632:
7521:
660:
38:
7591:
4185:
546:
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
562:
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
4304:
4685:
4034:
4861:
4582:
785:
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).
554:
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
4040:
3237:. 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 (
4191:
5406:
593:, 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.
2689:
3900:
3233:. 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
3745:
3852:
4934:
3396:
4601:
3912:
3545:
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
1055:
3802:
5024:
4753:
2934:
3450:
2890:
2478:
2413:
2307:
3522:
2988:
2560:
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.
991:
1591:
513:
3059:, 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
4963:
2748:
1691:
2775:
5091:
3636:
3304:
3190:
3022:
2841:
4745:
3607:, 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
2530:
2504:
2439:
2374:
2261:
1510:
1398:
1121:
3331:
3262:
3143:
3049:
2813:
2071:
2025:
1925:
1879:
223:
179:
2350:
4520:
2715:
1971:
1462:
1097:
348:
119:
3056:
459:
5698:
536:
402:
5114:
5048:
4721:
2232:
2208:
2184:
2160:
2117:
2093:
1828:
1804:
1780:
1756:
1713:
1641:
1617:
1534:
1484:
1422:
1372:
1348:
1324:
1300:
1276:
1248:
1224:
1196:
1172:
1145:
906:
882:
854:
830:
480:
430:
372:
300:
276:
83:
59:
4485:
4180:{\displaystyle {\mathfrak {c}}^{\aleph _{0}}=\left(2^{\aleph _{0}}\right)^{\aleph _{0}}=2^{{\aleph _{0}}\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
3608:
5900:
4299:{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=\left(2^{\aleph _{0}}\right)^{\mathfrak {c}}=2^{{\mathfrak {c}}\times \aleph _{0}}=2^{\mathfrak {c}}.}
4448:
8055:
6575:
4692:
8210:
5326:
5255:
3051:, 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
6658:
5799:
5451:
5300:
2632:
619:, 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.
581:, 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
4442:
3857:
3698:
3806:
4869:
563:
thing. The ancient Greek notion of infinity also considered the division of things into parts repeated without limit. In Euclid's
6972:
3589:
3230:
4680:{\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 .}
3344:
615:, a one-to-one correspondence between the elements of two sets based on a unique relationship. In 1891, with the publication of
5378:
4029:{\displaystyle {\mathfrak {c}}^{2}=\left(2^{\aleph _{0}}\right)^{2}=2^{2\times {\aleph _{0}}}=2^{\aleph _{0}}={\mathfrak {c}},}
3758:
3459:
2543:
7130:
5189:
5918:
7744:
7557:
6985:
6308:
1000:
3763:
3639:
4856:{\displaystyle |A|:={\mbox{Ord}}\cap \bigcap \{\alpha \in {\mbox{Ord}}|\exists (f:A\to \alpha ):(f{\mbox{ injective}})\}}
5712:
8072:
6990:
6980:
6717:
6570:
5923:
17:
5914:
4971:
7126:
5741:
5733:
5681:
5516:
3052:
2895:
6468:
3408:
2858:
1974:
7223:
6967:
5792:
2448:
2383:
2277:
8050:
7644:
6528:
6221:
3480:
2946:
2603:
7930:
5962:
5096:
This definition is natural since it agrees with the axiom of limitation of size which implies bijection between
938:
237:
on them. There are two notions often used when referring to cardinality: one which compares sets directly using
7484:
7186:
6949:
6944:
6769:
6190:
5874:
1541:
570:
486:
7824:
7703:
7479:
7262:
7179:
6892:
6823:
6700:
5942:
4497:
3455:
2852:
2539:
616:
8067:
7404:
7230:
6916:
6550:
6149:
3279:
3273:
2848:
2599:
31:
4939:
8060:
7698:
7661:
7282:
7277:
6887:
6626:
6555:
5884:
5785:
4696:
2720:
3229:
and others rejected the view that the whole cannot be the same size as the part. One example of this is
1648:
7211:
6801:
6195:
6163:
5854:
2753:
7715:
5056:
3595:
The second result was first demonstrated by Cantor in 1878, but it became more apparent in 1890, when
8205:
7749:
7634:
7622:
7617:
7501:
7450:
7347:
6845:
6806:
6283:
5928:
4592:
3617:
3577:
3285:
3171:
2993:
2822:
5957:
7550:
7342:
7272:
6811:
6663:
6646:
6369:
5849:
3076:
5337:
5273:
4726:
2513:
2487:
2422:
2357:
2244:
1493:
1381:
1104:
8169:
8087:
7962:
7914:
7728:
7651:
7174:
7151:
7112:
6998:
6939:
6585:
6505:
6349:
6293:
5906:
5312:
4577:{\displaystyle \left\vert A\cup B\right\vert =\left\vert A\right\vert +\left\vert B\right\vert .}
3309:
3240:
3121:
3027:
2791:
2030:
1984:
1884:
1838:
184:
140:
5297:
2314:
8121:
8002:
7814:
7627:
7464:
7191:
7169:
7136:
7029:
6875:
6860:
6833:
6784:
6668:
6603:
6428:
6394:
6389:
6263:
6094:
6071:
3581:
859:
556:
8037:
8007:
7951:
7871:
7851:
7829:
7394:
7247:
7039:
6757:
6493:
6399:
6258:
6243:
6124:
6099:
5456:
2700:
1930:
1429:
1064:
631:
307:
226:
8111:
8101:
7935:
7866:
7819:
7759:
7639:
7367:
7329:
7206:
7010:
6850:
6774:
6752:
6580:
6538:
6437:
6404:
6268:
6056:
5967:
5615:
5548:
5523:
5431:
5160:
3531:
3467:
2940:
2611:
2568:
565:
559:
and the manipulation of numbers without reference to a specific group of things or events.
88:
3474:
between the cardinality of the reals and the cardinality of the natural numbers, that is,
435:
8:
8106:
8017:
7925:
7920:
7734:
7676:
7607:
7543:
7496:
7387:
7372:
7352:
7309:
7196:
7146:
7072:
7017:
6954:
6747:
6742:
6690:
6458:
6447:
6119:
6019:
5947:
5938:
5934:
5869:
5864:
5140:
5135:
3903:
3643:
3600:
2583:
under this relation, then, consists of all those sets which have the same cardinality as
718:
518:
377:
234:
5619:
5552:
5362:
Kurt Von Fritz (1945). "The
Discovery of Incommensurability by Hippasus of Metapontum".
3527:
However, this hypothesis can neither be proved nor disproved within the widely accepted
8029:
8024:
7809:
7764:
7671:
7525:
7294:
7257:
7242:
7235:
7218:
7022:
7004:
6870:
6796:
6779:
6732:
6545:
6454:
6288:
6273:
6233:
6185:
6170:
6158:
6114:
6089:
5859:
5808:
5633:
5566:
5481:
5099:
5033:
4706:
4588:
3603:, curved lines that twist and turn enough to fill the whole of any square, or cube, or
3146:
2572:
2217:
2193:
2169:
2145:
2102:
2078:
1813:
1789:
1765:
1741:
1698:
1626:
1602:
1519:
1469:
1407:
1357:
1333:
1309:
1285:
1261:
1233:
1209:
1181:
1157:
1130:
891:
867:
839:
815:
465:
415:
357:
285:
261:
242:
68:
44:
6478:
5646:
5603:
5579:
5536:
4458:
1597:) is surjective, but not injective, since 0 and 1 for instance both map to 0. Neither
233:, which allows one to distinguish between different types of infinity, and to perform
7886:
7723:
7686:
7656:
7580:
7520:
7460:
7267:
7077:
7067:
6959:
6840:
6675:
6651:
6432:
6416:
6321:
6298:
6175:
6144:
6109:
6004:
5839:
5737:
5729:
5677:
5651:
5584:
5512:
5485:
5473:
5206:
5185:
4452:
3234:
2576:
590:
134:
5209:
2936:, this also being the cardinality of the set of all subsets of the natural numbers.
