystyle \aleph _{0}}
609:{\displaystyle \aleph _{1}}
578:{\displaystyle \aleph _{0}}
289:{\displaystyle \aleph _{0}}
173:are more numerous than the
5016:
4921:(Springer-Verlag edition).
4710:Gödel, Kurt (1940-12-31).
4025:the set of all subsets of
2985:, so it could be either a
2525:
2270:
1262:can be demonstrated using
851:Cantor's diagonal argument
738:introduced the concept of
589:). The second smallest is
2983:first uncountable ordinal
2126:{\displaystyle 2=\{0,1\}}
1853:digits in its expansion.
1772:one-to-one correspondence
1143:. Thus, we conclude that
2522:The continuum hypothesis
4551:Encyclopedia Britannica
4267:, i.e., the set of all
2442:The third beth number,
2158:{\displaystyle \{3,7\}}
718:within the widely used
60:, sometimes called the
4772:10.1073/pnas.50.6.1143
4508:
4460:
4438:
4416:
4388:
4366:
4338:
4316:
4287:
4261:
4233:
4211:
4195:of all functions from
4189:
4155:
4116:
4076:
4041:
4015:
3980:
3953:
3923:
3889:
3854:
3832:
3803:
3767:
3733:
3695:
3650:
3553:
3518:
3489:
3459:
3410:
3383:
3342:
3299:
3249:
3224:
3200:
3175:
3143:transcendental numbers
3125:
3089:
3057:
3027:
2975:
2948:
2914:
2887:
2863:
2818:
2792:
2765:
2720:
2652:
2628:
2601:
2581:
2550:
2513:
2462:
2434:
2388:
2364:
2311:
2258:
2206:
2159:
2127:
2087:
2023:
1847:
1820:
1790:
1733:
1690:
1659:
1538:
1387:
1256:
1221:The cardinal equality
1213:
1137:
1105:
1061:
1037:
1003:
972:
947:
916:
837:
795:
764:
708:
670:
645:
610:
579:
550:
456:
420:
353:
290:
262:
240:
214:
192:
167:
143:
91:
54:
4882:Was Cantor Surprised?
4839:10.1073/pnas.51.1.105
4722:10.1515/9781400881635
4579:mathworld.wolfram.com
4509:
4461:
4439:
4417:
4389:
4367:
4339:
4317:
4288:
4262:
4234:
4212:
4190:
4156:
4117:
4077:
4042:
4016:
3981:
3959:(i.e. the set of all
3954:
3924:
3895:(i.e. the set of all
3890:
3855:
3833:
3804:
3768:
3734:
3696:
3651:
3554:
3519:
3490:
3460:
3411:
3384:
3343:
3300:
3250:
3225:
3201:
3176:
3126:
3090:
3058:
3028:
2976:
2949:
2915:
2888:
2864:
2819:
2793:
2766:
2721:
2653:
2629:
2602:
2582:
2551:
2514:
2463:
2435:
2389:
2365:
2312:
2259:
2207:
2160:
2128:
2088:
2024:
1848:
1826:each real number has
1821:
1791:
1734:
1691:
1660:
1539:
1388:
1257:
1214:
1138:
1136:{\displaystyle [1,2)}
1106:
1104:{\displaystyle [0,1)}
1062:
1038:
1004:
973:
948:
917:
838:
796:
765:
709:
671:
646:
611:
580:
551:
457:
421:
354:
291:
263:
241:
215:
193:
168:
144:
92:
55:
4476:
4448:
4426:
4404:
4376:
4354:
4326:
4304:
4275:
4249:
4221:
4199:
4168:
4130:
4097:
4051:
4029:
4001:
3968:
3941:
3904:
3870:
3842:
3820:
3782:
3746:
3713:
3670:
3567:
3529:
3517:{\displaystyle a+bi}
3499:
3471:
3422:
3398:
3364:
3312:
3262:
3235:
3212:
3188:
3158:
3103:
3077:
3045:
3013:
2958:
2924:
2897:
2873:
2836:
2802:
2775:
2751:
2663:
2638:
2611:
2591:
2564:
2536:
2528:Continuum hypothesis
2478:
2450:
2404:
2374:
2321:
2281:
2220:
2171:
2137:
2099:
2037:
1862:
1830:
1800:
1778:
1704:
1671:
1556:
1401:
1272:
1225:
1149:
1115:
1083:
1047:
1013:
982:
960:
926:
873:
807:
778:
750:
744:uncountably infinite
681:
656:
628:
622:continuum hypothesis
593:
562:
472:
437:
408:
367:uncountability proof
303:
273:
250:
228:
202:
180:
155:
109:
75:
42:
4830:1964PNAS...51..105C
4763:1963PNAS...50.1143C
4573:Weisstein, Eric W.
