621:
7510:
649:
27:
7580:
4174:
535:
A crude sense of cardinality, an awareness that groups of things or events compare with other groups by containing more, fewer, or the same number of instances, is observed in a variety of present-day animal species, suggesting an origin millions of years ago. Human expression of cardinality is seen
551:
From the 6th century BCE, the writings of Greek philosophers show hints of the cardinality of infinite sets. While they considered the notion of infinity as an endless series of actions, such as adding 1 to a number repeatedly, they did not consider the size of an infinite set of numbers to be a
4293:
4674:
4023:
4850:
4571:
774:
While the cardinality of a finite set is simply comparable to its number of elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets (some of which are possibly infinite).
543:
years ago, with equating the size of a group with a group of recorded notches, or a representative collection of other things, such as sticks and shells. The abstraction of cardinality as a number is evident by 3000 BCE, in
Sumerian
4029:
3226:. Cantor introduced the cardinal numbers, and showedâaccording to his bijection-based definition of sizeâthat some infinite sets are greater than others. The smallest infinite cardinality is that of the natural numbers (
4180:
5395:
582:, it was seen that even the infinite set of all rational numbers was not enough to describe the length of every possible line segment. Still, there was no concept of infinite sets as something that had cardinality.
2678:
3889:
3222:. Indeed, Dedekind defined an infinite set as one that can be placed into a one-to-one correspondence with a strict subset (that is, having the same size in Cantor's sense); this notion of infinity is called
3734:
3841:
4923:
3385:
4590:
3901:
3534:
of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space. These results are highly counterintuitive, because they imply that there exist
1044:
3791:
5013:
4742:
2923:
3439:
2879:
2467:
2402:
2296:
3511:
2977:
2549:
In the above section, "cardinality" of a set was defined functionally. In other words, it was not defined as a specific object itself. However, such an object can be defined as follows.
980:
1580:
502:
3048:, a standard axiomatization of set theory; that is, it is impossible to prove the continuum hypothesis or its negation from ZFCâprovided that ZFC is consistent. For more detail, see
4952:
2737:
1680:
2764:
5080:
3625:
3293:
3179:
3011:
2830:
4734:
3596:, or finite-dimensional space. These curves are not a direct proof that a line has the same number of points as a finite-dimensional space, but they can be used to obtain
2519:
2493:
2428:
2363:
2250:
1499:
1387:
1110:
3320:
3251:
3132:
3038:
2802:
2060:
2014:
1914:
1868:
212:
168:
2339:
4509:
2704:
1960:
1451:
1086:
337:
108:
3045:
448:
5687:
525:
391:
5103:
5037:
4710:
2221:
2197:
2173:
2149:
2106:
2082:
1817:
1793:
1769:
1745:
1702:
1630:
1606:
1523:
1473:
1411:
1361:
1337:
1313:
1289:
1265:
1237:
1213:
1185:
1161:
1134:
895:
871:
843:
819:
469:
419:
361:
289:
265:
72:
48:
4474:
4169:{\displaystyle {\mathfrak {c}}^{\aleph _{0}}=\left(2^{\aleph _{0}}\right)^{\aleph _{0}}=2^{{\aleph _{0}}\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
3597:
5889:
4288:{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=\left(2^{\aleph _{0}}\right)^{\mathfrak {c}}=2^{{\mathfrak {c}}\times \aleph _{0}}=2^{\mathfrak {c}}.}
4437:
8037:
6564:
4681:
8192:
5315:
5244:
3040:, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis is
6647:
5788:
5440:
5289:
2621:
608:, he demonstrated that there are sets of numbers that cannot be placed in one-to-one correspondence with the set of natural numbers, i.e.
570:, as a ratio, as long as there were a third segment, no matter how small, that could be laid end-to-end a whole number of times into both
4431:
3846:
3687:
3795:
4858:
552:
thing. The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid's
6961:
3578:
3219:
4669:{\displaystyle \left\vert C\cup D\right\vert +\left\vert C\cap D\right\vert =\left\vert C\right\vert +\left\vert D\right\vert .}
3333:
604:, a one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of
5367:
4018:{\displaystyle {\mathfrak {c}}^{2}=\left(2^{\aleph _{0}}\right)^{2}=2^{2\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
3747:
3448:
2532:
7119:
5178:
5907:
7726:
7546:
6974:
6297:
989:
3752:
3628:
4845:{\displaystyle |A|:={\mbox{Ord}}\cap \bigcap \{\alpha \in {\mbox{Ord}}|\exists (f:A\to \alpha ):(f{\mbox{ injective}})\}}
5701:
8054:
6979:
6969:
6706:
6559:
5912:
5903:
4960:
7115:
5730:
5722:
5670:
5505:
3041:
2884:
6457:
3397:
2847:
1963:
7212:
6956:
5781:
2437:
2372:
2266:
8032:
6517:
6210:
3469:
2935:
2592:
7912:
5951:
5085:
This definition is natural since it agrees with the axiom of limitation of size which implies bijection between
927:
226:
on them. There are two notions often used when referring to cardinality: one which compares sets directly using
7473:
7175:
6938:
6933:
6758:
6179:
5863:
1530:
559:
475:
7806:
7685:
7468:
7251:
7168:
6881:
6812:
6689:
5931:
4486:
3444:
2841:
2528:
605:
8049:
7393:
7219:
6905:
6539:
6138:
3268:
3262:
2837:
2588:
20:
4928:
8042:
7680:
7643:
7271:
7266:
6876:
6615:
6544:
5873:
5774:
4685:
2709:
3218:
and others rejected the view that the whole cannot be the same size as the part. One example of this is
1637:
7200:
6790:
6184:
6152:
5843:
2742:
7697:
5045:
3584:
The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when
8187:
7731:
7623:
7611:
7606:
7490:
7439:
7336:
6834:
6795:
6272:
5917:
4581:
3606:
3566:
3274:
3160:
2982:
2811:
5946:
7539:
7331:
7261:
6800:
6652:
6635:
6358:
5838:
3065:
5326:
5262:
4715:
2502:
2476:
2411:
2346:
2233:
1482:
1370:
1093:
8151:
8069:
7944:
7896:
7710:
7633:
7163:
7140:
7101:
6987:
6928:
6574:
6494:
6338:
6282:
5895:
5301:
4566:{\displaystyle \left\vert A\cup B\right\vert =\left\vert A\right\vert +\left\vert B\right\vert .}
3298:
3229:
3110:
3016:
2780:
2019:
1973:
1873:
1827:
173:
129:
5286:
2303:
8103:
7984:
7796:
7616:
7453:
7180:
7158:
7125:
7018:
6864:
6849:
6822:
6773:
6657:
6592:
6417:
6383:
6378:
6252:
6083:
6060:
3570:
848:
545:
8019:
7989:
7933:
7853:
7833:
7811:
7383:
7236:
7028:
6746:
6482:
6388:
6247:
6232:
6113:
6088:
5445:
2689:
1919:
1418:
1053:
620:
296:
215:
8093:
8083:
7917:
7848:
7801:
7741:
7628:
7356:
7318:
7195:
6999:
6839:
6763:
6741:
6569:
6527:
6426:
6393:
6257:
6045:
5956:
5604:
5537:
5512:
5420:
5149:
3520:
3456:
2929:
2600:
2557:
554:
548:
and the manipulation of numbers without reference to a specific group of things or events.
77:
3463:
between the cardinality of the reals and the cardinality of the natural numbers, that is,
424:
8:
8088:
7999:
7907:
7902:
7716:
7658:
7596:
7532:
7485:
7376:
7361:
7341:
7298:
7185:
7135:
7061:
7006:
6943:
6736:
6731:
6679:
6447:
6436:
6108:
6008:
5936:
5927:
5923:
5858:
5853:
5129:
5124:
3892:
3632:
3589:
2572:
under this relation, then, consists of all those sets which have the same cardinality as
707:
507:
366:
223:
5608:
5541:
5351:
Kurt Von Fritz (1945). "The
Discovery of Incommensurability by Hippasus of Metapontum".
3516:
However, this hypothesis can neither be proved nor disproved within the widely accepted
8011:
8006:
7791:
7746:
7653:
7514:
7283:
7246:
7231:
7224:
7207:
7011:
6993:
6859:
6785:
6768:
6721:
6534:
6443:
6277:
6262:
6222:
6174:
6159:
6147:
6103:
6078:
5848:
5797:
5622:
5555:
5470:
5088:
5022:
4695:
4577:
3592:, curved lines that twist and turn enough to fill the whole of any square, or cube, or
3135:
2561:
2206:
2182:
2158:
2134:
2091:
2067:
1802:
1778:
1754:
1730:
1687:
1615:
1591:
1508:
1458:
1396:
1346:
1322:
1298:
1274:
1250:
1222:
1198:
1170:
1146:
1119:
880:
856:
828:
804:
454:
404:
346:
274:
250:
231:
57:
33:
6467:
5635:
5592:
5568:
5525:
4447:
1586:) is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither
222:, which allows one to distinguish between different types of infinity, and to perform
7868:
7705:
7668:
7638:
7569:
7509:
7449:
7256:
7066:
7056:
6948:
6829:
6664:
6640:
6421:
6405:
6310:
6287:
6164:
6133:
6098:
5993:
5828:
5726:
5718:
5666:
5640:
5573:
5501:
5474:
5462:
5195:
5174:
4441:
3223:
2565:
579:
123:
5198:
2925:, this also being the cardinality of the set of all subsets of the natural numbers.
8156:
8146:
8131:
8126:
7994:
7648:
7463:
7458:
7351:
7308:
7130:
7091:
7086:
7071:
6897:
6854:
6751:
6549:
6499:
6073:
6035:
5630:
5612:
5563:
5545:
5491:
5454:
5428:
3562:
3527:
3215:
2833:
1583:
3739:
3388:
612:
that contain more elements than there are in the infinite set of natural numbers.
8025:
7963:
7781:
7601:
7444:
7434:
7388:
7371:
7326:
7288:
7190:
7110:
6917:
6844:
6817:
6805:
6711:
6625:
6599:
6554:
6522:
6323:
6125:
6068:
6018:
5983:
5941:
5495:
5487:
5436:
5293:
5249:
3539:
3460:
3186:
3061:
2608:
2596:
2544:
2255:
1967:
1047:
648:
609:
235:
51:
3526:
Cardinal arithmetic can be used to show not only that the number of points in a
8161:
7958:
7939:
7843:
7828:
7785:
7721:
7663:
7429:
7408:
7366:
7346:
7241:
7096:
6694:
6684:
6674:
6669:
6603:
6477:
6353:
6242:
6237:
6215:
5816:
5597:
Proceedings of the
National Academy of Sciences of the United States of America
5530:
Proceedings of the
National Academy of Sciences of the United States of America
3585:
3077:
2770:
2684:
2553:
398:
1453:
is injective, but not surjective since 2, for instance, is not mapped to, and
26:
8181:
8166:
7968:
7882:
7877:
7403:
7081:
6588:
6373:
6363:
6333:
6318:
5988:
5760:. â A line of finite length is a set of points that has infinite cardinality.