8174:
8164:
8149:
8144:
8012:
7666:
7474:
7469:
7362:
7319:
7141:
7102:
7097:
7082:
6908:
6865:
6762:
6560:
6510:
6084:
6046:
5641:
5623:
5574:
5556:
5502:
5465:
5439:
3573:
3538:
3226:
2844:
1594:
3750:
3399:
623:
that contain more elements than there are in the infinite set of natural numbers.
8043:
7981:
7799:
7612:
7455:
7445:
7399:
7382:
7337:
7299:
7201:
7121:
6928:
6855:
6828:
6816:
6722:
6636:
6610:
6565:
6533:
6334:
6136:
6079:
6029:
5994:
5952:
5506:
5498:
5447:
5304:
5260:
3550:
3471:
3197:
3072:
2619:
2607:
2555:
2266:
1978:
1058:
659:
620:
246:
62:
3537:
Cardinal arithmetic can be used to show not only that the number of points in a
8179:
7976:
7957:
7861:
7846:
7803:
7739:
7681:
7440:
7419:
7377:
7357:
7252:
7107:
6705:
6695:
6685:
6680:
6614:
6488:
6364:
6253:
6248:
6226:
5827:
5608:
Proceedings of the
National Academy of Sciences of the United States of America
5541:
Proceedings of the
National Academy of Sciences of the United States of America
3596:
3088:
2781:
2695:
2564:
409:
1464:
is injective, but not surjective since 2, for instance, is not mapped to, and
37:
8199:
8184:
7986:
7900:
7895:
7414:
7092:
6599:
6384:
6374:
6344:
6329:
5999:
5771:. â A line of finite length is a set of points that has infinite cardinality.
5669:
5477:
5443:
5145:
4511:
3546:
3222:
8154:
5728:
Applied
Abstract Algebra, K.H. Kim, F.W. Roush, Ellis Horwood Series, 1983,
2602:
set is designated for each equivalence class. The most common choice is the
8134:
8129:
7947:
7876:
7834:
7693:
7590:
7314:
7161:
7062:
7054:
6934:
6882:
6791:
6727:
6710:
6641:
6500:
6359:
6061:
5844:
5655:
5588:
5561:
5125:
3542:
3218:
3214:
2785:
2623:
928:
604:
596:
To better understand infinite sets, a notion of cardinality was formulated
405:
230:
5628:
3565:
contains elements that do not belong to its subsets, and the supersets of
8159:
7424:
7304:
6483:
6473:
6104:
6024:
6009:
5889:
5435:
5130:
3572:
The first of these results is apparent by considering, for instance, the
2816:
2535:
994:
644:
573:
was described as the ability to compare the length of two line segments,
126:
5711:
Georg Cantor (1932), Adolf
Fraenkel (Lebenslauf); Ernst Zermelo (eds.),
4690:
8139:
7910:
7566:
6354:
6209:
6180:
5986:
5469:
5155:
3210:
3100:
915:
608:
3110:
that has the same cardinality as the set of the natural numbers, or |
229:. Beginning in the late 19th century, this concept was generalized to
7942:
7905:
7856:
7754:
7506:
7409:
6462:
6379:
6339:
6303:
6239:
6051:
6041:
6014:
5777:
5637:
5570:
5230:
5214:
5030:
This definition allows also obtain a cardinality of any proper class
3657:
3604:
2684:{\displaystyle \aleph _{0}<\aleph _{1}<\aleph _{2}<\ldots .}
2270:
2188:, if there is an injective function, but no bijective function, from
911:
807:
667:
612:
238:
3614:
Cantor also showed that sets with cardinality strictly greater than
7491:
7289:
6737:
6442:
6036:
5768:
5150:
3079:
holds for cardinality. Thus we can make the following definitions:
412:, and the meaning depends on context. The cardinal number of a set
5714:
Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
7087:
5879:
5674:
The Road to
Reality: A Complete guide to the Laws of the Universe
3895:{\displaystyle {\mathfrak {c}}^{\mathfrak {c}}=2^{\mathfrak {c}}}
5696:
Georg Cantor (1887), "Mitteilungen zur Lehre vom
Transfiniten",
3156:
with cardinality greater than that of the natural numbers, or |
7967:
7789:
5760:
5256:
Animals Count and Use Zero. How Far Does Their Number Sense Go?
3740:{\displaystyle 2^{\mathfrak {c}}=\beth _{2}>{\mathfrak {c}}}
4587:
From this, one can show that in general, the cardinalities of
3847:{\displaystyle {\mathfrak {c}}^{\aleph _{0}}={\mathfrak {c}},}
2508:
has cardinality strictly less than the cardinality of the set
7839:
7599:
7535:
6631:
5977:
5822:
4929:{\displaystyle (x\in \bigcap Q)\iff (\forall q\in Q:x\in q)}
253:, when no confusion with other notions of size is possible.
5764:
4441:|. This holds even for infinite cardinals, and is known as
2587:. There are two ways to define the "cardinality of a set":
3391:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}=\beth _{1}}
137:
which compares their relative size. For example, the sets
5511:(1. ed.), Berlin/Heidelberg: Springer, p. 587,
3528:
3066:
2595:
is defined as its equivalence class under equinumerosity.
1760:
has cardinality less than or equal to the cardinality of
5379:"Ueber eine elementare Frage der Mannigfaltigkeitslehre"
5204:
4866:
We use the intersection of a class which is defined by
2482:
can be bijective (see picture). By a similar argument,
5077:
5010:
4841:
4796:
4774:
4731:
2563:
The relation of having the same cardinality is called
2164:
has cardinality strictly less than the cardinality of
304:, have the same cardinality, it is usually written as
5699:
5102:
5059:
5036:
4974:
4942:
4872:
4756:
4729:
4709:
4604:
4523:
4461:
4194:
4043:
3915:
3860:
3809:
3766:
3701:
3620:
3483:
3411:
3347:
3312:
3288:
3243:
3174:
3124:
3030:
2996:
2949:
2898:
2861:
2825:
2794:
2756:
2723:
2703:
2635:
2516:
2490:
2451:
2425:
2386:
2360:
2317:
2280:
2247:
2220:
2196:
2172:
2148:
2105:
2081:
2033:
1987:
1933:
1887:
1841:
1816:
1792:
1768:
1744:
1701:
1651:
1629:
1605:
1544:
1522:
1496:
1472:
1432:
1410:
1384:
1360:
1336:
1312:
1288:
1264:
1236:
1212:
1184:
1160:
1133:
1107:
1067:
1050:{\displaystyle \mathbb {N} =\{0,1,2,3,{\text{...}}\}}
1003:
941:
894:
870:
842:
818:
806:
Two sets have the same cardinality if there exists a
521:
489:
468:
438:
418:
380:
360:
310:
288:
264:
187:
143:
91:
71:
47:
5430:
5386:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4451:
include the set of all real numbers, the set of all
3797:{\displaystyle {\mathfrak {c}}^{2}={\mathfrak {c}},}
3278:
One of Cantor's most important results was that the
1304:
is a bijection. This is no longer true for infinite
611:. He examined the process of equating two sets with
729:cannot be surjective. The picture shows an example
5604:"The Independence of the Continuum Hypothesis, II"
5108:
5085:
5042:
5018:
4957:
4928:
4855:
4739:
4715:
4679:
4576:
4479:
4298:
4179:
4028:
3894:
3846:
3796:
3739:
3630:
3516:
3444:
3390:
3325:
3298:
3256:
3184:
3137:
3043:
3016:
2982:
2928:
2884:
2835:
2807:
2769:
2742:
2709:
2683:
2524:
2498:
2472:
2433:
2407:
2368:
2344:
2301:
2255:
2226:
2202:
2178:
2154:
2111:
2087:
2065:
2019:
1965:
1919:
1873:
1822:
1798:
1774:
1750:
1707:
1685:
1635:
1611:
1585:
1528:
1504:
1478:
1456:
1416:
1392:
1366:
1342:
1318:
1294:
1270:
1242:
1218:
1190:
1166:
1139:
1115:
1091:
1049:
985:
900:
876:
848:
824:
530:
507:
474:
453:
424:
396:
366:
342:
294:
270:
249:. The cardinality of a set may also be called its
217:
173:
113:
77:
53:
5231:"Cardinality | Brilliant Math & Science Wiki"
3060:
8197:
5019:{\displaystyle (x\mapsto |x|):V\to {\mbox{Ord}}}
5497:
3306:) is greater than that of the natural numbers (
3267:
2929:{\displaystyle {\mathfrak {c}}=2^{\aleph _{0}}}
5537:"The Independence of the Continuum Hypothesis"
5361:
3569:contain elements that are not included in it.