4348:Lebesgue-measurable
4298:Lebesgue-integrable
4269:Lebesgue measurable
4091:indicator functions
3467:By definition, any
2817:{\displaystyle n=1}
2556:is also the second
1264:cardinal arithmetic
857:Cardinal equalities
464:space filling curve
375:bijective functions
361:This was proven by
4886:Fernando Q. GouvĂȘa
4504:
4456:
4434:
4412:
4384:
4362:
4334:
4312:
4283:
4257:
4243:Lebesgue Ï-algebra
4229:
4207:
4185:
4151:
4112:
4072:
4037:
4011:
3976:
3949:
3919:
3885:
3864:Euclidean topology
3850:
3828:
3799:
3763:
3729:
3691:
3646:
3644:
3549:
3514:
3485:
3455:
3406:
3379:
3338:
3295:
3245:
3220:
3196:
3171:
3137:irrational numbers
3121:
3085:
3053:
3023:
2987:successor cardinal
2971:
2944:
2910:
2883:
2869:). In particular,
2859:
2828:on the grounds of
2814:
2788:
2761:
2716:
2648:
2624:
2597:
2577:
2546:
2509:
2458:
2430:
2384:
2360:
2307:
2254:
2202:
2155:
2123:
2083:
2019:
1843:
1816:
1786:
1729:
1686:
1655:
1534:
1383:
1252:
1209:
1133:
1101:
1057:
1033:
999:
968:
943:
912:
833:
791:
760:
704:
666:
641:
606:
575:
546:
452:
416:
349:
286:
258:
236:
210:
188:
163:
139:
87:
71:and is denoted by
50:
4047:(i.e., power set
3147:algebraic numbers
3069:) closed or open
2999:singular cardinal
2600:{\displaystyle A}
1749:decimal expansion
1077:binary expansions
371:diagonal argument
151:The real numbers
31:or "size" of the
5007:
4990:Cardinal numbers
4911:Naive set theory
4895:
4879:
4870:
4869:
4859:
4841:
4809:
4803:
4802:
4792:
4774:
4757:(6): 1143â1148.