5658:
5466:
5432:
5134:
4500:
3535:
3211:
8136:
5717:
Applied
Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983,
2591:
set is designated for each equivalence class. The most common choice is the
8116:
8111:
7929:
7858:
7816:
7675:
7579:
7303:
7150:
7051:
7043:
6923:
6871:
6780:
6716:
6699:
6630:
6489:
6348:
6050:
5833:
5644:
5577:
5550:
5114:
3531:
3207:
3203:
2774:
2612:
917:
593:
585:
To better understand infinite sets, a notion of cardinality was formulated
394:
219:
5617:
3554:
contains elements that do not belong to its subsets, and the supersets of
8141:
7413:
7293:
6472:
6462:
6093:
6013:
5998:
5878:
5424:
5119:
3561:
The first of these results is apparent by considering, for instance, the
2805:
2524:
983:
633:
562:
was described as the ability to compare the length of two line segments,
115:
5700:
Georg Cantor (1932), Adolf
Fraenkel (Lebenslauf); Ernst Zermelo (eds.),
4679:
8121:
7892:
7555:
6343:
6198:
6169:
5975:
5458:
5144:
3199:
3089:
904:
597:
3099:
that has the same cardinality as the set of the natural numbers, or |
218:. Beginning in the late 19th century, this concept was generalized to
7924:
7887:
7838:
7736:
7495:
7398:
6451:
6368:
6328:
6292:
6228:
6040:
6030:
6003:
5766:
5626:
5559:
5219:
5203:
5019:
This definition allows also obtain a cardinality of any proper class
3646:
3593:
2673:{\displaystyle \aleph _{0}<\aleph _{1}<\aleph _{2}<\ldots .}
2259:
2177:, if there is an injective function, but no bijective function, from
900:
796:
656:
601:
227:
3603:
Cantor also showed that sets with cardinality strictly greater than
7480:
7278:
6726:
6431:
6025:
5757:
5139:
3068:
holds for cardinality. Thus we can make the following definitions:
401:, and the meaning depends on context. The cardinal number of a set
5703:
Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
7076:
5868:
5663:
The Road to
Reality: A Complete guide to the Laws of the Universe
3884:{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}}
5685:
Georg Cantor (1887), "Mitteilungen zur Lehre vom
Transfiniten",
3145:
with cardinality greater than that of the natural numbers, or |
7949:
7771:
5749:
5245:
Animals Count and Use Zero. How Far Does Their Number Sense Go?
3729:{\displaystyle 2^{\mathfrak {c}}=\beth _{2}>{\mathfrak {c}}}
4576:
From this, one can show that in general, the cardinalities of
3836:{\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},}
2497:
has cardinality strictly less than the cardinality of the set
7821:
7588:
7524:
6620:
5966:
5811:
4918:{\displaystyle (x\in \bigcap Q)\iff (\forall q\in Q:x\in q)}
242:, when no confusion with other notions of size is possible.
5753:
4430:|. This holds even for infinite cardinals, and is known as
2576:. There are two ways to define the "cardinality of a set":
3380:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}}
126:
which compares their relative size. For example, the sets
5500:(1. ed.), Berlin/Heidelberg: Springer, p. 587,
3517:
3055:
2584:
is defined as its equivalence class under equinumerosity.
1749:
has cardinality less than or equal to the cardinality of
5368:"Ueber eine elementare Frage der Mannigfaltigkeitslehre"
5193:
4855:
We use the intersection of a class which is defined by
2471:
can be bijective (see picture). By a similar argument,
5066:
4999:
4830:
4785:
4763:
4720:
2552:
The relation of having the same cardinality is called
2153:
has cardinality strictly less than the cardinality of
293:, have the same cardinality, it is usually written as
5688:
5091:
5048:
5025:
4963:
4931:
4861:
4745:
4718:
4698:
4593:
4512:
4450:
4183:
4032:
3904:
3849:
3798:
3755:
3690:
3609:
3472:
3400:
3336:
3301:
3277:
3232:
3163:
3113:
3019:
2985:
2938:
2887:
2850:
2814:
2783:
2745:
2712:
2692:
2624:
2505:
2479:
2440:
2414:
2375:
2349:
2306:
2269:
2236:
2209:
2185:
2161:
2137:
2094:
2070:
2022:
1976:
1922:
1876:
1830:
1805:
1781:
1757:
1733:
1690:
1640:
1618:
1594:
1533:
1511:
1485:
1461:
1421:
1399:
1373:
1349:
1325:
1301:
1277:
1253:
1225:
1201:
1173:
1149:
1122:
1096:
1056:
1039:{\displaystyle \mathbb {N} =\{0,1,2,3,{\text{...}}\}}
992:
930:
883:
859:
831:
807:
795:
Two sets have the same cardinality if there exists a
510:
478:
457:
427:
407:
369:
349:
299:
277:
253:
176:
132:
80:
60:
36:
5419:
5375:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4440:
include the set of all real numbers, the set of all
3786:{\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},}
3267:
One of Cantor's most important results was that the
1293:
is a bijection. This is no longer true for infinite
600:. He examined the process of equating two sets with
718:cannot be surjective. The picture shows an example
5593:"The Independence of the Continuum Hypothesis, II"
5097:
5074:
5031:
5007:
4946:
4917:
4844:
4728:
4704:
4668:
4565:
4468:
4287:
4168:
4017:
3883:
3835:
3785:
3728:
3619:
3505:
3433:
3379:
3314:
3287:
3245:
3173:
3126:
3032:
3005:
2971:
2917:
2873:
2824:
2796:
2758:
2731:
2698:
2672:
2513:
2487:
2461:
2422:
2396:
2357:
2333:
2290:
2244:
2215:
2191:
2167:
2143:
2100:
2076:
2054:
2008:
1954:
1908:
1862:
1811:
1787:
1763:
1739:
1696:
1674:
1624:
1600:
1574:
1517:
1493:
1467:
1445:
1405:
1381:
1355:
1331:
1307:
1283:
1259:
1231:
1207:
1179:
1155:
1128:
1104:
1080:
1038:
974:
889:
865:
837:
813:
519:
496:
463:
442:
413:
385:
355:
331:
283:
259:
238:. The cardinality of a set may also be called its
206:
162:
102:
66:
42:
5220:"Cardinality | Brilliant Math & Science Wiki"
3049:
8179:
5008:{\displaystyle (x\mapsto |x|):V\to {\mbox{Ord}}}
5486:
3295:) is greater than that of the natural numbers (
3256:
2918:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}
5526:"The Independence of the Continuum Hypothesis"
5350:
3558:contain elements that are not included in it.
3434:{\displaystyle 2^{\aleph _{0}}>\aleph _{0}}
2874:{\displaystyle {\mathfrak {c}}>\aleph _{0}}
7540:
5782:
2595:. This is usually taken as the definition of
2113:
1773:, if there exists an injective function from
16:Definition of the number of elements in a set
5699:
5684:
5494:; Srishti D. Chatterji; et al. (eds.),
5365:
4839:
4775:
4356:, peaches)} is a bijection between the sets
3013:is the smallest cardinal number bigger than
2462:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
2406:, and it can be shown that no function from
2397:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
2328:
2322:
2291:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
1682:, which was established by the existence of
1033:
1001:
969:
937:
201:
183:
157:
139:
5678:
5453:(4), Leipzig: B. G. Teubner: 438â443,
4680:Definition of cardinality in class theory (
3506:{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
2972:{\displaystyle \aleph _{1}=2^{\aleph _{0}}}
397:on each side; this is the same notation as
7547:
7533:
5974:
5789:
5775:
4884:
4880:
4736:denotes the class of all ordinal numbers.
4298:
2739:is the least cardinal number greater than
1709:
975:{\displaystyle E=\{0,2,4,6,{\text{...}}\}}
778:
655:does not have the same cardinality as its
5634:
5616:
5567:
5549:
5173:. San Francisco, CA: Dover Publications.
4480:
2507:
2481:
2452:
2416:
2387:
2351:
2281:
2238:
1663:
1575:{\displaystyle h(n)=n-(n{\text{ mod }}2)}
1487:
1375:
1098:
994:
799:(a.k.a., one-to-one correspondence) from
214:are the same size as they each contain 3
3530:is equal to the number of points in any
3322:); that is, there are more real numbers
647:
619:
497:{\displaystyle \operatorname {card} (A)}
54:has 5 elements. Thus the cardinality of
25:
5657:
4584:are related by the following equation:
3076:with cardinality less than that of the
2258:has cardinality strictly less than its
8180:
5796:
4438:Sets with cardinality of the continuum
3056:Finite, countable and uncountable sets
1245:injective or surjective function from
644:, both sets have the same cardinality.
8193:Basic concepts in infinite set theory
7528:
5770:
5590:
5523:
5390:
5388:
5313:
5194:
2836:"c"), and is also referred to as the
1970:is equivalent to the statement that
5706:, Berlin: Springer, pp. 378â439
5524:Cohen, Paul J. (December 15, 1963).
5307:
4947:{\displaystyle \bigcap \emptyset =V}
4324:= {apples, oranges, peaches}, where
3579:Hilbert's paradox of the Grand Hotel
3220:Hilbert's paradox of the Grand Hotel
986:has the same cardinality as the set
5591:Cohen, Paul J. (January 15, 1964).