3445:{\displaystyle 2^{\aleph _{0}}>\aleph _{0}}
2885:{\displaystyle {\mathfrak {c}}>\aleph _{0}}
7551:
5793:
2606:. This is usually taken as the definition of
2124:
1784:, if there exists an injective function from
27:Definition of the number of elements in a set
5710:
5695:
5505:; Srishti D. Chatterji; et al. (eds.),
5376:
4850:
4786:
4367:, peaches)} is a bijection between the sets
3024:is the smallest cardinal number bigger than
2473:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
2417:, and it can be shown that no function from
2408:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
2339:
2333:
2302:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
1693:, which was established by the existence of
1044:
1012:
980:
948:
212:
194:
168:
150:
5689:
5464:(4), Leipzig: B. G. Teubner: 438â443,
4691:Definition of cardinality in class theory (
3517:{\displaystyle 2^{\aleph _{0}}=\aleph _{1}}
2983:{\displaystyle \aleph _{1}=2^{\aleph _{0}}}
408:on each side; this is the same notation as
7558:
7544:
5985:
5800:
5786:
4895:
4891:
4747:denotes the class of all ordinal numbers.
4309:
2750:is the least cardinal number greater than
1720:
986:{\displaystyle E=\{0,2,4,6,{\text{...}}\}}
789:
666:does not have the same cardinality as its
5645:
5627:
5578:
5560:
5184:. San Francisco, CA: Dover Publications.
4491:
2518:
2492:
2463:
2427:
2398:
2362:
2292:
2249:
1674:
1586:{\displaystyle h(n)=n-(n{\text{ mod }}2)}
1498:
1386:
1109:
1005:
810:(a.k.a., one-to-one correspondence) from
225:are the same size as they each contain 3
3541:is equal to the number of points in any
3333:); that is, there are more real numbers
658:
630:
508:{\displaystyle \operatorname {card} (A)}
65:has 5 elements. Thus the cardinality of
36:
5668:
4595:are related by the following equation:
3087:with cardinality less than that of the
2269:has cardinality strictly less than its
14:
8198:
5807:
4449:Sets with cardinality of the continuum
3067:Finite, countable and uncountable sets
1256:injective or surjective function from
655:, both sets have the same cardinality.
8211:Basic concepts in infinite set theory
7539:
5781:
5601:
5534:
5401:
5399:
5324:
5205:
2847:"c"), and is also referred to as the
1981:is equivalent to the statement that
5717:, Berlin: Springer, pp. 378â439
5535:Cohen, Paul J. (December 15, 1963).
5318:
4958:{\displaystyle \bigcap \emptyset =V}
4335:= {apples, oranges, peaches}, where
3590:Hilbert's paradox of the Grand Hotel
3231:Hilbert's paradox of the Grand Hotel
997:has the same cardinality as the set
5602:Cohen, Paul J. (January 15, 1964).
4287:
4258:
4242:
4205:
4198:
4169:
4047:
4018:
3919:
3886:
3871:
3864:
3836:
3813:
3786:
3770:
3732:
3708:
3623:
3460:Cantor's first uncountability proof
3350:
3291:
3177:
2901:
2864:
2828:
2743:{\displaystyle \aleph _{\alpha +1}}
2549:
2544:Cantor's first uncountability proof
717:)} disagrees with every set in the
24:
5452:"Ăber das Problem der Wohlordnung"
5396:
4946:
4899:
4807:
4443:CantorâBernsteinâSchroeder theorem
4267:
4225:
4153:
4132:
4117:
4096:
4079:
4054:
4002:
3981:
3944:
3820:
3505:
3490:
3433:
3418:
3364:
3314:
3245:
3126:
3032:
3003:
2969:
2951:
2915:
2873:
2796:
2758:
2725:
2663:
2650:
2637:
2454:
2389:
2283:
1686:{\displaystyle |E|=|\mathbb {N} |}
522:
25:
8222:
5179:
2770:{\displaystyle \aleph _{\alpha }}
626:
133:describes a relationship between
7589:
7519:
5086:{\displaystyle |P|={\mbox{Ord}}}
4723:denote a class of all sets, and
3204:
2815:), while the cardinality of the
432:may alternatively be denoted by
5722:
5662:
5595:
5528:
5491:
5424:
5407:"Infinite Sets and Cardinality"
5370:
3646:). They include, for instance:
3631:{\displaystyle {\mathfrak {c}}}
3299:{\displaystyle {\mathfrak {c}}}
3185:{\displaystyle {\mathfrak {c}}}
3017:{\displaystyle 2^{\aleph _{0}}}
2836:{\displaystyle {\mathfrak {c}}}
350:; however, if referring to the
85:is 5 or, written symbolically,
7565:
5753:
5355:
5290:
5266:
5247:
5223:
5198:
5173:
5069:
5061:
5006:
4997:
4993:
4985:
4981:
4975:
4923:
4896:
4892:
4888:
4873:
4847:
4834:
4828:
4822:
4810:
4803:
4766:
4758:
4474:
4462:
3213:breaks down when dealing with
3061:§ Cardinality of the continuum
2467:
2459:
2402:
2394:
2352:is an injective function from
2327:
2321:
2296:
2288:
2059:
2051:
2043:
2035:
2013:
2005:
1997:
1989:
1959:
1951:
1943:
1935:
1913:
1905:
1897:
1889:
1867:
1859:
1851:
1843:
1679:
1669:
1661:
1653:
1580:
1566:
1554:
1548:
1442:
1436:
1077:
1071:
502:
496:
448:
442:
390:
382:
336:
328:
320:
312:
101:
93:
13:
1:
7480:History of mathematical logic
5166:
4498:Inclusion-exclusion principle
4375:. The cardinality of each of
3640:generalized diagonal argument
3341:. Namely, Cantor showed that
2604:initial ordinal in that class
597:
7405:Primitive recursive function
5309:Third Millennium Mathematics
5274:"Early Human Counting Tools"
4740:{\displaystyle {\mbox{Ord}}}
3280:cardinality of the continuum
3274:Cardinality of the continuum
3268:Cardinality of the continuum
2849:cardinality of the continuum
2525:{\displaystyle \mathbb {R} }
2499:{\displaystyle \mathbb {N} }
2434:{\displaystyle \mathbb {N} }
2369:{\displaystyle \mathbb {N} }
2256:{\displaystyle \mathbb {N} }
1505:{\displaystyle \mathbb {N} }
1393:{\displaystyle \mathbb {N} }
1352:. For example, the function
1116:{\displaystyle \mathbb {N} }
589:. But with the discovery of
32:Cardinality (disambiguation)
7:
5298:Third Millennium Chronology
5296:Duncan J. Melville (2003).