4742:
4736:
4735:
4707:
4701:
4700:
4674:
4661:
4660:
4624:
4613:
4612:
4599:"Cantor theorem"
4595:
4589:
4588:
4586:
4585:
4570:
4561:
4560:
4558:
4557:
4543:
4513:
4511:
4510:
4505:
4503:
4502:
4490:
4489:
4488:
4465:
4463:
4462:
4457:
4455:
4443:
4441:
4440:
4435:
4433:
4421:
4419:
4418:
4413:
4411:
4393:
4391:
4390:
4385:
4383:
4371:
4369:
4368:
4363:
4361:
4343:
4341:
4340:
4335:
4333:
4321:
4319:
4318:
4313:
4311:
4292:
4290:
4289:
4284:
4282:
4266:
4264:
4263:
4258:
4256:
4238:
4236:
4235:
4230:
4228:
4216:
4214:
4213:
4208:
4206:
4194:
4192:
4191:
4186:
4184:
4183:
4182:
4176:
4160:
4158:
4157:
4152:
4147:
4139:
4138:
4121:
4119:
4118:
4113:
4111:
4110:
4109:
4081:
4079:
4078:
4073:
4068:
4060:
4059:
4046:
4044:
4043:
4038:
4036:
4020:
4018:
4017:
4012:
4010:
4009:
3987:
3985:
3983:
3982:
3977:
3975:
3958:
3956:
3955:
3950:
3948:
3930:
3928:
3926:
3925:
3920:
3918:
3917:
3912:
3894:
3892:
3891:
3886:
3884:
3883:
3878:
3859:
3857:
3856:
3851:
3849:
3837:
3835:
3834:
3829:
3827:
3809:
3808:
3806:
3805:
3800:
3798:
3797:
3796:
3790:
3774:
3772:
3770:
3769:
3764:
3762:
3761:
3760:
3754:
3740:
3738:
3736:
3735:
3730:
3728:
3720:
3700:
3698:
3697:
3692:
3687:
3679:
3678:
3655:
3653:
3652:
3647:
3645:
3603:
3591:
3590:
3585:
3560:
3558:
3556:
3555:
3550:
3548:
3523:
3521:
3520:
3515:
3494:
3492:
3491:
3486:
3484:
3466:
3464:
3462:
3461:
3456:
3454:
3453:
3444:
3440:
3439:
3434:
3415:
3413:
3412:
3407:
3405:
3388:
3386:
3385:
3380:
3378:
3377:
3372:
3349:
3347:
3345:
3344:
3339:
3337:
3336:
3327:
3323:
3306:
3304:
3302:
3301:
3296:
3294:
3293:
3284:
3283:
3271:
3270:
3256:
3254:
3252:
3251:
3246:
3244:
3243:
3229:
3227:
3226:
3221:
3219:
3207:
3205:
3203:
3202:
3197:
3195:
3182:
3180:
3178:
3177:
3172:
3170:
3169:
3145:The set of real
3132:
3130:
3128:
3127:
3124:{\displaystyle }
3122:
3094:
3092:
3091:
3086:
3084:
3062:
3060:
3059:
3054:
3052:
3032:
3030:
3029:
3024:
3022:
3021:
2995:regular cardinal
2980:
2978:
2977:
2972:
2970:
2969:
2953:
2951:
2950:
2945:
2943:
2942:
2941:
2940:
2919:
2917:
2916:
2911:
2909:
2908:
2893:could be either
2892:
2890:
2889:
2884:
2882:
2881:
2868:
2866:
2865:
2860:
2858:
2857:
2845:
2844:
2823:
2821:
2820:
2815:
2797:
2795:
2794:
2789:
2787:
2786:
2770:
2768:
2767:
2762:
2760:
2759:
2725:
2723:
2722:
2717:
2712:
2711:
2702:
2694:
2686:
2685:
2657:
2655:
2654:
2649:
2647:
2646:
2633:
2631:
2630:
2625:
2623:
2622:
2606:
2604:
2603:
2598:
2586:
2584:
2583:
2578:
2576:
2575:
2555:
2553:
2552:
2547:
2545:
2544:
2518:
2516:
2515:
2510:
2505:
2504:
2492:
2491:
2490:
2467:
2465:
2464:
2459:
2457:
2439:
2437:
2436:
2431:
2426:
2425:
2413:
2412:
2393:
2391:
2390:
2385:
2383:
2382:
2369:
2367:
2366:
2361:
2359:
2358:
2357:
2356:
2339:
2338:
2316:
2314:
2313:
2308:
2306:
2305:
2293:
2292:
2263:
2261:
2260:
2255:
2249:
2248:
2247:
2246:
2229:
2228:
2211:
2209:
2208:
2203:
2200:
2199:
2190:
2189:
2188:
2187:
2164:
2162:
2161:
2156:
2132:
2130:
2129:
2124:
2092:
2090:
2089:
2084:
2081:
2080:
2068:
2067:
2049:
2048:
2028:
2026:
2025:
2020:
2018:
2017:
2016:
2015:
1998:
1997:
1996:
1995:
1977:
1976:
1959:
1958:
1957:
1956:
1946:
1942:
1941:
1924:
1923:
1922:
1921:
1904:
1903:
1902:
1901:
1884:
1883:
1871:
1870:
1852:
1850:
1849:
1844:
1842:
1841:
1825:
1823:
1822:
1817:
1812:
1811:
1795:
1793:
1792:
1787:
1785:
1757:1/3 = 0.33333...