4276:
4247:
4231:
4194:
4187:
4158:
4036:
4007:
3908:
3875:
3860:
3853:
3825:
3802:
3775:
3759:
3721:
3697:
3612:
3449:Cantor's first uncountability proof
3339:
3280:
3166:
2890:
2853:
2817:
2732:{\displaystyle \aleph _{\alpha +1}}
2538:
2533:Cantor's first uncountability proof
706:)} disagrees with every set in the
13:
5441:"Ăber das Problem der Wohlordnung"
5385:
4935:
4888:
4796:
4432:CantorâBernsteinâSchroeder theorem
4256:
4214:
4142:
4121:
4106:
4085:
4068:
4043:
3991:
3970:
3933:
3809:
3494:
3479:
3422:
3407:
3353:
3303:
3234:
3115:
3021:
2992:
2958:
2940:
2904:
2862:
2785:
2747:
2714:
2652:
2639:
2626:
2443:
2378:
2272:
1675:{\displaystyle |E|=|\mathbb {N} |}
511:
14:
8204:
5168:
2759:{\displaystyle \aleph _{\alpha }}
615:
122:describes a relationship between
7578:
7508:
5075:{\displaystyle |P|={\mbox{Ord}}}
4712:denote a class of all sets, and
3193:
2804:), while the cardinality of the
421:may alternatively be denoted by
5711:
5651:
5584:
5517:
5480:
5413:
5396:"Infinite Sets and Cardinality"
5359:
3635:). They include, for instance:
3620:{\displaystyle {\mathfrak {c}}}
3288:{\displaystyle {\mathfrak {c}}}
3174:{\displaystyle {\mathfrak {c}}}
3006:{\displaystyle 2^{\aleph _{0}}}
2825:{\displaystyle {\mathfrak {c}}}
339:; however, if referring to the
74:is 5 or, written symbolically,
7554:
5742:
5344:
5279:
5255:
5236:
5212:
5187:
5162:
5058:
5050:
4995:
4986:
4982:
4974:
4970:
4964:
4912:
4885:
4881:
4877:
4862:
4836:
4823:
4817:
4811:
4799:
4792:
4755:
4747:
4463:
4451:
3202:breaks down when dealing with
3050:§ Cardinality of the continuum
2456:
2448:
2391:
2383:
2341:is an injective function from
2316:
2310:
2285:
2277:
2048:
2040:
2032:
2024:
2002:
1994:
1986:
1978:
1948:
1940:
1932:
1924:
1902:
1894:
1886:
1878:
1856:
1848:
1840:
1832:
1668:
1658:
1650:
1642:
1569:
1555:
1543:
1537:
1431:
1425:
1066:
1060:
491:
485:
437:
431:
379:
371:
325:
317:
309:
301:
90:
82:
1:
7469:History of mathematical logic
5155:
4487:Inclusion-exclusion principle
4364:. The cardinality of each of
3629:generalized diagonal argument
3330:. Namely, Cantor showed that
2593:initial ordinal in that class
586:
7394:Primitive recursive function
5298:Third Millennium Mathematics
5263:"Early Human Counting Tools"
4729:{\displaystyle {\mbox{Ord}}}
3269:cardinality of the continuum
3263:Cardinality of the continuum
3257:Cardinality of the continuum
2838:cardinality of the continuum
2514:{\displaystyle \mathbb {R} }
2488:{\displaystyle \mathbb {N} }
2423:{\displaystyle \mathbb {N} }
2358:{\displaystyle \mathbb {N} }
2245:{\displaystyle \mathbb {N} }
1494:{\displaystyle \mathbb {N} }
1382:{\displaystyle \mathbb {N} }
1341:. For example, the function
1105:{\displaystyle \mathbb {N} }
578:. But with the discovery of
21:Cardinality (disambiguation)
7:
5287:Third Millennium Chronology
5285:Duncan J. Melville (2003).
5108:
3546:that have the same size as
3315:{\displaystyle \aleph _{0}}
3246:{\displaystyle \aleph _{0}}
3206:. In the late 19th century
3127:{\displaystyle \aleph _{0}}
3033:{\displaystyle \aleph _{0}}
2840:. Cantor showed, using the
2797:{\displaystyle \aleph _{0}}
2611:, the cardinalities of the
2055:{\displaystyle |B|\leq |A|}
2009:{\displaystyle |A|\leq |B|}
1909:{\displaystyle |B|\leq |A|}
1863:{\displaystyle |A|\leq |B|}
907:. Such sets are said to be
207:{\displaystyle B=\{2,4,6\}}
163:{\displaystyle A=\{1,2,3\}}
10:
8209:
8038:von NeumannâBernaysâGödel
6458:SchröderâBernstein theorem
6185:Monadic predicate calculus
5844:Foundations of mathematics
4484:
3891:can be demonstrated using
3641:the set of all subsets of
3445:Cantor's diagonal argument
3260:
3198:Our intuition gained from
2542:
2529:Cantor's diagonal argument
2334:{\displaystyle g(n)=\{n\}}
1964:SchröderâBernstein theorem
606:Cantor's diagonal argument
530:
18:
8102:
8065:
7977:
7867:
7839:One-to-one correspondence
7755:
7696:
7587:
7576:
7562:
7504:
7491:Philosophy of mathematics
7440:Automated theorem proving
7422:
7317:
7149:
7042:
6894:
6611:
6587:
6565:Von NeumannâBernaysâGödel
6510:
6404:
6308:
6206:
6197:
6124:
6059:
5965:
5887:
5804:
5497:GrundzĂŒge der Mengenlehre
5353:The Annals of Mathematics
5323:Texas A&M Mathematics
5316:"The History of Infinity"
3567:one-to-one correspondence
2580:The cardinality of a set
234:, and another which uses
3523:, if ZFC is consistent.
3459:states that there is no
624:Bijective function from
7141:Self-verifying theories
6962:Tarski's axiomatization
5913:Tarski's undefinability
5908:incompleteness theorems
5513:Original edition (1914)
5302:St. Lawrence University
4299:Examples and properties
2769:The cardinality of the
2699:{\displaystyle \alpha }
1955:{\displaystyle |A|=|B|}
1446:{\displaystyle g(n)=4n}
1081:{\displaystyle f(n)=2n}
363:, it is simply denoted
332:{\displaystyle |A|=|B|}
7797:Constructible universe
7624:Constructibility (V=L)
7515:Mathematics portal
7126:Proof of impossibility
6774:propositional variable
6084:Propositional calculus
5551:10.1073/pnas.50.6.1143
5400:Mathematics LibreTexts
5314:Allen, Donald (2003).
5242:Cepelewicz, Jordana
5105:and any proper class.
5099:
5076:
5033:
5009:
4948:
4919:
4846:
4730:
4706:
4670:
4567:
4481:Union and intersection
4470:
4289:
4170:
4019:
3885:
3837:
3787:
3730:
3681:Both have cardinality
3668:of all functions from
3621:
3507:
3435:
3381:
3316:
3289:
3247:
3175:
3128:
3034:
3007:
2973:
2919:
2875:
2826:
2798:
2760:
2733:
2700:
2674:
2515:
2489:
2463:
2424:
2398:
2359:
2335:
2292:
2246:
2217:
2193:
2169:
2145:
2102:
2078:
2056:
2010:
1956:
1910:
1864:
1813:
1789:
1765:
1741:
1698:
1676:
1626:
1602:
1576:
1519:
1495:
1469:
1447:
1407:
1383:
1357:
1333:
1309:
1285:
1261:
1233:
1209:
1193:bijection exists from
1181:
1157:
1130:
1106:
1082:
1040:
976:
891:
867:
839:
815:
771:
722:and the corresponding
666:): For every function
645:
640:is a proper subset of
521:
498:
465:
444:
415:
387:
357:
333:
285:
261:
208:
164:
111:
104:
68:
44:
8020:Principia Mathematica
7854:Transfinite induction
7713:(i.e. set difference)
7384:Kolmogorov complexity
7337:Computably enumerable
7237:Model complete theory
7029:Principia Mathematica
6089:Propositional formula
5918:BanachâTarski paradox
5618:10.1073/pnas.51.1.105
5446:Mathematische Annalen
5366:Georg Cantor (1891).
5100:
5077:
5034:
5010:
4949:
4920:
4847:
4731:
4707:
4671:
4568:
4471:
4383:|, then there exists
4336:are distinct, then |
4290:
4171:
4020:
3886:
3838:
3788:
3731:
3622:
3508:
3436:
3382:
3326:than natural numbers
3317:
3290:
3248:
3176:
3129:
3035:
3008:
2974:
2920:
2876:
2827:
2799:
2761:
2734:
2701:
2675:
2516:
2490:
2464:
2425:
2399:
2360:
2336:
2293:
2247:
2228:For example, the set
2218:
2194:
2170:
2146:
2103:
2079:
2057:
2011:
1957:
1911:
1865:
1814:
1790:
1766:
1742:
1699:
1677:
1627:
1603:
1577:
1520:
1496:
1470:
1448:
1408:
1384:
1358:
1334:
1310:
1286:
1262:
1234:
1210:
1182:
1158:
1131:
1107:
1083:
1050:, since the function
1041:
977:
924:For example, the set
892:
868:
840:
816:
651:
623:
522:
499:
466:
445:
416:
388:
358:
343:of an individual set
334:
286:
262:
209:
165:
105:
103:{\displaystyle |S|=5}
69:
45:
29:
8094:Burali-Forti paradox
7849:Set-builder notation
7802:Continuum hypothesis
7742:Symmetric difference
7332:ChurchâTuring thesis
7319:Computability theory
6528:continuum hypothesis
6046:Square of opposition
5904:Gödel's completeness
5421:Friedrich M. Hartogs
5171:Set Theory and Logic
5150:Pigeonhole principle
5089:
5046:
5023:
4961:
4929:
4859:
4743:
4716:
4696:
4591:
4510:
4448:
4181:
4030:
3902:
3847:
3796:
3753:
3688:
3607:
3590:space-filling curves
3521:axiomatic set theory
3470:
3457:continuum hypothesis
3398:
3334:
3299:
3275:
3230:
3161:
3111:
3017:
2983:
2936:
2930:continuum hypothesis
2885:
2848:
2812:
2781:
2743:
2710:
2690:
2622:
2601:axiomatic set theory
2558:equivalence relation
2503:
2477:
2438:
2412:
2373:
2347:
2304:
2267:
2234:
2207:
2183:
2159:
2135:
2114:Definition 3: |
2092:
2068:
2020:
1974:
1920:
1874:
1828:
1803:
1779:
1755:
1731:
1710:Definition 2: |
1688:
1638:
1616:
1592:
1531:
1509:
1483:
1459:
1419:
1397:
1371:
1347:
1323:
1299:
1275:
1251:
1223:
1199:
1171:
1147:
1120:
1094:
1088:is a bijection from
1054:
990:
928:
881:
857:
829:
805:
779:Definition 1: |
596:, the originator of
508:
476:
455:
443:{\displaystyle n(A)}
425:
405:
367:
347:
297:
275:
251:
174:
130:
78:
58:
34:
19:For other uses, see
8055:TarskiâGrothendieck
7486:Mathematical object
7377:P versus NP problem
7342:Computable function
7136:Reverse mathematics
7062:Logical consequence
6939:primitive recursive
6934:elementary function
6707:Free/bound variable
6560:TarskiâGrothendieck
6079:Logical connectives
6009:Logical equivalence
5859:Logical consequence
5725:(student edition),
5609:1964PNAS...51..105C
5542:1963PNAS...50.1143C
3893:cardinal arithmetic
3748:cardinal equalities
3565:, which provides a
3542:of an infinite set
3088:|, is said to be a
2881:. We can show that
520:{\displaystyle \#A}
386:{\displaystyle |A|}
7644:Limitation of size
7284:Transfer principle
7247:Semantics of logic
7232:Categorical theory
7208:Non-standard model
6722:Logical connective
5849:Information theory
5798:Mathematical logic
5708:Here: p.413 bottom
5459:10.1007/bf01458215
5292:2018-07-07 at the
5196:Weisstein, Eric W.