5119:
3557:that have the same size as
3326:{\displaystyle \aleph _{0}}
3257:{\displaystyle \aleph _{0}}
3217:. In the late 19th century
3138:{\displaystyle \aleph _{0}}
3044:{\displaystyle \aleph _{0}}
2851:. Cantor showed, using the
2808:{\displaystyle \aleph _{0}}
2622:, the cardinalities of the
2066:{\displaystyle |B|\leq |A|}
2020:{\displaystyle |A|\leq |B|}
1920:{\displaystyle |B|\leq |A|}
1874:{\displaystyle |A|\leq |B|}
918:. Such sets are said to be
218:{\displaystyle B=\{2,4,6\}}
174:{\displaystyle A=\{1,2,3\}}
10:
8227:
8056:von NeumannâBernaysâGödel
6469:SchröderâBernstein theorem
6196:Monadic predicate calculus
5855:Foundations of mathematics
4495:
3902:can be demonstrated using
3652:the set of all subsets of
3456:Cantor's diagonal argument
3271:
3209:Our intuition gained from
2553:
2540:Cantor's diagonal argument
2345:{\displaystyle g(n)=\{n\}}
1975:SchröderâBernstein theorem
617:Cantor's diagonal argument
541:
29:
8120:
8083:
7995:
7885:
7857:One-to-one correspondence
7773:
7714:
7598:
7587:
7573:
7515:
7502:Philosophy of mathematics
7451:Automated theorem proving
7433:
7328:
7160:
7053:
6905:
6622:
6598:
6576:Von NeumannâBernaysâGödel
6521:
6415:
6319:
6217:
6208:
6135:
6070:
5976:
5898:
5815:
5508:GrundzĂŒge der Mengenlehre
5364:The Annals of Mathematics
5334:Texas A&M Mathematics
5327:"The History of Infinity"
3578:one-to-one correspondence
2591:The cardinality of a set
245:, and another which uses
3534:, if ZFC is consistent.
3470:states that there is no
635:Bijective function from
7152:Self-verifying theories
6973:Tarski's axiomatization
5924:Tarski's undefinability
5919:incompleteness theorems
5524:Original edition (1914)
5313:St. Lawrence University
4310:Examples and properties
2780:The cardinality of the
2710:{\displaystyle \alpha }
1966:{\displaystyle |A|=|B|}
1457:{\displaystyle g(n)=4n}
1092:{\displaystyle f(n)=2n}
374:, it is simply denoted
343:{\displaystyle |A|=|B|}
7815:Constructible universe
7635:Constructibility (V=L)
7526:Mathematics portal
7137:Proof of impossibility
6785:propositional variable
6095:Propositional calculus
5562:10.1073/pnas.50.6.1143
5411:Mathematics LibreTexts
5325:Allen, Donald (2003).
5253:Cepelewicz, Jordana
5116:and any proper class.
5110:
5087:
5044:
5020:
4959:
4930:
4857:
4741:
4717:
4681:
4578:
4492:Union and intersection
4481:
4300:
4181:
4030:
3896:
3848:
3798:
3741:
3692:Both have cardinality
3679:of all functions from
3632:
3518:
3446:
3392:
3327:
3300:
3258:
3186:
3139:
3045:
3018:
2984:
2930:
2886:
2837:
2809:
2771:
2744:
2711:
2685:
2526:
2500:
2474:
2435:
2409:
2370:
2346:
2303:
2257:
2228:
2204:
2180:
2156:
2113:
2089:
2067:
2021:
1967:
1921:
1875:
1824:
1800:
1776:
1752:
1709:
1687:
1637:
1613:
1587:
1530:
1506:
1480:
1458:
1418:
1394:
1368:
1344:
1320:
1296:
1272:
1244:
1220:
1204:bijection exists from
1192:
1168:
1141:
1117:
1093:
1051:
987:
902:
878:
850:
826:
782:
733:and the corresponding
677:): For every function
656:
651:is a proper subset of
532:
509:
476:
455:
426:
398:
368:
344:
296:
272:
219:
175:
122:
115:
79:
55:
8038:Principia Mathematica
7872:Transfinite induction
7731:(i.e. set difference)
7395:Kolmogorov complexity
7348:Computably enumerable
7248:Model complete theory
7040:Principia Mathematica
6100:Propositional formula
5929:BanachâTarski paradox
5629:10.1073/pnas.51.1.105
5457:Mathematische Annalen
5377:Georg Cantor (1891).
5111:
5088:
5045:
5021:
4960:
4931:
4858:
4742:
4718:
4682:
4579:
4482:
4394:|, then there exists
4347:are distinct, then |
4301:
4182:
4031:
3897:
3849:
3799:
3742:
3633:
3519:
3447:
3393:
3337:than natural numbers
3328:
3301:
3259:
3187:
3140:
3046:
3019:
2985:
2931:
2887:
2838:
2810:
2772:
2745:
2712:
2686:
2527:
2501:
2475:
2436:
2410:
2371:
2347:
2304:
2258:
2239:For example, the set
2229:
2205:
2181:
2157:
2114:
2090:
2068:
2022:
1968:
1922:
1876:
1825:
1801:
1777:
1753:
1710:
1688:
1638:
1614:
1588:
1531:
1507:
1481:
1459:
1419:
1395:
1369:
1345:
1321:
1297:
1273:
1245:
1221:
1193:
1169:
1142:
1118:
1094:
1061:, since the function
1052:
988:
935:For example, the set
903:
879:
851:
827:
662:
634:
533:
510:
477:
456:
427:
399:
369:
354:of an individual set
345:
297:
273:
220:
176:
116:
114:{\displaystyle |S|=5}
80:
56:
40:
8112:Burali-Forti paradox
7867:Set-builder notation
7820:Continuum hypothesis
7760:Symmetric difference
7343:ChurchâTuring thesis
7330:Computability theory
6539:continuum hypothesis
6057:Square of opposition
5915:Gödel's completeness
5432:Friedrich M. Hartogs
5182:Set Theory and Logic
5161:Pigeonhole principle
5100:
5057:
5034:
4972:
4940:
4870:
4754:
4727:
4707:
4602:
4521:
4459:
4192:
4041:
3913:
3858:
3807:
3764:
3699:
3618:
3601:space-filling curves
3532:axiomatic set theory
3481:
3468:continuum hypothesis
3409:
3345:
3310:
3286:
3241:
3172:
3122:
3028:
2994:
2947:
2941:continuum hypothesis
2896:
2859:
2823:
2792:
2754:
2721:
2701:
2633:
2612:axiomatic set theory
2569:equivalence relation
2514:
2488:
2449:
2423:
2384:
2358:
2315:
2278:
2245:
2218:
2194:
2170:
2146:
2125:Definition 3: |
2103:
2079:
2031:
1985:
1931:
1885:
1839:
1814:
1790:
1766:
1742:
1721:Definition 2: |
1699:
1649:
1627:
1603:
1542:
1520:
1494:
1470:
1430:
1408:
1382:
1358:
1334:
1310:
1286:
1262:
1234:
1210:
1182:
1158:
1131:
1105:
1099:is a bijection from
1065:
1001:
939:
892:
868:
840:
816:
790:Definition 1: |
607:, the originator of
519:
487:
466:
454:{\displaystyle n(A)}
436:
416:
378:
358:
308:
286:
262:
185:
141:
89:
69:
45:
30:For other uses, see
8073:TarskiâGrothendieck
7497:Mathematical object
7388:P versus NP problem
7353:Computable function
7147:Reverse mathematics
7073:Logical consequence
6950:primitive recursive
6945:elementary function
6718:Free/bound variable
6571:TarskiâGrothendieck
6090:Logical connectives
6020:Logical equivalence
5870:Logical consequence
5736:(student edition),
5620:1964PNAS...51..105C
5553:1963PNAS...50.1143C
3904:cardinal arithmetic
3759:cardinal equalities
3576:, which provides a
3553:of an infinite set
3099:|, is said to be a
2892:. We can show that
531:{\displaystyle \#A}
397:{\displaystyle |A|}
7662:Limitation of size
7295:Transfer principle
7258:Semantics of logic
7243:Categorical theory
7219:Non-standard model
6733:Logical connective
5860:Information theory
5809:Mathematical logic
5719:Here: p.413 bottom
5470:10.1007/bf01458215
5303:2018-07-07 at the
5207:Weisstein, Eric W.