1754:1/2 = 0.50000...
1738:
1736:
1735:
1730:
1728:
1727:
1718:
1717:
1716:
1695:
1693:
1692:
1687:
1685:
1684:
1683:
1664:
1662:
1661:
1656:
1654:
1653:
1652:
1639:
1638:
1637:
1636:
1624:
1623:
1609:
1608:
1607:
1597:
1596:
1595:
1594:
1574:
1573:
1572:
1566:
1565:
1543:
1541:
1540:
1535:
1533:
1532:
1523:
1522:
1510:
1509:
1503:
1502:
1490:
1489:
1484:
1483:
1473:
1472:
1471:
1470:
1453:
1452:
1451:
1450:
1440:
1439:
1438:
1424:
1423:
1422:
1421:
1411:
1410:
1392:
1390:
1389:
1384:
1379:
1378:
1369:
1368:
1367:
1366:
1349:
1348:
1347:
1346:
1345:
1321:
1320:
1311:
1310:
1309:
1308:
1288:
1287:
1282:
1281:
1261:
1259:
1258:
1253:
1251:
1250:
1241:
1240:
1235:
1234:
1218:
1216:
1215:
1210:
1205:
1204:
1203:
1202:
1185:
1177:
1166:
1158:
1157:
1142:
1140:
1139:
1134:
1110:
1108:
1107:
1102:
1066:
1064:
1063:
1058:
1056:
1055:
1042:
1040:
1039:
1034:
1032:
1031:
1030:
1029:
1009:, by definition
1008:
1006:
1005:
1000:
995:
977:
975:
974:
969:
967:
952:
950:
949:
944:
939:
921:
919:
918:
913:
911:
910:
909:
901:
888:
880:
863:Cantor's theorem
842:
840:
839:
834:
829:
828:
819:
818:
800:
798:
797:
792:
790:
789:
769:
767:
766:
761:
759:
758:
716:cannot be proven
713:
711:
710:
705:
703:
702:
690:
689:
676:
675:
673:
672:
667:
665:
664:
650:
648:
647:
642:
640:
639:
615:
613:
612:
607:
605:
604:
584:
582:
581:
576:
574:
573:
555:
553:
552:
547:
542:
537:
536:
531:
525:
517:
512:
507:
499:
479:
461:
459:
458:
453:
451:
450:
445:
425:
423:
422:
417:
415:
384: <
358:
356:
355:
350:
345:
344:
332:
331:
330:
329:
312:
311:
295:
293:
292:
287:
285:
284:
267:
265:
264:
259:
257:
245:
243:
242:
237:
235:
219:
217:
216:
211:
209:
197:
195:
194:
189:
187:
172:
170:
169:
164:
162:
148:
146:
145:
140:
138:
137:
136:
129:
128:
127:
120:
119:
118:
96:
94:
93:
88:
86:
85:
84:
59:
57:
56:
51:
49:
5015:
5014:
5010:
5009:
5008:
5006:
5005:
5004:
4980:
4979:
4903:
4898:
4880:
4873:
4811:
4810:
4806:
4744:
4743:
4739:
4732:
4709:
4708:
4704:
4697:
4676:
4675:
4664:
4626:
4625:
4616:
4597:
4596:
4592:
4583:
4581:
4572:
4571:
4564:
4555:
4553:
4545:
4544:
4540:
4536:
4524:
4494:
4479:
4474:
4473:
4446:
4445:
4424:
4423:
4402:
4401:
4374:
4373:
4352:
4351:
4350:functions from
4346:the set of all
4324:
4323:
4302:
4301:
4300:functions from
4296:the set of all
4273:
4272:
4247:
4246:
4219:
4218:
4197:
4196:
4171:
4166:
4165:
4128:
4127:
4100:
4095:
4094:
4049:
4048:
4027:
4026:
3999:
3998:
3995:
3990:
3966:
3965:
3964:
3939:
3938:
3935:Borel Ï-algebra
3907:
3902:
3901:
3900:
3873:
3868:
3867:
3840:
3839:
3818:
3817:
3816:functions from
3812:the set of all
3785:
3780:
3779:
3778:
3749:
3744:
3743:
3742:
3711:
3710:
3709:
3668:
3667:
3665:natural numbers
3656:
3643:
3642:
3623:
3605:
3604:
3592:
3580:
3565:
3564:
3527:
3526:
3525:
3497:
3496:
3469:
3468:
3429:
3425:
3420:
3419:
3418:
3396:
3395:
3393:complex numbers
3367:
3362:
3361:
3359:Euclidean space
3315:
3310:
3309:
3308:
3275:
3260:
3259:
3258:
3233:
3232:
3231:
3210:
3209:
3186:
3185:
3184:
3161:
3156:
3155:
3154:
3101:
3100:
3099:
3075:
3074:
3043:
3042:
3011:
3010:
3007:
2993:, and either a
2961:
2956:
2955:
2932:
2927:
2922:
2921:
2900:
2895:
2894:
2871:
2870:
2849:
2834:
2833:
2826:König's theorem
2800:
2799:
2778:
2773:
2772:
2749:
2748:
2747:, the equality
2726:
2677:
2661:
2660:
2636:
2635:
2614:
2609:
2608:
2589:
2588:
2567:
2562:
2561:
2534:
2533:
2530:
2524:
2519:
2496:
2481:
2476:
2475:
2448:
2447:
2440:
2417:
2402:
2401:
2372:
2371:
2348:
2343:
2324:
2319:
2318:
2297:
2284:
2279:
2278:
2275:
2269:
2264:
2238:
2233:
2218:
2217:
2212:
2179:
2174:
2169:
2168:
2135:
2134:
2097:
2096:
2093:
2072:
2059:
2040:
2035:
2034:
2029:
2007:
2002:
1987:
1968:
1963:
1948:
1933:
1928:
1913:
1908:
1893:
1888:
1875:
1860:
1859:
1833:
1828:
1827:
1803:
1798:
1797:
1776:
1775:
1761:
1760:Ï = 3.14159....
1758:
1755:
1751:. For example,
1745:
1707:
1702:
1701:
1674:
1669:
1668:
1665:
1643:
1628:
1613:
1598:
1586:
1581:
1559:
1554:
1553:
1544:
1494:
1477:
1462:
1457:
1442:
1430:
1428:
1413:
1404:
1399:
1398:
1393:
1358:
1353:
1337:
1325:
1312:
1300:
1295:
1275:
1270:
1269:
1228:
1223:
1222:
1219:
1194:
1189:
1147:
1146:
1113:
1112:
1081:
1080:
1045:
1044:
1021:
1016:
1011:
1010:
980:
979:
958:
957:
955:natural numbers
924:
923:
892:
871:
870:
859:
843:
810:
805:
804:
781:
776:
775:
772:natural numbers
748:
747:
733:
728:
694:
679:
678:
654:
653:
652:
631:
626:
625:
596:
591:
590:
565:
560:
559:
556:
526:
470:
469:
440:
435:
434:
432:Euclidean space
406:
405:
359:
336:
321:
316:
301:
300:
276:
271:
270:
248:
247:
226:
225:
200:
199:
178:
177:
175:natural numbers
153:
152:
107:
106:
73:
72:
69:cardinal number
40:
39:
17:
12:
11:
5:
5013:
5011:
5003:
5002:
4997:
4992:
4982:
4981:
4961:
4960:
4943:Kunen, Kenneth
4940:
4922:
4902:
4899:
4897:
4896:
4871:
4824:(1): 105â110.
4804:
4737:
4730:
4702:
4695:
4662:
4635:(3): 286â297.