5095:
5072:
5070:
5029:
5005:
5003:
4944:
4915:
4842:
4834:
4789:
4767:
4726:
4724:
4702:
4666:
4563:
4466:
4442:irrational numbers
4285:
4166:
4015:
3881:
3833:
3783:
3726:
3617:
3503:
3431:
3377:
3312:
3285:
3243:
3171:
3136:countably infinite
3134:, is said to be a
3124:
3030:
3003:
2969:
2915:
2871:
2822:
2794:
2756:
2729:
2696:
2670:
2527:. For proofs, see
2511:
2485:
2459:
2420:
2394:
2355:
2331:
2288:
2242:
2213:
2189:
2165:
2141:
2120:| < |
2098:
2074:
2052:
2006:
1952:
1906:
1860:
1809:
1785:
1761:
1737:
1694:
1672:
1622:
1598:
1572:
1515:
1491:
1465:
1443:
1403:
1379:
1353:
1329:
1305:
1281:
1257:
1229:
1205:
1177:
1153:
1126:
1102:
1078:
1036:
972:
887:
863:
835:
811:
772:
646:
580:irrational numbers
517:
494:
461:
440:
411:
383:
353:
329:
281:
257:
204:
160:
112:
100:
64:
40:
8175:
8174:
8084:Russell's paradox
8033:ZermeloâFraenkel
7934:Dedekind-infinite
7807:Diagonal argument
7706:Cartesian product
7570:Set (mathematics)
7522:
7521:
7454:Abstract category
7257:Theories of truth
7067:Rule of inference
7057:Natural deduction
7038:
7037:
6583:
6582:
6288:Cartesian product
6193:
6192:
6099:Many-valued logic
6074:Boolean functions
5957:Russell's paradox
5932:diagonal argument
5829:First-order logic
5733:(library edition)
5665:, Vintage Books,
5332:on August 1, 2020
5199:"Cardinal Number"
5180:978-0-486-63829-4
5169:Stoll, Robert R.
5098:{\displaystyle V}
5069:
5032:{\displaystyle P}
5002:
4833:
4788:
4766:
4723:
4705:{\displaystyle V}
4444:and the interval
3224:Dedekind infinite
3185:|, is said to be
3153:|, for example |
3066:law of trichotomy
2842:diagonal argument
2566:equivalence class
2564:of all sets. The
2556:, and this is an
2216:{\displaystyle B}
2192:{\displaystyle A}
2168:{\displaystyle B}
2144:{\displaystyle A}
2101:{\displaystyle B}
2077:{\displaystyle A}
1962:(a fact known as
1812:{\displaystyle B}
1788:{\displaystyle A}
1764:{\displaystyle B}
1740:{\displaystyle A}
1697:{\displaystyle f}
1625:{\displaystyle h}
1601:{\displaystyle g}
1564:
1518:{\displaystyle E}
1468:{\displaystyle h}
1406:{\displaystyle E}
1356:{\displaystyle g}
1332:{\displaystyle B}
1308:{\displaystyle A}
1284:{\displaystyle B}
1260:{\displaystyle A}
1232:{\displaystyle B}
1208:{\displaystyle A}
1180:{\displaystyle B}
1156:{\displaystyle A}
1129:{\displaystyle E}
1031:
967:
890:{\displaystyle B}
866:{\displaystyle A}
838:{\displaystyle B}
814:{\displaystyle A}
464:{\displaystyle A}
414:{\displaystyle A}
356:{\displaystyle A}
284:{\displaystyle B}
260:{\displaystyle A}
67:{\displaystyle S}
43:{\displaystyle S}
8200:
8188:Cardinal numbers
8157:Bertrand Russell
8147:John von Neumann
8132:Abraham Fraenkel
8127:Richard Dedekind
8089:Suslin's problem
8000:Cantor's theorem
7717:De Morgan's laws
7582:
7549:
7542:
7535:
7526:
7525:
7513:
7512:
7464:History of logic
7459:Category of sets
7352:Decision problem
7131:Ordinal analysis
7072:Sequent calculus
6970:Boolean algebras
6910:
6909:
6884:
6855:logical/constant
6609:
6608:
6595:
6518:ZermeloâFraenkel
6269:Set operations:
6204:
6203:
6141:
5972:
5971:
5952:LöwenheimâSkolem
5839:Formal semantics
5791:
5784:
5777:
5768:
5767:
5761:
5746:
5734:
5715:
5709:
5707:
5696:
5682:
5676:
5675:
5655:
5649:
5648:
5638:
5620:
5588:
5582:
5581:
5571:
5553:
5536:(6): 1143â1148.
5521:
5515:
5510:
5492:Egbert Brieskorn
5484:
5478:
5477:
5429:Walther von Dyck
5417:
5411:
5410:
5408:
5407:
5392:
5383:
5382:
5372:
5363:
5357:
5356:
5348:
5342:
5341:
5339:
5337:
5331:
5325:. Archived from
5320:
5311:
5305:
5283:
5277:
5276:
5274:
5273:
5259:
5253:
5252:, August 9, 2021
5240:
5234:
5233:
5231:
5230:
5216:
5210:
5209:
5208:
5191:
5185:
5184:
5166:
5130:Cantor's theorem
5125:Cantor's paradox
5104:
5102:
5101:
5096:
5081:
5079:
5078:
5073:
5071:
5067:
5061:
5053:
5039:, in particular
5038:
5036:
5035:
5030:
5014:
5012:
5011:
5006:
5004:
5000:
4985:
4977:
4954:. In this case
4953:
4951:
4950:
4945:
4924:
4922:
4921:
4916:
4851:
4849:
4848:
4843:
4835:
4831:
4795:
4790:
4786:
4768:
4764:
4758:
4750:
4735:
4733:
4732:
4727:
4725:
4721:
4711:
4709:
4708:
4703:
4675:
4673:
4672:
4667:
4662:
4648:
4634:
4630:
4612:
4608:
4572:
4570:
4569:
4564:
4559:
4545:
4531:
4527:
4475:
4473:
4472:
4469:{\displaystyle }
4467:
4294:
4292:
4291:
4286:
4281:
4280:
4279:
4266:
4265:
4264:
4263:
4251:
4250:
4236:
4235:
4234:
4228:
4224:
4223:
4222:
4221:
4199:
4198:
4197:
4191:
4190:
4175:
4173:
4172:
4167:
4162:
4161:
4152:
4151:
4150:
4149:
4132:
4131:
4130:
4129:
4128:
4115:
4114:
4113:
4095:
4094:
4093:
4092:
4082:
4078:
4077:
4076:
4075:
4053:
4052:
4051:
4050:
4040:
4039:
4024:
4022:
4021:
4016:
4011:
4010:
4001:
4000:
3999:
3998:
3981:
3980:
3979:
3978:
3977:
3953:
3952:
3947:
3943:
3942:
3941:
3940:
3918:
3917:
3912:
3911:
3890:
3888:
3887:
3882:
3880:
3879:
3878:
3865:
3864:
3863:
3857:
3856:
3842:
3840:
3839:
3834:
3829:
3828:
3819:
3818:
3817:
3816:
3806:
3805:
3792:
3790:
3789:
3784:
3779:
3778:
3769:
3768:
3763:
3762:
3735:
3733:
3732:
3727:
3725:
3724:
3715:
3714:
3702:
3701:
3700:
3626:
3624:
3623:
3618:
3616:
3615:
3563:tangent function
3540:proper supersets
3528:real number line
3512:
3510:
3509:
3504:
3502:
3501:
3489:
3488:
3487:
3486:
3440:
3438:
3437:
3432:
3430:
3429:
3417:
3416:
3415:
3414:
3386:
3384:
3383:
3378:
3376:
3375:
3363:
3362:
3361:
3360:
3343:
3342:
3321:
3319:
3318:
3313:
3311:
3310:
3294:
3292:
3291:
3286:
3284:
3283:
3252:
3250:
3249:
3244:
3242:
3241:
3216:Richard Dedekind
3180:
3178:
3177:
3172:
3170:
3169:
3133:
3131:
3130:
3125:
3123:
3122:
3039:
3037:
3036:
3031:
3029:
3028:
3012:
3010:
3009:
3004:
3002:
3001:
3000:
2999:
2978:
2976:
2975:
2970:
2968:
2967:
2966:
2965:
2948:
2947:
2924:
2922:
2921:
2916:
2914:
2913:
2912:
2911:
2894:
2893:
2880:
2878:
2877:
2872:
2870:
2869:
2857:
2856:
2831:
2829:
2828:
2823:
2821:
2820:
2803:
2801:
2800:
2795:
2793:
2792:
2765:
2763:
2762:
2757:
2755:
2754:
2738:
2736:
2735:
2730:
2728:
2727:
2705:
2703:
2702:
2697:
2679:
2677:
2676:
2671:
2660:
2659:
2647:
2646:
2634:
2633:
2539:Cardinal numbers
2522:
2520:
2518:
2517:
2512:
2510:
2496:
2494:
2492:
2491:
2486:
2484:
2470:
2468:
2466:
2465:
2460:
2455:
2447:
2446:
2431:
2429:
2427:
2426:
2421:
2419:
2405:
2403:
2401:
2400:
2395:
2390:
2382:
2381:
2366:
2364:
2362:
2361:
2356:
2354:
2340:
2338:
2337:
2332:
2299:
2297:
2295:
2294:
2289:
2284:
2276:
2275:
2253:
2251:
2249:
2248:
2243:
2241:
2224:
2222:
2220:
2219:
2214:
2200:
2198:
2196:
2195:
2190:
2176:
2174:
2172:
2171:
2166:
2152:
2150:
2148:
2147:
2142:
2125:
2119:
2109:
2107:
2105:
2104:
2099:
2085:
2083:
2081:
2080:
2075:
2061:
2059:
2058:
2053:
2051:
2043:
2035:
2027:
2015:
2013:
2012:
2007:
2005:
1997:
1989:
1981:
1961:
1959:
1958:
1953:
1951:
1943:
1935:
1927:
1915:
1913:
1912:
1907:
1905:
1897:
1889:
1881:
1869:
1867:
1866:
1861:
1859:
1851:
1843:
1835:
1820:
1818:
1816:
1815:
1810:
1796:
1794:
1792:
1791:
1786:
1772:
1770:
1768:
1767:
1762:
1748:
1746:
1744:
1743:
1738:
1721:
1715:
1705:
1703:
1701:
1700:
1695:
1681:
1679:
1678:
1673:
1671:
1666:
1661:
1653:
1645:
1633:
1631:
1629:
1628:
1623:
1609:
1607:
1605:
1604:
1599:
1584:modulo operation
1581:
1579:
1578:
1573:
1565:
1562:
1526:
1524:
1522:
1521:
1516:
1502:
1500:
1498:
1497:
1492:
1490:
1476:
1474:
1472:
1471:
1466:
1452:
1450:
1449:
1444:
1414:
1412:
1410:
1409:
1404:
1390:
1388:
1386:
1385:
1380:
1378:
1364:
1362:
1360:
1359:
1354:
1340:
1338:
1336:
1335:
1330:
1316:
1314:
1312:
1311:
1306:
1292:
1290:
1288:
1287:
1282:
1268:
1266:
1264:
1263:
1258:
1240:
1238:
1236:
1235:
1230:
1216:
1214:
1212:
1211:
1206:
1188:
1186:
1184:
1183:
1178:
1164:
1162:
1160:
1159:
1154:
1141:For finite sets
1137:
1135:
1133:
1132:
1127:
1113:
1111:
1109:
1108:
1103:
1101:
1087:
1085:
1084:
1079:
1045:
1043:
1042:
1037:
1032:
1029:
997:
982:of non-negative
981:
979:
978:
973:
968:
965:
898:
896:
894:
893:
888:
874:
872:
870:
869:
864:
846:
844:
842:
841:
836:
822:
820:
818:
817:
812:
790:
784:
753:
731:
610:uncountable sets
591:
588:
560:commensurability
542:
541:
526:
524:
523:
518:
503:
501:
500:
495:
471:
470:
468:
467:
462:
449:
447:
446:
441:
420:
418:
417:
412:
392:
390:
389:
384:
382:
374:
362:
360:
359:
354:
338:
336:
335:
330:
328:
320:
312:
304:
292:
290:
288:
287:
282:
268:
266:
264:
263:
258:
236:cardinal numbers
213:
211:
210:
205:
169:
167:
166:
161:
109:
107:
106:
101:
93:
85:
73:
71:
70:
65:
49:
47:
46:
41:
8208:
8207:
8203:
8202:
8201:
8199:
8198:
8197:
8178:
8177:
8176:
8171:
8098:
8077:
8061:
8026:New Foundations
7973:
7863:
7782:Cardinal number
7765:
7751:
7692:
7583:
7574:
7558:
7553:
7523:
7518:
7507:
7500:
7445:Category theory
7435:Algebraic logic
7418:
7389:Lambda calculus
7327:Church encoding
7313:
7289:Truth predicate
7145:
7111:Complete theory
7034:
6903:
6899:
6895:
6890:
6882:
6602: and
6598:
6593:
6579:
6555:New Foundations
6523:axiom of choice
6506:
6468:Gödel numbering
6408: and
6400:
6304:
6189:
6139:
6120:
6069:Boolean algebra
6055:
6019:Equiconsistency
5984:Classical logic
5961:
5942:Halting problem
5930: and
5906: and
5894: and
5893:
5888:Theorems (
5883:
5800:
5795:
5765:
5764:
5747:
5743:
5738:
5737:
5716:
5712:
5697:
5683:
5679:
5673:
5656:
5652:
5589:
5585:
5522:
5518:
5508:
5488:Felix Hausdorff
5485:
5481:
5437:Otto Blumenthal
5418:
5414:
5405:
5403:
5394:
5393:
5386:
5370:
5364:
5360:
5349:
5345:
5335:
5333:
5329:
5318:
5312:
5308:
5294:Wayback Machine
5284:
5280:
5271:
5269:
5261:
5260:
5256:
5241:
5237:
5228:
5226:
5218:
5217:
5213:
5192:
5188:
5181:
5167:
5163:
5158:
5111:
5090:
5087:
5086:
5065:
5057:
5049:
5047:
5044:
5043:
5024:
5021:
5020:
4998:
4981:
4973:
4962:
4959:
4958:
4930:
4927:
4926:
4860:
4857:
4856:
4832: injective
4829:
4791:
4784:
4762:
4754:
4746:
4744:
4741:
4740:
4719:
4717:
4714:
4713:
4697:
4694:
4693:
4690:
4652:
4638:
4620:
4616:
4598:
4594:
4592:
4589:
4588:
4549:
4535:
4517:
4513:
4511:
4508:
4507:
4489:
4483:
4449:
4446:
4445:
4301:
4275:
4274:
4270:
4259:
4255:
4246:
4245:
4244:
4240:
4230:
4229:
4217:
4213:
4212:
4208:
4204:
4203:
4193:
4192:
4186:
4185:
4184:
4182:
4179:
4178:
4157:
4156:
4145:
4141:
4140:
4136:
4124:
4120:
4119:
4109:
4105:
4104:
4103:
4099:
4088:
4084:
4083:
4071:
4067:
4066:
4062:
4058:
4057:
4046:
4042:
4041:
4035:
4034:
4033:
4031:
4028:
4027:
4006:
4005:
3994:
3990:
3989:
3985:
3973:
3969:
3968:
3961:
3957:
3948:
3936:
3932:
3931:
3927:
3923:
3922:
3913:
3907:
3906:
3905:
3903:
3900:
3899:
3874:
3873:
3869:
3859:
3858:
3852:
3851:
3850:
3848:
3845:
3844:
3824:
3823:
3812:
3808:
3807:
3801:
3800:
3799:
3797:
3794:
3793:
3774:
3773:
3764:
3758:
3757:
3756:
3754:
3751:
3750:
3720:
3719:
3710:
3706:
3696:
3695:
3691:
3689:
3686:
3685:
3627:exist (see his
3611:
3610:
3608:
3605:
3604:
3588:introduced the
3497:
3493:
3482:
3478:
3477:
3473:
3471:
3468:
3467:
3461:cardinal number
3425:
3421:
3410:
3406:
3405:
3401:
3399:
3396:
3395:
3371:
3367:
3356:
3352:
3351:
3347:
3338:
3337:
3335:
3332:
3331:
3306:
3302:
3300:
3297:
3296:
3279:
3278:
3276:
3273:
3272:
3265:
3259:
3237:
3233:
3231:
3228:
3227:
3196:
3165:
3164:
3162:
3159:
3158:
3118:
3114:
3112:
3109:
3108:
3078:natural numbers
3062:axiom of choice
3058:
3024:
3020:
3018:
3015:
3014:
2995:
2991:
2990:
2986:
2984:
2981:
2980:
2961:
2957:
2956:
2952:
2943:
2939:
2937:
2934:
2933:
2907:
2903:
2902:
2898:
2889:
2888:
2886:
2883:
2882:
2865:
2861:
2852:
2851:
2849:
2846:
2845:
2832:" (a lowercase
2816:
2815:
2813:
2810:
2809:
2808:is denoted by "
2788:
2784:
2782:
2779:
2778:
2771:natural numbers
2750:
2746:
2744:
2741:
2740:
2717:
2713:
2711:
2708:
2707:
2691:
2688:
2687:
2655:
2651:
2642:
2638:
2629:
2625:
2623:
2620:
2619:
2609:axiom of choice
2597:cardinal number
2547:
2545:Cardinal number
2541:
2506:
2504:
2501:
2500:
2498:
2480:
2478:
2475:
2474:
2472:
2451:
2442:
2441:
2439:
2436:
2435:
2433:
2415:
2413:
2410:
2409:
2407:
2386:
2377:
2376:
2374:
2371:
2370:
2368:
2350:
2348:
2345:
2344:
2342:
2305:
2302:
2301:
2280:
2271:
2270:
2268:
2265:
2264:
2262:
2256:natural numbers
2237:
2235:
2232:
2231:
2229:
2208:
2205:
2204:
2202:
2184:
2181:
2180:
2178:
2160:
2157:
2156:
2154:
2136:
2133:
2132:
2130:
2128:
2121:
2115:
2093:
2090:
2089:
2087:
2069:
2066:
2065:
2063:
2047:
2039:
2031:
2023:
2021:
2018:
2017:
2001:
1993:
1985:
1977:
1975:
1972:
1971:
1968:axiom of choice
1947:
1939:
1931:
1923:
1921:
1918:
1917:
1901:
1893:
1885:
1877:
1875:
1872:
1871:
1855:
1847:
1839:
1831:
1829:
1826:
1825:
1804:
1801:
1800:
1798:
1780:
1777:
1776:
1774:
1756:
1753:
1752:
1750:
1732:
1729:
1728:
1726:
1724:
1717:
1716:| †|
1711:
1689:
1686:
1685:
1683:
1667:
1662:
1657:
1649:
1641:
1639:
1636:
1635:
1617:
1614:
1613:
1611:
1593:
1590:
1589:
1587:
1563: mod
1561:
1532:
1529:
1528:
1510:
1507:
1506:
1504:
1486:
1484:
1481:
1480:
1478:
1460:
1457:
1456:
1454:
1420:
1417:
1416:
1398:
1395:
1394:
1392:
1374:
1372:
1369:
1368:
1366:
1348:
1345:
1344:
1342:
1324:
1321:
1320:
1318:
1300:
1297:
1296:
1294:
1276:
1273:
1272:
1270:
1252:
1249:
1248:
1246:
1224:
1221:
1220:
1218:
1200:
1197:
1196:
1194:
1172:
1169:
1168:
1166:
1148:
1145:
1144:
1142:
1138:(see picture).