5106:
5083:
5081:
5040:
5016:
5014:
4955:
4926:
4853:
4845:
4800:
4778:
4737:
4735:
4713:
4677:
4574:
4477:
4453:irrational numbers
4296:
4177:
4026:
3892:
3844:
3794:
3737:
3628:
3514:
3442:
3388:
3323:
3296:
3254:
3182:
3147:countably infinite
3145:, is said to be a
3135:
3041:
3014:
2980:
2926:
2882:
2833:
2805:
2767:
2740:
2707:
2681:
2538:. For proofs, see
2522:
2496:
2470:
2431:
2405:
2366:
2342:
2299:
2253:
2224:
2200:
2176:
2152:
2131:| < |
2109:
2085:
2063:
2017:
1963:
1917:
1871:
1820:
1796:
1772:
1748:
1705:
1683:
1633:
1609:
1583:
1526:
1502:
1476:
1454:
1414:
1390:
1364:
1340:
1316:
1292:
1268:
1240:
1216:
1188:
1164:
1137:
1113:
1089:
1047:
983:
898:
874:
846:
822:
783:
657:
591:irrational numbers
528:
505:
472:
451:
422:
394:
364:
340:
292:
268:
215:
171:
123:
111:
75:
51:
18:Finite cardinality
8193:
8192:
8102:Russell's paradox
8051:ZermeloâFraenkel
7952:Dedekind-infinite
7825:Diagonal argument
7724:Cartesian product
7581:Set (mathematics)
7533:
7532:
7465:Abstract category
7268:Theories of truth
7078:Rule of inference
7068:Natural deduction
7049:
7048:
6594:
6593:
6299:Cartesian product
6204:
6203:
6110:Many-valued logic
6085:Boolean functions
5968:Russell's paradox
5943:diagonal argument
5840:First-order logic
5744:(library edition)
5676:, Vintage Books,
5343:on August 1, 2020
5210:"Cardinal Number"
5191:978-0-486-63829-4
5180:Stoll, Robert R.
5109:{\displaystyle V}
5080:
5043:{\displaystyle P}
5013:
4844:
4799:
4777:
4734:
4716:{\displaystyle V}
4455:and the interval
3235:Dedekind infinite
3196:|, is said to be
3164:|, for example |
3077:law of trichotomy
2853:diagonal argument
2577:equivalence class
2575:of all sets. The
2567:, and this is an
2227:{\displaystyle B}
2203:{\displaystyle A}
2179:{\displaystyle B}
2155:{\displaystyle A}
2112:{\displaystyle B}
2088:{\displaystyle A}
1973:(a fact known as
1823:{\displaystyle B}
1799:{\displaystyle A}
1775:{\displaystyle B}
1751:{\displaystyle A}
1708:{\displaystyle f}
1636:{\displaystyle h}
1612:{\displaystyle g}
1575:
1529:{\displaystyle E}
1479:{\displaystyle h}
1417:{\displaystyle E}
1367:{\displaystyle g}
1343:{\displaystyle B}
1319:{\displaystyle A}
1295:{\displaystyle B}
1271:{\displaystyle A}
1243:{\displaystyle B}
1219:{\displaystyle A}
1191:{\displaystyle B}
1167:{\displaystyle A}
1140:{\displaystyle E}
1042:
978:
901:{\displaystyle B}
877:{\displaystyle A}
849:{\displaystyle B}
825:{\displaystyle A}
475:{\displaystyle A}
425:{\displaystyle A}
367:{\displaystyle A}
295:{\displaystyle B}
271:{\displaystyle A}
78:{\displaystyle S}
54:{\displaystyle S}
16:(Redirected from
8218:
8206:Cardinal numbers
8175:Bertrand Russell
8165:John von Neumann
8150:Abraham Fraenkel
8145:Richard Dedekind
8107:Suslin's problem
8018:Cantor's theorem
7735:De Morgan's laws
7593:
7560:
7553:
7546:
7537:
7536:
7524:
7523:
7475:History of logic
7470:Category of sets
7363:Decision problem
7142:Ordinal analysis
7083:Sequent calculus
6981:Boolean algebras
6921:
6920:
6895:
6866:logical/constant
6620:
6619:
6606:
6529:ZermeloâFraenkel
6280:Set operations:
6215:
6214:
6152:
5983:
5982:
5963:LöwenheimâSkolem
5850:Formal semantics
5802:
5795:
5788:
5779:
5778:
5772:
5757:
5745:
5726:
5720:
5718:
5707:
5693:
5687:
5686:
5666:
5660:
5659:
5649:
5631:
5599:
5593:
5592:
5582:
5564:
5547:(6): 1143â1148.
5532:
5526:
5521:
5503:Egbert Brieskorn
5495:
5489:
5488:
5440:Walther von Dyck
5428:
5422:
5421:
5419:
5418:
5403:
5394:
5393:
5383:
5374:
5368:
5367:
5359:
5353:
5352:
5350:
5348:
5342:
5336:. Archived from
5331:
5322:
5316:
5294:
5288:
5287:
5285:
5284:
5270:
5264:
5263:, August 9, 2021
5251:
5245:
5244:
5242:
5241:
5227:
5221:
5220:
5219:
5202:
5196:
5195:
5177:
5141:Cantor's theorem
5136:Cantor's paradox
5115:
5113:
5112:
5107:
5092:
5090:
5089:
5084:
5082:
5078:
5072:
5064:
5050:, in particular
5049:
5047:
5046:
5041:
5025:
5023:
5022:
5017:
5015:
5011:
4996:
4988:
4965:. In this case
4964:
4962:
4961:
4956:
4935:
4933:
4932:
4927:
4862:
4860:
4859:
4854:
4846:
4842:
4806:
4801:
4797:
4779:
4775:
4769:
4761:
4746:
4744:
4743:
4738:
4736:
4732:
4722:
4720:
4719:
4714:
4686:
4684:
4683:
4678:
4673:
4659:
4645:
4641:
4623:
4619:
4583:
4581:
4580:
4575:
4570:
4556:
4542:
4538:
4486:
4484:
4483:
4480:{\displaystyle }
4478:
4305:
4303:
4302:
4297:
4292:
4291:
4290:
4277:
4276:
4275:
4274:
4262:
4261:
4247:
4246:
4245:
4239:
4235:
4234:
4233:
4232:
4210:
4209:
4208:
4202:
4201:
4186:
4184:
4183:
4178:
4173:
4172:
4163:
4162:
4161:
4160:
4143:
4142:
4141:
4140:
4139:
4126:
4125:
4124:
4106:
4105:
4104:
4103:
4093:
4089:
4088:
4087:
4086:
4064:
4063:
4062:
4061:
4051:
4050:
4035:
4033:
4032:
4027:
4022:
4021:
4012:
4011:
4010:
4009:
3992:
3991:
3990:
3989:
3988:
3964:
3963:
3958:
3954:
3953:
3952:
3951:
3929:
3928:
3923:
3922:
3901:
3899:
3898:
3893:
3891:
3890:
3889:
3876:
3875:
3874:
3868:
3867:
3853:
3851:
3850:
3845:
3840:
3839:
3830:
3829:
3828:
3827:
3817:
3816:
3803:
3801:
3800:
3795:
3790:
3789:
3780:
3779:
3774:
3773:
3746:
3744:
3743:
3738:
3736:
3735:
3726:
3725:
3713:
3712:
3711:
3637:
3635:
3634:
3629:
3627:
3626:
3574:tangent function
3551:proper supersets
3539:real number line
3523:
3521:
3520:
3515:
3513:
3512:
3500:
3499:
3498:
3497:
3451:
3449:
3448:
3443:
3441:
3440:
3428:
3427:
3426:
3425:
3397:
3395:
3394:
3389:
3387:
3386:
3374:
3373:
3372:
3371:
3354:
3353:
3332:
3330:
3329:
3324:
3322:
3321:
3305:
3303:
3302:
3297:
3295:
3294:
3263:
3261:
3260:
3255:
3253:
3252:
3227:Richard Dedekind
3191:
3189:
3188:
3183:
3181:
3180:
3144:
3142:
3141:
3136:
3134:
3133:
3050:
3048:
3047:
3042:
3040:
3039:
3023:
3021:
3020:
3015:
3013:
3012:
3011:
3010:
2989:
2987:
2986:
2981:
2979:
2978:
2977:
2976:
2959:
2958:
2935:
2933:
2932:
2927:
2925:
2924:
2923:
2922:
2905:
2904:
2891:
2889:
2888:
2883:
2881:
2880:
2868:
2867:
2842:
2840:
2839:
2834:
2832:
2831:
2814:
2812:
2811:
2806:
2804:
2803:
2776:
2774:
2773:
2768:
2766:
2765:
2749:
2747:
2746:
2741:
2739:
2738:
2716:
2714:
2713:
2708:
2690:
2688:
2687:
2682:
2671:
2670:
2658:
2657:
2645:
2644:
2550:Cardinal numbers
2533:
2531:
2529:
2528:
2523:
2521:
2507:
2505:
2503:
2502:
2497:
2495:
2481:
2479:
2477:
2476:
2471:
2466:
2458:
2457:
2442:
2440:
2438:
2437:
2432:
2430:
2416:
2414:
2412:
2411:
2406:
2401:
2393:
2392:
2377:
2375:
2373:
2372:
2367:
2365:
2351:
2349:
2348:
2343:
2310:
2308:
2306:
2305:
2300:
2295:
2287:
2286:
2264:
2262:
2260:
2259:
2254:
2252:
2235:
2233:
2231:
2230:
2225:
2211:
2209:
2207:
2206:
2201:
2187:
2185:
2183:
2182:
2177:
2163:
2161:
2159:
2158:
2153:
2136:
2130:
2120:
2118:
2116:
2115:
2110:
2096:
2094:
2092:
2091:
2086:
2072:
2070:
2069:
2064:
2062:
2054:
2046:
2038:
2026:
2024:
2023:
2018:
2016:
2008:
2000:
1992:
1972:
1970:
1969:
1964:
1962:
1954:
1946:
1938:
1926:
1924:
1923:
1918:
1916:
1908:
1900:
1892:
1880:
1878:
1877:
1872:
1870:
1862:
1854:
1846:
1831:
1829:
1827:
1826:
1821:
1807:
1805:
1803:
1802:
1797:
1783:
1781:
1779:
1778:
1773:
1759:
1757:
1755:
1754:
1749:
1732:
1726:
1716:
1714:
1712:
1711:
1706:
1692:
1690:
1689:
1684:
1682:
1677:
1672:
1664:
1656:
1644:
1642:
1640:
1639:
1634:
1620:
1618:
1616:
1615:
1610:
1595:modulo operation
1592:
1590:
1589:
1584:
1576:
1573:
1537:
1535:
1533:
1532:
1527:
1513:
1511:
1509:
1508:
1503:
1501:
1487:
1485:
1483:
1482:
1477:
1463:
1461:
1460:
1455:
1425:
1423:
1421:
1420:
1415:
1401:
1399:
1397:
1396:
1391:
1389:
1375:
1373:
1371:
1370:
1365:
1351:
1349:
1347:
1346:
1341:
1327:
1325:
1323:
1322:
1317:
1303:
1301:
1299:
1298:
1293:
1279:
1277:
1275:
1274:
1269:
1251:
1249:
1247:
1246:
1241:
1227:
1225:
1223:
1222:
1217:
1199:
1197:
1195:
1194:
1189:
1175:
1173:
1171:
1170:
1165:
1152:For finite sets
1148:
1146:
1144:
1143:
1138:
1124:
1122:
1120:
1119:
1114:
1112:
1098:
1096:
1095:
1090:
1056:
1054:
1053:
1048:
1043:
1040:
1008:
993:of non-negative
992:
990:
989:
984:
979:
976:
909:
907:
905:
904:
899:
885:
883:
881:
880:
875:
857:
855:
853:
852:
847:
833:
831:
829:
828:
823:
801:
795:
764:
742:
621:uncountable sets
602:
599:
571:commensurability
553:
552:
537:
535:
534:
529:
514:
512:
511:
506:
482:
481:
479:
478:
473:
460:
458:
457:
452:
431:
429:
428:
423:
403:
401:
400:
395:
393:
385:
373:
371:
370:
365:
349:
347:
346:
341:
339:
331:
323:
315:
303:
301:
299:
298:
293:
279:
277:
275:
274:
269:
247:cardinal numbers
224:
222:
221:
216:
180:
178:
177:
172:
120:
118:
117:
112:
104:
96:
84:
82:
81:
76:
60:
58:
57:
52:
21:
8226:
8225:
8221:
8220:
8219:
8217:
8216:
8215:
8196:
8195:
8194:
8189:
8116:
8095:
8079:
8044:New Foundations
7991:
7881:
7800:Cardinal number
7783:
7769:
7710:
7594:
7585:
7569:
7564:
7534:
7529:
7518:
7511:
7456:Category theory
7446:Algebraic logic
7429:
7400:Lambda calculus
7338:Church encoding
7324:
7300:Truth predicate
7156:
7122:Complete theory
7045:
6914:
6910:
6906:
6901:
6893:
6613: and
6609:
6604:
6590:
6566:New Foundations
6534:axiom of choice
6517:
6479:Gödel numbering
6419: and
6411:
6315:
6200:
6150:
6131:
6080:Boolean algebra
6066:
6030:Equiconsistency
5995:Classical logic
5972:
5953:Halting problem
5941: and
5917: and
5905: and
5904:
5899:Theorems (
5894:
5811:
5806:
5776:
5775:
5758:
5754:
5749:
5748:
5727:
5723:
5708:
5694:
5690:
5684:
5667:
5663:
5600:
5596:
5533:
5529:
5519:
5499:Felix Hausdorff
5496:
5492:
5448:Otto Blumenthal
5429:
5425:
5416:
5414:
5405:
5404:
5397:
5381:
5375:
5371:
5360:
5356:
5346:
5344:
5340:
5329:
5323:
5319:
5305:Wayback Machine
5295:
5291:
5282:
5280:
5272:
5271:
5267:
5252:
5248:
5239:
5237:
5229:
5228:
5224:
5203:
5199:
5192:
5178:
5174:
5169:
5122:
5101:
5098:
5097:
5076:
5068:
5060:
5058:
5055:
5054:
5035:
5032:
5031:
5009:
4992:
4984:
4973:
4970:
4969:
4941:
4938:
4937:
4871:
4868:
4867:
4843: injective
4840:
4802:
4795:
4773:
4765:
4757:
4755:
4752:
4751:
4730:
4728:
4725:
4724:
4708:
4705:
4704:
4701:
4663:
4649:
4631:
4627:
4609:
4605:
4603:
4600:
4599:
4560:
4546:
4528:
4524:
4522:
4519:
4518:
4500:
4494:
4460:
4457:
4456:
4312:
4286:
4285:
4281:
4270:
4266:
4257:
4256:
4255:
4251:
4241:
4240:
4228:
4224:
4223:
4219:
4215:
4214:
4204:
4203:
4197:
4196:
4195:
4193:
4190:
4189:
4168:
4167:
4156:
4152:
4151:
4147:
4135:
4131:
4130:
4120:
4116:
4115:
4114:
4110:
4099:
4095:
4094:
4082:
4078:
4077:
4073:
4069:
4068:
4057:
4053:
4052:
4046:
4045:
4044:
4042:
4039:
4038:
4017:
4016:
4005:
4001:
4000:
3996:
3984:
3980:
3979:
3972:
3968:
3959:
3947:
3943:
3942:
3938:
3934:
3933:
3924:
3918:
3917:
3916:
3914:
3911:
3910:
3885:
3884:
3880:
3870:
3869:
3863:
3862:
3861:
3859:
3856:
3855:
3835:
3834:
3823:
3819:
3818:
3812:
3811:
3810:
3808:
3805:
3804:
3785:
3784:
3775:
3769:
3768:
3767:
3765:
3762:
3761:
3731:
3730:
3721:
3717:
3707:
3706:
3702:
3700:
3697:
3696:
3638:exist (see his
3622:
3621:
3619:
3616:
3615:
3599:introduced the
3508:
3504:
3493:
3489:
3488:
3484:
3482:
3479:
3478:
3472:cardinal number
3436:
3432:
3421:
3417:
3416:
3412:
3410:
3407:
3406:
3382:
3378:
3367:
3363:
3362:
3358:
3349:
3348:
3346:
3343:
3342:
3317:
3313:
3311:
3308:
3307:
3290:
3289:
3287:
3284:
3283:
3276:
3270:
3248:
3244:
3242:
3239:
3238:
3207:
3176:
3175:
3173:
3170:
3169:
3129:
3125:
3123:
3120:
3119:
3089:natural numbers
3073:axiom of choice
3069:
3035:
3031:
3029:
3026:
3025:
3006:
3002:
3001:
2997:
2995:
2992:
2991:
2972:
2968:
2967:
2963:
2954:
2950:
2948:
2945:
2944:
2918:
2914:
2913:
2909:
2900:
2899:
2897:
2894:
2893:
2876:
2872:
2863:
2862:
2860:
2857:
2856:
2843:" (a lowercase
2827:
2826:
2824:
2821:
2820:
2819:is denoted by "
2799:
2795:
2793:
2790:
2789:
2782:natural numbers
2761:
2757:
2755:
2752:
2751:
2728:
2724:
2722:
2719:
2718:
2702:
2699:
2698:
2666:
2662:
2653:
2649:
2640:
2636:
2634:
2631:
2630:
2620:axiom of choice
2608:cardinal number
2558:
2556:Cardinal number