4614:
4590:
4562:
4537:
4535:
4532:
4531:
4530:
4523:
4520:
4501:
4497:
4493:
4487:
4482:
4470:
4469:
4466:
4454:
4432:
4410:
4394:
4382:
4360:
4344:
4332:
4310:
4294:
4281:
4255:
4239:
4227:
4205:
4181:
4175:
4162:
4150:
4146:
4142:
4137:
4108:
4103:
4083:
4071:
4067:
4063:
4058:
4035:
4008:
3994:
3991:
3989:
3988:
3974:
3947:
3931:
3916:
3911:
3882:
3877:
3860:
3848:
3826:
3810:
3795:
3789:
3775:
3759:
3753:
3741:often denoted
3727:
3723:
3719:
3702:
3690:
3686:
3682:
3677:
3657:
3641:
3638:
3635:
3632:
3629:
3626:
3624:
3622:
3619:
3616:
3613:
3610:
3607:
3606:
3602:
3598:
3595:
3593:
3589:
3584:
3579:
3576:
3573:
3572:
3563:
3547:
3543:
3540:
3537:
3534:
3513:
3510:
3507:
3504:
3483:
3479:
3476:
3452:
3447:
3443:
3438:
3433:
3428:
3404:
3389:
3376:
3371:
3356:
3350:
3335:
3330:
3326:
3322:
3318:
3292:
3287:
3282:
3278:
3274:
3269:
3242:
3218:
3194:
3168:
3164:
3139:
3133:
3120:
3117:
3114:
3111:
3108:
3083:
3063:
3051:
3035:
3020:
3006:
3003:
2991:limit cardinal
2968:
2964:
2939:
2935:
2930:
2907:
2903:
2880:
2856:
2852:
2848:
2843:
2813:
2810:
2807:
2785:
2781:
2758:
2742:natural number
2715:
2710:
2705:
2701:
2697:
2693:
2689:
2684:
2680:
2675:
2671:
2668:
2659:
2645:
2621:
2617:
2596:
2574:
2570:
2543:
2526:Main article:
2523:
2520:
2508:
2503:
2499:
2495:
2489:
2484:
2474:
2456:
2429:
2424:
2420:
2416:
2411:
2400:
2381:
2355:
2351:
2346:
2342:
2337:
2334:
2331:
2327:
2304:
2300:
2296:
2291:
2287:
2271:Main article:
2268:
2265:
2253:
2245:
2241:
2236:
2232:
2227:
2216:
2198:
2193:
2186:
2182:
2177:
2167:
2154:
2151:
2148:
2145:
2142:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2079:
2075:
2071:
2066:
2062:
2058:
2055:
2052:
2047:
2043:
2033:
2014:
2010:
2005:
2001:
1994:
1990:
1986:
1983:
1980:
1975:
1971:
1966:
1962:
1955:
1951:
1945:
1940:
1936:
1932:
1927:
1920:
1916:
1911:
1907:
1900:
1896:
1891:
1887:
1882:
1878:
1874:
1869:
1858:
1840:
1836:
1815:
1810:
1806:
1784:
1759:
1756:
1753:
1744:
1741:
1726:
1721:
1715:
1710:
1682:
1677:
1651:
1646:
1642:
1635:
1631:
1627:
1622:
1616:
1612:
1606:
1601:
1593:
1589:
1584:
1580:
1577:
1571:
1564:
1552:
1531:
1526:
1521:
1516:
1513:
1508:
1501:
1497:
1493:
1488:
1482:
1476:
1469:
1465:
1460:
1456:
1449:
1445:
1437:
1433:
1427:
1420:
1416:
1409:
1397:
1382:
1377:
1372:
1365:
1361:
1356:
1352:
1344:
1340:
1335:
1332:
1328:
1324:
1319:
1315:
1307:
1303:
1298:
1294:
1291:
1286:
1280:
1268:
1249:
1244:
1239:
1233:
1208:
1201:
1197:
1192:
1188:
1184:
1180:
1176:
1172:
1169:
1165:
1161:
1156:
1145:
1132:
1129:
1126:
1123:
1120:
1100:
1097:
1094:
1091:
1088:
1054:
1043:, is equal to
1028:
1024:
1019:
998:
994:
990:
987:
966:
942:
938:
934:
931:
908:
904:
900:
895:
891:
887:
883:
879:
858:
855:
832:
827:
822:
817:
813:
803:
788:
784:
757:
732:
731:Uncountability
729:
727:
724:
701:
697:
693:
688:
663:
638:
634:
603:
599:
572:
568:
545:
541:
535:
530:
524:
520:
516:
511:
506:
502:
498:
494:
491:
488:
485:
482:
478:
468:
449:
444:
414:
348:
343:
339:
335:
328:
324:
319:
315:
310:
299:
283:
279:
268:is denoted as
256:
234:
208:
186:
161:
135:
126:
117:
83:
48:
15:
13:
10:
9:
6:
4:
3:
2:
5012:
5001:
4998:
4996:
4993:
4991:
4988:
4987:
4985:
4978:
4977:
4975:
4971:
4967:
4958:
4957:0-444-86839-9
4954:
4950:
4949:
4944:
4941:
4938:
4937:3-540-44085-2
4934:
4930:
4926:
4923:
4920:
4919:0-387-90092-6
4916:
4912:
4908:
4905:
4904:
4900:
4894:, March 2011.
4893:
4892:
4887:
4883:
4878:
4876:
4872:
4867:
4863:
4858:
4853:
4849:
4845:
4840:
4835:
4831:
4827:
4823:
4819:
4815:
4808:
4805:
4800:
4796:
4791:
4786:
4782:
4778:
4773:
4768:
4764:
4760:
4756:
4752:
4748:
4741:
4738:
4733:
4731:9781400881635
4727:
4723:
4719:
4715:
4714:
4706:
4703:
4698:
4696:9781447106036
4692:
4688:
4684:
4680:
4673:
4671:
4669:
4667:
4663:
4658:
4654:
4650:
4646:
4642:
4638:
4634:
4630:
4623:
4621:
4619:
4615:
4610:
4606:
4605:
4600:
4594:
4591:
4580:
4576:
4569:
4567:
4563:
4552:
4548:
4542:
4539:
4533:
4529:
4526:
4525:
4521:
4519:
4517:
4499:
4495:
4491:
4480:
4467:
4399:
4395:
4349:
4345:
4299:
4295:
4270:
4244:
4240:
4163:
4125:
4101:
4092:
4088:
4084:
4024:
4023:
4022:
3992:
3962:
3936:
3932:
3914:
3898:
3880:
3865:
3861:
3815:
3811:
3776:
3707:
3703:
3666:
3662:
3658:
3639:
3636:
3633:
3630:
3625:
3617:
3614:
3611:
3594:
3587:
3577:
3574:
3562:
3541:
3538:
3535:
3532:
3511:
3508:
3505:
3502:
3477:
3474:
3445:
3441:
3436:
3426:
3394:
3390:
3374:
3360:
3357:
3355:
3351:
3328:
3324:
3316:
3285:
3280:
3272:
3166:
3152:
3148:
3144:
3140:
3138:
3134:
3115:
3112:
3109:
3098:
3097:unit interval
3095:(such as the
3072:
3068:
3067:nondegenerate
3064:
3041:
3037:
3036:
3034:
3004:
3002:
3000:
2996:
2992:
2988:
2984:
2966:
2962:
2937:
2933:
2905:
2854:
2846:
2831:
2827:
2811:
2808:
2805:
2783:
2746:
2743:
2739:
2735:
2731:
2713:
2703:
2695:
2687:
2682:
2673:
2669:
2666:
2658:
2619:
2594:
2572:
2559:
2529:
2521:
2506:
2501:
2497:
2493:
2482:
2473:
2471:
2445:
2427:
2422:
2418:
2414:
2399:
2397:
2353:
2349:
2344:
2340:
2335:
2332:
2329:
2325:
2302:
2294:
2289:
2285:
2274:
2266:
2251:
2243:
2234:
2230:
2215:
2191:
2184:
2175:
2166:
2149:
2146:
2143:
2117:
2114:
2111:
2105:
2102:
2077:
2069:
2064:
2056:
2053:
2050:
2045:
2032:
2012:
2003:
1999:
1992:
1984:
1981:
1978:
1973:
1964:
1960:
1953:
1938:
1934:
1925:
1918:
1909:
1905:
1898:
1889:
1885:
1880:
1872:
1857:
1854:
1838:
1813:
1808:
1773:
1769:
1764:
1752:
1750:
1742:
1740:
1719:
1708:
1699:
1675:
1644:
1640:
1633:
1625:
1614:
1610:
1591:
1582:
1575:
1551:
1549:
1524:
1514:
1511:
1499:
1491:
1486:
1474:
1467:
1458:
1454:
1447:
1435:
1425:
1418:
1396:
1380:
1370:
1363:
1354:
1350:
1342:
1333:
1330:
1326:
1322:
1317:
1305:
1296:
1289:
1284:
1267:
1265:
1242:
1237:
1206:
1199:
1190:
1186:
1159:
1144:
1127:
1124:
1121:
1095:
1092:
1089:
1078:
1074:
1073:Dedekind cuts
1070:
1026:
1017:
956:
902:
893:
889:
881:
868:
864:
856:
854:
852:
848:
830:
820:
815:
802:
786:
773:
745:
741:
737:
730:
725:
723:
721:
717:
699:
691:
677:, means that
636:
623:
619:
601:
588:
570:
543:
533:
518:
500:
489:
486:
483:
467:
465:
447:
433:
430:-dimensional
429:
403:
399:
395:
391:
390:open interval
387:
383:
378:
376:
372:
368:
364:
346:
341:
333:
326:
317:
313:
298:
296:
281:
223:
176:
149:
104:
100:
70:
67:
63:
38:
34:
30:
26:
22:
4963:
4962:
4951:. Elsevier.
4946:
4931:. Springer.
4928:
4925:Jech, Thomas
4910:
4901:Bibliography
4889:
4821:
4817:
4807:
4754:
4750:
4740:
4712:
4705:
4678:
4632:
4628:
4602:
4593:
4582:. Retrieved
4578:
4554:. Retrieved
4550:
4541:
4471:
3996:
3416:
3151:Gödel number
3040:real numbers
3008:
2744:
2727:
2558:aleph number
2531:
2443:
2441:
2395:
2276:
2267:Beth numbers
2213:
2094:
2030:
1855:
1765:
1762:
1746:
1697:
1666:
1547:
1545:
1394:
1220:
860:
844:
736:Georg Cantor
734:
557:
466:). That is,
427:
402:equinumerous
397:
393:
385:
381:
379:
363:Georg Cantor
360:
198:. Moreover,
150:
102:
37:real numbers
24:
18:
4907:Paul Halmos
4575:"Continuum"
3704:the set of
2273:Beth number
869:. That is,
746:. That is,
740:cardinality
97:(lowercase
64:. It is an
29:cardinality
4995:Set theory
4984:Categories
4970:PlanetMath
4584:2020-08-12
4556:2020-08-12
4534:References
4124:isomorphic
3961:Borel sets
3814:continuous
3354:Cantor set
2830:cofinality
2738:Paul Cohen
2734:Kurt Gödel
726:Properties
587:aleph-null
21:set theory
4848:0027-8424
4781:0027-8424
4609:EMS Press
4496:ℶ
4021:include:
3897:open sets
3722:→
3706:sequences
3661:power set
3628:↦
3597:→
3578::
3542:∈
3524:for some
3478:∈
3277:ℵ
3273:−
3163:ℵ
2963:ω
2934:ω
2929:ℵ
2902:ℵ
2855:ω
2851:ℵ
2847:≠
2780:ℵ
2679:ℵ
2667:∄
2616:ℵ
2569:ℵ
2498:ℶ
2470:real line
2419:ℶ
2350:ℶ
2326:ℶ
2299:ℵ
2286:ℶ
2240:ℵ
2214:and thus
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