1121:
1118:
1117:
1115:
1097:
1095:
1092:
1091:
1089:
1055:
1052:
1051:
1048:natural numbers
1028:
993:
991:
988:
987:
964:
929:
926:
925:
882:
879:
878:
876:
858:
855:
854:
852:
830:
827:
826:
824:
806:
803:
802:
800:
793:
786:
785:| = |
780:
749:
727:
618:
589:
539:
537:
533:
509:
506:
505:
477:
474:
473:
456:
453:
452:
451:
426:
423:
422:
406:
403:
402:
378:
370:
368:
365:
364:
348:
345:
344:
341:cardinal number
324:
316:
308:
300:
298:
295:
294:
276:
273:
272:
270:
252:
249:
248:
246:
245:When two sets,
175:
172:
171:
131:
128:
127:
89:
81:
79:
76:
75:
59:
56:
55:
52:Platonic solids
35:
32:
31:
24:
17:
12:
11:
5:
8206:
8196:
8195:
8190:
8173:
8172:
8170:
8169:
8164:
8162:Thoralf Skolem
8159:
8154:
8149:
8144:
8139:
8134:
8129:
8124:
8119:
8114:
8108:
8106:
8100:
8099:
8097:
8096:
8091:
8086:
8080:
8078:
8076:
8075:
8072:
8066:
8063:
8062:
8060:
8059:
8058:
8057:
8052:
8047:
8046:
8045:
8030:
8029:
8028:
8016:
8015:
8014:
8003:
8002:
7997:
7992:
7987:
7981:
7979:
7975:
7974:
7972:
7971:
7966:
7961:
7956:
7947:
7942:
7937:
7927:
7922:
7921:
7920:
7915:
7910:
7900:
7890:
7885:
7880:
7874:
7872:
7865:
7864:
7862:
7861:
7856:
7851:
7846:
7844:Ordinal number
7841:
7836:
7831:
7826:
7825:
7824:
7819:
7809:
7804:
7799:
7794:
7789:
7779:
7774:
7768:
7766:
7764:
7763:
7760:
7756:
7753:
7752:
7750:
7749:
7744:
7739:
7734:
7729:
7724:
7722:Disjoint union
7719:
7714:
7708:
7702:
7700:
7694:
7693:
7691:
7690:
7689:
7688:
7683:
7672:
7671:
7669:Martin's axiom
7666:
7661:
7656:
7651:
7646:
7641:
7636:
7634:Extensionality
7631:
7626:
7621:
7620:
7619:
7614:
7609:
7599:
7593:
7591:
7585:
7584:
7577:
7575:
7573:
7572:
7566:
7564:
7560:
7559:
7552:
7551:
7544:
7537:
7529:
7520:
7519:
7505:
7502:
7501:
7499:
7498:
7493:
7488:
7483:
7478:
7477:
7476:
7466:
7461:
7456:
7447:
7442:
7437:
7432:
7430:Abstract logic
7426:
7424:
7420:
7419:
7417:
7416:
7411:
7409:Turing machine
7406:
7401:
7396:
7391:
7386:
7381:
7380:
7379:
7374:
7369:
7364:
7359:
7349:
7347:Computable set
7344:
7339:
7334:
7329:
7323:
7321:
7315:
7314:
7312:
7311:
7306:
7301:
7296:
7291:
7286:
7281:
7276:
7275:
7274:
7269:
7264:
7254:
7249:
7244:
7242:Satisfiability
7239:
7234:
7229:
7228:
7227:
7217:
7216:
7215:
7205:
7204:
7203:
7198:
7193:
7188:
7183:
7173:
7172:
7171:
7166:
7159:Interpretation
7155:
7153:
7147:
7146:
7144:
7143:
7138:
7133:
7128:
7123:
7113:
7108:
7107:
7106:
7105:
7104:
7094:
7089:
7079:
7074:
7069:
7064:
7059:
7054:
7048:
7046:
7040:
7039:
7036:
7035:
7033:
7032:
7024:
7023:
7022:
7021:
7016:
7015:
7014:
7009:
7004:
6984:
6983:
6982:
6980:minimal axioms
6977:
6966:
6965:
6964:
6953:
6952:
6951:
6946:
6941:
6936:
6931:
6926:
6913:
6911:
6892:
6891:
6889:
6888:
6887:
6886:
6874:
6869:
6868:
6867:
6862:
6857:
6852:
6842:
6837:
6832:
6827:
6826:
6825:
6820:
6810:
6809:
6808:
6803:
6798:
6793:
6783:
6778:
6777:
6776:
6771:
6766:
6756:
6755:
6754:
6749:
6744:
6739:
6734:
6729:
6719:
6714:
6709:
6704:
6703:
6702:
6697:
6692:
6687:
6677:
6672:
6670:Formation rule
6667:
6662:
6661:
6660:
6655:
6645:
6644:
6643:
6633:
6628:
6623:
6618:
6612:
6606:
6589:Formal systems
6585:
6584:
6581:
6580:
6578:
6577:
6572:
6567:
6562:
6557:
6552:
6547:
6542:
6537:
6532:
6531:
6530:
6525:
6514:
6512:
6508:
6507:
6505:
6504:
6503:
6502:
6492:
6487:
6486:
6485:
6478:Large cardinal
6475:
6470:
6465:
6460:
6455:
6441:
6440:
6439:
6434:
6429:
6414:
6412:
6402:
6401:
6399:
6398:
6397:
6396:
6391:
6386:
6376:
6371:
6366:
6361:
6356:
6351:
6346:
6341:
6336:
6331:
6326:
6321:
6315:
6313:
6306:
6305:
6303:
6302:
6301:
6300:
6295:
6290:
6285:
6280:
6275:
6267:
6266:
6265:
6260:
6250:
6245:
6243:Extensionality
6240:
6238:Ordinal number
6235:
6225:
6220:
6219:
6218:
6207:
6201:
6195:
6194:
6191:
6190:
6188:
6187:
6182:
6177:
6172:
6167:
6162:
6157:
6156:
6155:
6145:
6144:
6143:
6130:
6128:
6122:
6121:
6119:
6118:
6117:
6116:
6111:
6106:
6096:
6091:
6086:
6081:
6076:
6071:
6065:
6063:
6057:
6056:
6054:
6053:
6048:
6043:
6038:
6033:
6028:
6023:
6022:
6021:
6011:
6006:
6001:
5996:
5991:
5986:
5980:
5978:
5969:
5963:
5962:
5960:
5959:
5954:
5949:
5944:
5939:
5934:
5922:Cantor's
5920:
5915:
5910:
5900:
5898:
5885:
5884:
5882:
5881:
5876:
5871:
5866:
5861:
5856:
5851:
5846:
5841:
5836:
5831:
5826:
5821:
5820:
5819:
5808:
5806:
5802:
5801:
5794:
5793:
5786:
5779:
5771:
5763:
5762:
5740:
5739:
5736:
5735:
5710:
5698:Reprinted in:
5677:
5671:
5650:
5603:(1): 105â110.
5583:
5516:
5506:
5479:
5412:
5384:
5358:
5343:
5306:
5278:
5254:
5235:
5211:
5186:
5179:
5160:
5159:
5157:
5154:
5153:
5152:
5147:
5142:
5137:
5132:
5127:
5122:
5117:
5110:
5107:
5094:
5083:
5082:
5064:
5060:
5056:
5052:
5028:
5017:
5016:
4997:
4994:
4991:
4988:
4984:
4980:
4976:
4972:
4969:
4966:
4943:
4940:
4937:
4934:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4883:
4879:
4876:
4873:
4870:
4867:
4864:
4853:
4852:
4841:
4838:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4794:
4783:
4780:
4777:
4774:
4771:
4761:
4757:
4753:
4749:
4701:
4689:
4678:
4677:
4676:
4665:
4661:
4658:
4655:
4651:
4647:
4644:
4641:
4637:
4633:
4629:
4626:
4623:
4619:
4615:
4611:
4607:
4604:
4601:
4597:
4574:
4573:
4562:
4558:
4555:
4552:
4548:
4544:
4541:
4538:
4534:
4530:
4526:
4523:
4520:
4516:
4485:Main article:
4482:
4479:
4478:
4477:
4465:
4462:
4459:
4456:
4453:
4435:
4404:
4373:
4300:
4297:
4296:
4295:
4284:
4278:
4273:
4269:
4262:
4258:
4254:
4249:
4243:
4239:
4233:
4227:
4220:
4216:
4211:
4207:
4202:
4196:
4189:
4176:
4165:
4160:
4155:
4148:
4144:
4139:
4135:
4127:
4123:
4118:
4112:
4108:
4102:
4098:
4091:
4087:
4081:
4074:
4070:
4065:
4061:
4056:
4049:
4045:
4038:
4025:
4014:
4009:
4004:
3997:
3993:
3988:
3984:
3976:
3972:
3967:
3964:
3960:
3956:
3951:
3946:
3939:
3935:
3930:
3926:
3921:
3916:
3910:
3877:
3872:
3868:
3862:
3855:
3832:
3827:
3822:
3815:
3811:
3804:
3782:
3777:
3772:
3767:
3761:
3744:
3743:
3736:
3723:
3718:
3713:
3709:
3705:
3699:
3694:
3679:
3678:
3677:
3676:
3662:
3614:
3586:Giuseppe Peano
3573:(âÂœÏ, ÂœÏ) and
3536:proper subsets
3514:
3513:
3500:
3496:
3492:
3485:
3481:
3476:
3453:
3452:
3441:
3428:
3424:
3420:
3413:
3409:
3404:
3374:
3370:
3366:
3359:
3355:
3350:
3346:
3341:
3309:
3305:
3282:
3261:Main article:
3258:
3255:
3240:
3236:
3195:
3192:
3191:
3190:
3168:
3139:
3121:
3117:
3093:
3057:
3054:
3027:
3023:
2998:
2994:
2989:
2964:
2960:
2955:
2951:
2946:
2942:
2910:
2906:
2901:
2897:
2892:
2868:
2864:
2860:
2855:
2834:fraktur script
2819:
2791:
2787:
2753:
2749:
2726:
2723:
2720:
2716:
2695:
2681:
2680:
2669:
2666:
2663:
2658:
2654:
2650:
2645:
2641:
2637:
2632:
2628:
2605:
2604:
2589:representative
2585:
2554:equinumerosity
2543:Main article:
2540:
2537:
2509:
2483:
2458:
2454:
2450:
2445:
2418:
2393:
2389:
2385:
2380:
2353:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2287:
2283:
2279:
2274:
2240:
2212:
2188:
2164:
2140:
2127:
2112:
2097:
2073:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1808:
1784:
1760:
1736:
1723:
1708:
1693:
1670:
1665:
1660:
1656:
1652:
1648:
1644:
1634:can challenge
1621:
1597:
1571:
1568:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1514:
1489:
1464:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1402:
1377:
1352:
1328:
1304:
1280:
1256:
1228:
1204:
1176:
1152:
1125:
1100:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1035:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
996:
971:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
886:
862:
834:
810:
792:
777:
617:
616:Comparing sets
614:
532:
529:
516:
513:
493:
490:
487:
484:
481:
460:
439:
436:
433:
430:
410:
399:absolute value
381:
377:
373:
352:
327:
323:
319:
315:
311:
307:
303:
280:
256:
203:
200:
197:
194:
191:
188:
185:
182:
179:
159:
156:
153:
150:
147:
144:
141:
138:
135:
99:
96:
92:
88:
84:
63:
39:
15:
9:
6:
4:
3:
2:
8205:
8194:
8191:
8189:
8186:
8185:
8183:
8168:
8167:Ernst Zermelo
8165:
8163:
8160:
8158:
8155:
8153:
8152:Willard Quine
8150:
8148:
8145:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8118:
8115:
8113:
8110:
8109:
8107:
8105:
8104:Set theorists
8101:
8095:
8092:
8090:
8087:
8085:
8082:
8081:
8079:
8073:
8071:
8068:
8067:
8064:
8056:
8053:
8051:
8050:KripkeâPlatek
8048:
8044:
8041:
8040:
8039:
8036:
8035:
8034:
8031:
8027:
8024:
8023:
8022:
8021:
8017:
8013:
8010:
8009:
8008:
8005:
8004:
8001:
7998:
7996:
7993:
7991:
7988:
7986:
7983:
7982:
7980:
7976:
7970:
7967:
7965:
7962:
7960:
7957:
7955:
7953:
7948:
7946:
7943:
7941:
7938:
7935:
7931:
7928:
7926:
7923:
7919:
7916:
7914:
7911:
7909:
7906:
7905:
7904:
7901:
7898:
7894:
7891:
7889:
7886:
7884:
7881:
7879:
7876:
7875:
7873:
7870:
7866:
7860:
7857:
7855:
7852:
7850:
7847:
7845:
7842:
7840:
7837:
7835:
7832:
7830:
7827:
7823:
7820:
7818:
7815:
7814:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7790:
7787:
7783:
7780:
7778:
7775:
7773:
7770:
7769:
7767:
7761:
7758:
7757:
7754:
7748:
7745:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7712:
7709:
7707:
7704:
7703:
7701:
7699:
7695:
7687:
7686:specification
7684:
7682:
7679:
7678:
7677:
7674:
7673:
7670:
7667:
7665:
7662:
7660:
7657:
7655:
7652:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7630:
7627:
7625:
7622:
7618:
7615:
7613:
7610:
7608:
7605:
7604:
7603:
7600:
7598:
7595:
7594:
7592:
7590:
7586:
7581:
7571:
7568:
7567:
7565:
7561:
7557:
7550:
7545:
7543:
7538:
7536:
7531:
7530:
7527:
7517:
7516:
7511:
7503:
7497:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7475:
7472:
7471:
7470:
7467:
7465:
7462:
7460:
7457:
7455:
7451:
7448:
7446:
7443:
7441:
7438:
7436:
7433:
7431:
7428:
7427:
7425:
7421:
7415:
7412:
7410:
7407:
7405:
7404:Recursive set
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7360:
7358:
7355:
7354:
7353:
7350:
7348:
7345:
7343:
7340:
7338:
7335:
7333:
7330:
7328:
7325:
7324:
7322:
7320:
7316:
7310:
7307:
7305:
7302:
7300:
7297:
7295:
7292:
7290:
7287:
7285:
7282:
7280:
7277:
7273:
7270:
7268:
7265:
7263:
7260:
7259:
7258:
7255:
7253:
7250:
7248:
7245:
7243:
7240:
7238:
7235:
7233:
7230:
7226:
7223:
7222:
7221:
7218:
7214:
7213:of arithmetic
7211:
7210:
7209:
7206:
7202:
7199:
7197:
7194:
7192:
7189:
7187:
7184:
7182:
7179:
7178:
7177:
7174:
7170:
7167:
7165:
7162:
7161:
7160:
7157:
7156:
7154:
7152:
7148:
7142:
7139:
7137:
7134:
7132:
7129:
7127:
7124:
7121:
7120:from ZFC
7117:
7114:
7112:
7109:
7103:
7100:
7099:
7098:
7095:
7093:
7090:
7088:
7085:
7084:
7083:
7080:
7078:
7075:
7073:
7070:
7068:
7065:
7063:
7060:
7058:
7055:
7053:
7050:
7049:
7047:
7045:
7041:
7031:
7030:
7026:
7025:
7020:
7019:non-Euclidean
7017:
7013:
7010:
7008:
7005:
7003:
7002:
6998:
6997:
6995:
6992:
6991:
6989:
6985:
6981:
6978:
6976:
6973:
6972:
6971:
6967:
6963:
6960:
6959:
6958:
6954:
6950:
6947:
6945:
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6921:
6919:
6915:
6914:
6912:
6907:
6901:
6896:Example
6893:
6885:
6880:
6879:
6878:
6875:
6873:
6870:
6866:
6863:
6861:
6858:
6856:
6853:
6851:
6848:
6847:
6846:
6843:
6841:
6838:
6836:
6833:
6831:
6828:
6824:
6821:
6819:
6816:
6815:
6814:
6811:
6807:
6804:
6802:
6799:
6797:
6794:
6792:
6789:
6788:
6787:
6784:
6782:
6779:
6775:
6772:
6770:
6767:
6765:
6762:
6761:
6760:
6757:
6753:
6750:
6748:
6745:
6743:
6740:
6738:
6735:
6733:
6730:
6728:
6725:
6724:
6723:
6720:
6718:
6715:
6713:
6710:
6708:
6705:
6701:
6698:
6696:
6693:
6691:
6688:
6686:
6683:
6682:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6659:
6656:
6654:
6653:by definition
6651:
6650:
6649:
6646:
6642:
6639:
6638:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6613:
6610:
6607:
6605:
6601:
6596:
6590:
6586:
6576:
6573:
6571:
6568:
6566:
6563:
6561:
6558:
6556:
6553:
6551:
6548:
6546:
6543:
6541:
6540:KripkeâPlatek
6538:
6536:
6533:
6529:
6526:
6524:
6521:
6520:
6519:
6516:
6515:
6513:
6509:
6501:
6498:
6497:
6496:
6493:
6491:
6488:
6484:
6481:
6480:
6479:
6476:
6474:
6471:
6469:
6466:
6464:
6461:
6459:
6456:
6453:
6449:
6445:
6442:
6438:
6435:
6433:
6430:
6428:
6425:
6424:
6423:
6419:
6416:
6415:
6413:
6411:
6407:
6403:
6395:
6392:
6390:
6387:
6385:
6384:constructible
6382:
6381:
6380:
6377:
6375:
6372:
6370:
6367:
6365:
6362:
6360:
6357:
6355:
6352:
6350:
6347:
6345:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6316:
6314:
6312:
6307:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6281:
6279:
6276:
6274:
6271:
6270:
6268:
6264:
6261:
6259:
6256:
6255:
6254:
6251:
6249:
6246:
6244:
6241:
6239:
6236:
6234:
6230:
6226:
6224:
6221:
6217:
6214:
6213:
6212:
6209:
6208:
6205:
6202:
6200:
6196:
6186:
6183:
6181:
6178:
6176:
6173:
6171:
6168:
6166:
6163:
6161:
6158:
6154:
6151:
6150:
6149:
6146:
6142:
6137:
6136:
6135:
6132:
6131:
6129:
6127:
6123:
6115:
6112:
6110:
6107:
6105:
6102:
6101:
6100:
6097:
6095:
6092:
6090:
6087:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6066:
6064:
6062:
6061:Propositional
6058:
6052:
6049:
6047:
6044:
6042:
6039:
6037:
6034:
6032:
6029:
6027:
6024:
6020:
6017:
6016:
6015:
6012:
6010:
6007:
6005:
6002:
6000:
5997:
5995:
5992:
5990:
5989:Logical truth
5987:
5985:
5982:
5981:
5979:
5977:
5973:
5970:
5968:
5964:
5958:
5955:
5953:
5950:
5948:
5945:
5943:
5940:
5938:
5935:
5933:
5929:
5925:
5921:
5919:
5916:
5914:
5911:
5909:
5905:
5902:
5901:
5899:
5897:
5891:
5886:
5880:
5877:
5875:
5872:
5870:
5867:
5865:
5862:
5860:
5857:
5855:
5852:
5850:
5847:
5845:
5842:
5840:
5837:
5835:
5832:
5830:
5827:
5825:
5822:
5818:
5815:
5814:
5813:
5810:
5809:
5807:
5803:
5799:
5792:
5787:
5785:
5780:
5778:
5773:
5772:
5769:
5759:
5755:
5751:
5745:
5741:
5732:
5731:0-85312-563-5
5728:
5724:
5723:0-85312-612-7
5720:
5714:
5705:
5704:
5694:
5690:
5689:
5681:
5674:
5672:0-09-944068-7
5668:
5664:
5660:
5654:
5646:
5642:
5637:
5632:
5628:
5624:
5619:
5614:
5610:
5606:
5602:
5598:
5594:
5587:
5579:
5575:
5570:
5565:
5561:
5557:
5552:
5547:
5543:
5539:
5535:
5531:
5527:
5520:
5514:
5509:
5507:3-540-42224-2
5503:
5499:
5498:
5493:
5489:
5483:
5476:
5472:
5468:
5464:
5460:
5456:
5452:
5448:
5447:
5442:
5438:
5434:
5433:David Hilbert
5430:
5426:
5422:
5416:
5401:
5397:
5391:
5389:
5380:
5376:
5369:
5362:
5354:
5347:
5328:
5324:
5317:
5310:
5303:
5299:
5295:
5291:
5288:
5282:
5268:
5267:Math Timeline
5264:
5258:
5251:
5247:
5246:
5239:
5225:
5224:brilliant.org
5221:
5215:
5206:
5205:
5200:
5197:
5190:
5182:
5176:
5172:
5165:
5161:
5151:
5148:
5146:
5143:
5141:
5138:
5136:
5135:Countable set
5133:
5131:
5128:
5126:
5123:
5121:
5118:
5116:
5113:
5112:
5106:
5092:
5062:
5054:
5042:
5041:
5040:
5026:
4992:
4989:
4978:
4967:
4957:
4956:
4955:
4941:
4938:
4932:
4909:
4906:
4903:
4900:
4897:
4894:
4891:
4874:
4871:
4868:
4865:
4826:
4820:
4814:
4808:
4805:
4802:
4781:
4778:
4772:
4769:
4759:
4751:
4739:
4738:
4737:
4699:
4687:
4683:
4663:
4659:
4656:
4653:
4649:
4645:
4642:
4639:
4635:
4631:
4627:
4624:
4621:
4617:
4613:
4609:
4605:
4602:
4599:
4595:
4587:
4586:
4585:
4583:
4582:intersections
4579:
4560:
4556:
4553:
4550:
4546:
4542:
4539:
4536:
4532:
4528:
4524:
4521:
4518:
4514:
4506:
4505:
4504:
4502:
4501:disjoint sets
4498:
4494:
4488:
4460:
4457:
4454:
4443:
4439:
4436:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4405:
4402:
4398:
4394:
4390:
4386:
4382:
4378:
4374:
4371:
4367:
4363:
4359:
4355:
4352:, oranges), (
4351:
4347:
4344:| because { (
4343:
4339:
4335:
4331:
4327:
4323:
4319:
4315:
4311:
4307:
4303:
4302:
4282:
4271:
4267:
4260:
4252:
4241:
4237:
4225:
4218:
4209:
4205:
4200:
4177:
4163:
4153:
4146:
4137:
4133:
4125:
4116:
4110:
4100:
4096:
4089:
4079:
4072:
4063:
4059:
4054:
4047:
4026:
4012:
4002:
3995:
3986:
3982:
3974:
3965:
3962:
3958:
3954:
3949:
3944:
3937:
3928:
3924:
3919:
3914:
3898:
3897:
3896:
3894:
3870:
3866:
3830:
3820:
3813:
3780:
3770:
3765:
3749:
3741:
3737:
3716:
3711:
3707:
3703:
3692:
3684:
3683:
3682:
3675:
3671:
3667:
3663:
3660:
3656:
3652:
3648:
3644:
3640:
3639:
3638:
3637:
3636:
3634:
3630:
3601:
3599:
3595:
3591:
3587:
3582:
3580:
3576:
3572:
3568:
3564:
3559:
3557:
3553:
3549:
3545:
3541:
3537:
3533:
3529:
3524:
3522:
3519:
3498:
3490:
3483:
3474:
3466:
3465:
3464:
3462:
3458:
3450:
3446:
3442:
3426:
3418:
3411:
3402:
3394:
3393:
3392:
3391:) satisfies:
3390:
3372:
3368:
3364:
3357:
3348:
3344:
3329:
3325:
3307:
3270:
3264:
3254:
3238:
3225:
3221:
3217:
3213:
3212:Gottlob Frege
3209:
3205:
3204:infinite sets
3201:
3194:Infinite sets
3188:
3184:
3156:
3152:
3148:
3144:
3140:
3137:
3119:
3106:
3102:
3098:
3094:
3091:
3087:
3083:
3079:
3075:
3071:
3070:
3069:
3067:
3063:
3053:
3051:
3047:
3043:
3025:
2996:
2987:
2962:
2953:
2949:
2944:
2931:
2926:
2908:
2899:
2895:
2866:
2858:
2843:
2839:
2835:
2807:
2789:
2776:
2772:
2767:
2751:
2724:
2721:
2718:
2693:
2686:
2667:
2664:
2661:
2656:
2648:
2643:
2635:
2630:
2618:
2617:
2616:
2614:
2613:infinite sets
2610:
2607:Assuming the
2602:
2598:
2594:
2590:
2586:
2583:
2579:
2578:
2577:
2575:
2571:
2567:
2563:
2559:
2555:
2550:
2546:
2536:
2534:
2530:
2526:
2325:
2319:
2313:
2307:
2261:
2257:
2226:
2210:
2186:
2162:
2138:
2124:
2118:
2111:
2095:
2071:
2044:
2036:
2028:
1998:
1990:
1982:
1969:
1965:
1944:
1936:
1928:
1898:
1890:
1882:
1852:
1844:
1836:
1822:
1806:
1782:
1758:
1734:
1720:
1714:
1707:
1691:
1654:
1646:
1619:
1595:
1585:
1566:
1558:
1552:
1549:
1546:
1540:
1534:
1527:, defined by
1512:
1462:
1440:
1437:
1434:
1428:
1422:
1415:, defined by
1400:
1350:
1326:
1302:
1278:
1254:
1244:
1226:
1202:
1192:
1174:
1150:
1139:
1123:
1075:
1072:
1069:
1063:
1057:
1049:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
998:
985:
961:
958:
955:
952:
949:
946:
943:
940:
934:
931:
922:
920:
919:
914:
910:
906:
902:
899:that is both
884:
860:
850:
847:, that is, a
832:
808:
798:
789:
783:
776:
769:
765:
761:
757:
752:
747:
743:
739:
735:
730:
725:
721:
717:
713:
709:
705:
701:
697:
693:
689:
685:
681:
677:
673:
669:
665:
661:
658:
654:
650:
643:
639:
635:
631:
627:
622:
613:
611:
607:
603:
599:
595:
583:
581:
577:
573:
569:
565:
561:
557:
556:
549:
547:
528:
514:
488:
482:
479:
458:
434:
428:
408:
400:
396:
375:
350:
342:
321:
313:
305:
278:
254:
243:
241:
237:
233:
229:
225:
221:
220:infinite sets
217:
198:
195:
192:
189:
186:
180:
177:
154:
151:
148:
145:
142:
136:
133:
125:
121:
117:
97:
94:
86:
61:
53:
37:
28:
22:
8117:Georg Cantor
8112:Paul Bernays
8043:MorseâKelley
8018:
7951:
7950:Subset
7897:hereditarily
7859:Venn diagram
7817:ordered pair
7776:
7732:Intersection
7676:Axiom schema
7506:
7304:Ultraproduct
7151:Model theory
7116:Independence
7052:Formal proof
7044:Proof theory
7027:
7000:
6957:real numbers
6929:second-order
6840:Substitution
6717:Metalanguage
6658:conservative
6631:Axiom schema
6575:Constructive
6545:MorseâKelley
6511:Set theories
6490:Aleph number
6483:inaccessible
6409:
6389:Grothendieck
6273:intersection
6160:Higher-order
6148:Second-order
6094:Truth tables
6051:Venn diagram
5834:Formal proof
5823:
5744:
5713:
5702:
5692:
5686:
5680:
5662:
5653:
5600:
5596:
5586:
5533:
5529:
5519:
5496:
5482:
5450:
5444:
5415:
5404:. Retrieved
5402:. 2019-12-05
5399:
5378:
5374:
5361:
5352:
5346:
5334:. Retrieved
5327:the original
5322:
5309:
5297:
5281:
5270:. Retrieved
5266:
5257:
5243:
5238:
5227:. Retrieved
5223:
5214:
5202:
5189:
5170:
5164:
5115:Aleph number
5084:
5018:
4925:, therefore
4854:
4691:
4575:
4496:
4492:
4490:
4427:
4423:
4419:
4415:
4411:
4407:
4400:
4396:
4392:
4388:
4387:such that |
4384:
4380:
4376:
4369:
4365:
4361:
4357:
4353:
4349:
4348:, apples), (
4345:
4341:
4337:
4333:
4329:
4325:
4321:
4317:
4313:
4309:
4305:
3745:
3680:
3673:
3669:
3665:
3658:
3654:
3650:
3645:, i.e., the
3642:
3602:
3598:such a proof
3583:
3574:
3569:between the
3560:
3555:
3551:
3547:
3543:
3525:
3515:
3454:
3327:
3323:
3266:
3208:Georg Cantor
3197:
3182:
3154:
3150:
3146:
3142:
3104:
3100:
3096:
3085:
3081:
3073:
3059:
2927:
2806:real numbers
2768:
2682:
2615:are denoted
2606:
2581:
2573:
2569:
2551:
2548:
2525:real numbers
2227:
2129:
2122:
2116:
1823:
1725:
1718:
1712:
1242:
1190:
1140:
984:even numbers
923:
918:equinumerous
916:
912:
908:
794:
787:
781:
773:
767:
763:
759:
755:
750:
745:
741:
737:
733:
728:
723:
719:
715:
711:
703:
699:
695:
691:
687:
683:
679:
675:
671:
667:
663:
659:
652:
641:
637:
634:even numbers
629:
625:
594:Georg Cantor
584:
575:
571:
567:
563:
553:
550:
536:as early as
534:
395:vertical bar
340:
244:
239:
119:
113:
8142:Thomas Jech
7985:Alternative
7964:Uncountable
7918:Ultrafilter
7777:Cardinality
7681:replacement
7629:Determinacy
7414:Type theory
7362:undecidable
7294:Truth value
7181:equivalence
6860:non-logical
6473:Enumeration
6463:Isomorphism
6410:cardinality
6394:Von Neumann
6359:Ultrafilter
6324:Uncountable
6258:equivalence
6175:Quantifiers
6165:Fixed-point
6134:First-order
6014:Consistency
5999:Proposition
5976:Traditional
5947:Lindström's
5937:Compactness
5879:Type theory
5824:Cardinality
5425:Felix Klein
5120:Beth number
3550:, although
3200:finite sets
3187:uncountable
3064:holds, the
3042:independent
2773:is denoted
913:equipollent
682:), the set
636:. Although
628:to the set
590: 1880
546:mathematics
120:cardinality
116:mathematics
8182:Categories
8137:Kurt Gödel
8122:Paul Cohen
7959:Transitive
7727:Identities
7711:Complement
7698:Operations
7659:Regularity
7597:Adjunction
7556:Set theory
7225:elementary
6918:arithmetic
6786:Quantifier
6764:functional
6636:Expression
6354:Transitive
6298:identities
6283:complement
6216:hereditary
6199:Set theory
5659:Penrose, R
5406:2020-08-23
5272:2018-04-26
5229:2020-08-23
5156:References
5145:Ordinality
4422:|, then |
3653:, written
3577:(see also
3090:finite set
2932:says that
2775:aleph-null
2300:, because
2062:for every
909:equipotent
905:surjective
598:set theory
232:injections
228:bijections
224:arithmetic
8070:Paradoxes
7990:Axiomatic
7969:Universal
7945:Singleton
7940:Recursive
7883:Countable
7878:Amorphous
7737:Power set
7654:Power set
7612:dependent
7607:countable
7496:Supertask
7399:Recursion
7357:decidable
7191:saturated
7169:of models
7092:deductive
7087:axiomatic
7007:Hilbert's
6994:Euclidean
6975:canonical
6898:axiomatic
6830:Signature
6759:Predicate
6648:Extension
6570:Ackermann
6495:Operation
6374:Universal
6364:Recursive
6339:Singleton
6334:Inhabited
6319:Countable
6309:Types of
6293:power set
6263:partition
6180:Predicate
6126:Predicate
6041:Syllogism
6031:Soundness
6004:Inference
5994:Tautology
5896:paradoxes
5475:121598654
5467:0025-5831
5204:MathWorld
4996:→
4971:↦
4936:∅
4933:⋂
4907:∈
4895:∈
4889:∀
4882:⟺
4872:⋂
4869:∈
4815:α
4812:→
4797:∃
4782:∈
4779:α
4773:⋂
4770:∩
4625:∩
4603:∪
4522:∪
4257:ℵ
4253:×
4215:ℵ
4143:ℵ
4122:ℵ
4117:×
4107:ℵ
4086:ℵ
4069:ℵ
4044:ℵ
3992:ℵ
3971:ℵ
3966:×
3934:ℵ
3810:ℵ
3708:ℶ
3647:power set
3594:hypercube
3495:ℵ
3480:ℵ
3423:ℵ
3408:ℵ
3369:ℶ
3354:ℵ
3304:ℵ
3235:ℵ
3149:| > |
3116:ℵ
3084:| < |
3022:ℵ
2993:ℵ
2959:ℵ
2941:ℵ
2905:ℵ
2863:ℵ
2786:ℵ
2752:α
2748:ℵ
2719:α
2715:ℵ
2694:α
2683:For each
2665:…
2653:ℵ
2640:ℵ
2627:ℵ
2568:of a set
2260:power set
2037:≤
1991:≤
1891:≤
1845:≤
1553:−
901:injective
797:bijection
657:power set
602:bijection
512:#
483:
393:, with a
8074:Problems
7978:Theories
7954:Superset
7930:Infinite
7759:Concepts
7639:Infinity
7563:Overview
7481:Logicism
7474:timeline
7450:Concrete
7309:Validity
7279:T-schema
7272:Kripke's
7267:Tarski's
7262:semantic
7252:Strength
7201:submodel
7196:spectrum
7164:function
7012:Tarski's
7001:Elements
6988:geometry
6944:Robinson
6865:variable
6850:function
6823:spectrum
6813:Sentence
6769:variable
6712:Language
6665:Relation
6626:Automata
6616:Alphabet
6600:language
6454:-jection
6432:codomain
6418:Function
6379:Universe
6349:Infinite
6253:Relation
6036:Validity
6026:Argument
5924:theorem,
5758:geometry
5748:Such as
5695:: 81â125
5661:(2005),
5645:16591132
5578:16578557
5490:(2002),
5439:(eds.),
5423:(1915),
5381:: 75â78.
5290:Archived
5140:Counting
5109:See also
4414:| and |
3740:Beth two
3664:the set
3571:interval
3389:Beth one
3141:Any set
3095:Any set
3072:Any set
849:function
714:, hence
555:Elements
216:elements
30:The set
8012:General
8007:Zermelo
7913:subbase
7895: (
7834:Forcing
7812:Element
7784: (
7762:Methods
7649:Pairing
7423:Related
7220:Diagram
7118: (
7097:Hilbert
7082:Systems
7077:Theorem
6955:of the
6900:systems
6680:Formula
6675:Grammar
6591: (
6535:General
6248:Forcing
6233:Element
6153:Monadic
5928:paradox
5869:Theorem
5805:General
5605:Bibcode
5538:Bibcode
5336:Nov 15,
4503:, then
3633:theorem
3532:segment
3181:> |
3080:, or |
3060:If the
3052:below.
2979:, i.e.
2844:, that
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