2552:
2517:
2515:
2512:
2511:
2509:
2491:
2489:
2486:
2485:
2483:
2462:
2453:
2452:
2450:
2447:
2446:
2444:
2426:
2424:
2421:
2420:
2418:
2397:
2388:
2387:
2385:
2382:
2381:
2379:
2361:
2359:
2356:
2355:
2353:
2316:
2313:
2312:
2291:
2282:
2281:
2279:
2276:
2275:
2273:
2267:natural numbers
2248:
2246:
2243:
2242:
2240:
2219:
2216:
2215:
2213:
2195:
2192:
2191:
2189:
2171:
2168:
2167:
2165:
2147:
2144:
2143:
2141:
2139:
2132:
2126:
2104:
2101:
2100:
2098:
2080:
2077:
2076:
2074:
2058:
2050:
2042:
2034:
2032:
2029:
2028:
2012:
2004:
1996:
1988:
1986:
1983:
1982:
1979:axiom of choice
1958:
1950:
1942:
1934:
1932:
1929:
1928:
1912:
1904:
1896:
1888:
1886:
1883:
1882:
1866:
1858:
1850:
1842:
1840:
1837:
1836:
1815:
1812:
1811:
1809:
1791:
1788:
1787:
1785:
1767:
1764:
1763:
1761:
1743:
1740:
1739:
1737:
1735:
1728:
1727:| †|
1722:
1700:
1697:
1696:
1694:
1678:
1673:
1668:
1660:
1652:
1650:
1647:
1646:
1628:
1625:
1624:
1622:
1604:
1601:
1600:
1598:
1574: mod
1572:
1543:
1540:
1539:
1521:
1518:
1517:
1515:
1497:
1495:
1492:
1491:
1489:
1471:
1468:
1467:
1465:
1431:
1428:
1427:
1409:
1406:
1405:
1403:
1385:
1383:
1380:
1379:
1377:
1359:
1356:
1355:
1353:
1335:
1332:
1331:
1329:
1311:
1308:
1307:
1305:
1287:
1284:
1283:
1281:
1263:
1260:
1259:
1257:
1235:
1232:
1231:
1229:
1211:
1208:
1207:
1205:
1183:
1180:
1179:
1177:
1159:
1156:
1155:
1153:
1149:(see picture).
1132:
1129:
1128:
1126:
1108:
1106:
1103:
1102:
1100:
1066:
1063:
1062:
1059:natural numbers
1039:
1004:
1002:
999:
998:
975:
940:
937:
936:
893:
890:
889:
887:
869:
866:
865:
863:
841:
838:
837:
835:
817:
814:
813:
811:
804:
797:
796:| = |
791:
760:
738:
629:
600:
550:
548:
544:
520:
517:
516:
488:
485:
484:
467:
464:
463:
462:
437:
434:
433:
417:
414:
413:
389:
381:
379:
376:
375:
359:
356:
355:
352:cardinal number
335:
327:
319:
311:
309:
306:
305:
287:
284:
283:
281:
263:
260:
259:
257:
256:When two sets,
186:
183:
182:
142:
139:
138:
100:
92:
90:
87:
86:
70:
67:
66:
63:Platonic solids
46:
43:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8224:
8214:
8213:
8208:
8191:
8190:
8188:
8187:
8182:
8180:Thoralf Skolem
8177:
8172:
8167:
8162:
8157:
8152:
8147:
8142:
8137:
8132:
8126:
8124:
8118:
8117:
8115:
8114:
8109:
8104:
8098:
8096:
8094:
8093:
8090:
8084:
8081:
8080:
8078:
8077:
8076:
8075:
8070:
8065:
8064:
8063:
8048:
8047:
8046:
8034:
8033:
8032:
8021:
8020:
8015:
8010:
8005:
7999:
7997:
7993:
7992:
7990:
7989:
7984:
7979:
7974:
7965:
7960:
7955:
7945:
7940:
7939:
7938:
7933:
7928:
7918:
7908:
7903:
7898:
7892:
7890:
7883:
7882:
7880:
7879:
7874:
7869:
7864:
7862:Ordinal number
7859:
7854:
7849:
7844:
7843:
7842:
7837:
7827:
7822:
7817:
7812:
7807:
7797:
7792:
7786:
7784:
7782:
7781:
7778:
7774:
7771:
7770:
7768:
7767:
7762:
7757:
7752:
7747:
7742:
7740:Disjoint union
7737:
7732:
7726:
7720:
7718:
7712:
7711:
7709:
7708:
7707:
7706:
7701:
7690:
7689:
7687:Martin's axiom
7684:
7679:
7674:
7669:
7664:
7659:
7654:
7652:Extensionality
7649:
7648:
7647:
7637:
7632:
7631:
7630:
7625:
7620:
7610:
7604:
7602:
7596:
7595:
7588:
7586:
7584:
7583:
7577:
7575:
7571:
7570:
7563:
7562:
7555:
7548:
7540:
7531:
7530:
7516:
7513:
7512:
7510:
7509:
7504:
7499:
7494:
7489:
7488:
7487:
7477:
7472:
7467:
7458:
7453:
7448:
7443:
7441:Abstract logic
7437:
7435:
7431:
7430:
7428:
7427:
7422:
7420:Turing machine
7417:
7412:
7407:
7402:
7397:
7392:
7391:
7390:
7385:
7380:
7375:
7370:
7360:
7358:Computable set
7355:
7350:
7345:
7340:
7334:
7332:
7326:
7325:
7323:
7322:
7317:
7312:
7307:
7302:
7297:
7292:
7287:
7286:
7285:
7280:
7275:
7265:
7260:
7255:
7253:Satisfiability
7250:
7245:
7240:
7239:
7238:
7228:
7227:
7226:
7216:
7215:
7214:
7209:
7204:
7199:
7194:
7184:
7183:
7182:
7177:
7170:Interpretation
7166:
7164:
7158:
7157:
7155:
7154:
7149:
7144:
7139:
7134:
7124:
7119:
7118:
7117:
7116:
7115:
7105:
7100:
7090:
7085:
7080:
7075:
7070:
7065:
7059:
7057:
7051:
7050:
7047:
7046:
7044:
7043:
7035:
7034:
7033:
7032:
7027:
7026:
7025:
7020:
7015:
6995:
6994:
6993:
6991:minimal axioms
6988:
6977:
6976:
6975:
6964:
6963:
6962:
6957:
6952:
6947:
6942:
6937:
6924:
6922:
6903:
6902:
6900:
6899:
6898:
6897:
6885:
6880:
6879:
6878:
6873:
6868:
6863:
6853:
6848:
6843:
6838:
6837:
6836:
6831:
6821:
6820:
6819:
6814:
6809:
6804:
6794:
6789:
6788:
6787:
6782:
6777:
6767:
6766:
6765:
6760:
6755:
6750:
6745:
6740:
6730:
6725:
6720:
6715:
6714:
6713:
6708:
6703:
6698:
6688:
6683:
6681:Formation rule
6678:
6673:
6672:
6671:
6666:
6656:
6655:
6654:
6644:
6639:
6634:
6629:
6623:
6617:
6600:Formal systems
6596:
6595:
6592:
6591:
6589:
6588:
6583:
6578:
6573:
6568:
6563:
6558:
6553:
6548:
6543:
6542:
6541:
6536:
6525:
6523:
6519:
6518:
6516:
6515:
6514:
6513:
6503:
6498:
6497:
6496:
6489:Large cardinal
6486:
6481:
6476:
6471:
6466:
6452:
6451:
6450:
6445:
6440:
6425:
6423:
6413:
6412:
6410:
6409:
6408:
6407:
6402:
6397:
6387:
6382:
6377:
6372:
6367:
6362:
6357:
6352:
6347:
6342:
6337:
6332:
6326:
6324:
6317:
6316:
6314:
6313:
6312:
6311:
6306:
6301:
6296:
6291:
6286:
6278:
6277:
6276:
6271:
6261:
6256:
6254:Extensionality
6251:
6249:Ordinal number
6246:
6236:
6231:
6230:
6229:
6218:
6212:
6206:
6205:
6202:
6201:
6199:
6198:
6193:
6188:
6183:
6178:
6173:
6168:
6167:
6166:
6156:
6155:
6154:
6141:
6139:
6133:
6132:
6130:
6129:
6128:
6127:
6122:
6117:
6107:
6102:
6097:
6092:
6087:
6082:
6076:
6074:
6068:
6067:
6065:
6064:
6059:
6054:
6049:
6044:
6039:
6034:
6033:
6032:
6022:
6017:
6012:
6007:
6002:
5997:
5991:
5989:
5980:
5974:
5973:
5971:
5970:
5965:
5960:
5955:
5950:
5945:
5933:Cantor's
5931:
5926:
5921:
5911:
5909:
5896:
5895:
5893:
5892:
5887:
5882:
5877:
5872:
5867:
5862:
5857:
5852:
5847:
5842:
5837:
5832:
5831:
5830:
5819:
5817:
5813:
5812:
5805:
5804:
5797:
5790:
5782:
5774:
5773:
5751:
5750:
5747:
5746:
5721:
5709:Reprinted in:
5688:
5682:
5661:
5614:(1): 105â110.
5594:
5527:
5517:
5490:
5423:
5395:
5369:
5354:
5317:
5289:
5265:
5246:
5222:
5197:
5190:
5171:
5170:
5168:
5165:
5164:
5163:
5158:
5153:
5148:
5143:
5138:
5133:
5128:
5121:
5118:
5105:
5094:
5093:
5075:
5071:
5067:
5063:
5039:
5028:
5027:
5008:
5005:
5002:
4999:
4995:
4991:
4987:
4983:
4980:
4977:
4954:
4951:
4948:
4945:
4925:
4922:
4919:
4916:
4913:
4910:
4907:
4904:
4901:
4898:
4894:
4890:
4887:
4884:
4881:
4878:
4875:
4864:
4863:
4852:
4849:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4805:
4794:
4791:
4788:
4785:
4782:
4772:
4768:
4764:
4760:
4712:
4700:
4689:
4688:
4687:
4676:
4672:
4669:
4666:
4662:
4658:
4655:
4652:
4648:
4644:
4640:
4637:
4634:
4630:
4626:
4622:
4618:
4615:
4612:
4608:
4585:
4584:
4573:
4569:
4566:
4563:
4559:
4555:
4552:
4549:
4545:
4541:
4537:
4534:
4531:
4527:
4496:Main article:
4493:
4490:
4489:
4488:
4476:
4473:
4470:
4467:
4464:
4446:
4415:
4384:
4311:
4308:
4307:
4306:
4295:
4289:
4284:
4280:
4273:
4269:
4265:
4260:
4254:
4250:
4244:
4238:
4231:
4227:
4222:
4218:
4213:
4207:
4200:
4187:
4176:
4171:
4166:
4159:
4155:
4150:
4146:
4138:
4134:
4129:
4123:
4119:
4113:
4109:
4102:
4098:
4092:
4085:
4081:
4076:
4072:
4067:
4060:
4056:
4049:
4036:
4025:
4020:
4015:
4008:
4004:
3999:
3995:
3987:
3983:
3978:
3975:
3971:
3967:
3962:
3957:
3950:
3946:
3941:
3937:
3932:
3927:
3921:
3888:
3883:
3879:
3873:
3866:
3843:
3838:
3833:
3826:
3822:
3815:
3793:
3788:
3783:
3778:
3772:
3755:
3754:
3747:
3734:
3729:
3724:
3720:
3716:
3710:
3705:
3690:
3689:
3688:
3687:
3673:
3625:
3597:Giuseppe Peano
3584:(âÂœÏ, ÂœÏ) and
3547:proper subsets
3525:
3524:
3511:
3507:
3503:
3496:
3492:
3487:
3464:
3463:
3452:
3439:
3435:
3431:
3424:
3420:
3415:
3385:
3381:
3377:
3370:
3366:
3361:
3357:
3352:
3320:
3316:
3293:
3272:Main article:
3269:
3266:
3251:
3247:
3206:
3203:
3202:
3201:
3179:
3150:
3132:
3128:
3104:
3068:
3065:
3038:
3034:
3009:
3005:
3000:
2975:
2971:
2966:
2962:
2957:
2953:
2921:
2917:
2912:
2908:
2903:
2879:
2875:
2871:
2866:
2845:fraktur script
2830:
2802:
2798:
2764:
2760:
2737:
2734:
2731:
2727:
2706:
2692:
2691:
2680:
2677:
2674:
2669:
2665:
2661:
2656:
2652:
2648:
2643:
2639:
2616:
2615:
2600:representative
2596:
2565:equinumerosity
2554:Main article:
2551:
2548:
2520:
2494:
2469:
2465:
2461:
2456:
2429:
2404:
2400:
2396:
2391:
2364:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2298:
2294:
2290:
2285:
2251:
2223:
2199:
2175:
2151:
2138:
2123:
2108:
2084:
2061:
2057:
2053:
2049:
2045:
2041:
2037:
2015:
2011:
2007:
2003:
1999:
1995:
1991:
1961:
1957:
1953:
1949:
1945:
1941:
1937:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1869:
1865:
1861:
1857:
1853:
1849:
1845:
1819:
1795:
1771:
1747:
1734:
1719:
1704:
1681:
1676:
1671:
1667:
1663:
1659:
1655:
1645:can challenge
1632:
1608:
1582:
1579:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1525:
1500:
1475:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1413:
1388:
1363:
1339:
1315:
1291:
1267:
1239:
1215:
1187:
1163:
1136:
1111:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1046:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1007:
982:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
897:
873:
845:
821:
803:
788:
628:
627:Comparing sets
625:
543:
540:
527:
524:
504:
501:
498:
495:
492:
471:
450:
447:
444:
441:
421:
410:absolute value
392:
388:
384:
363:
338:
334:
330:
326:
322:
318:
314:
291:
267:
214:
211:
208:
205:
202:
199:
196:
193:
190:
170:
167:
164:
161:
158:
155:
152:
149:
146:
110:
107:
103:
99:
95:
74:
50:
26:
9:
6:
4:
3:
2:
8223:
8212:
8209:
8207:
8204:
8203:
8201:
8186:
8185:Ernst Zermelo
8183:
8181:
8178:
8176:
8173:
8171:
8170:Willard Quine
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8136:
8133:
8131:
8128:
8127:
8125:
8123:
8122:Set theorists
8119:
8113:
8110:
8108:
8105:
8103:
8100:
8099:
8097:
8091:
8089:
8086:
8085:
8082:
8074:
8071:
8069:
8068:KripkeâPlatek
8066:
8062:
8059:
8058:
8057:
8054:
8053:
8052:
8049:
8045:
8042:
8041:
8040:
8039:
8035:
8031:
8028:
8027:
8026:
8023:
8022:
8019:
8016:
8014:
8011:
8009:
8006:
8004:
8001:
8000:
7998:
7994:
7988:
7985:
7983:
7980:
7978:
7975:
7973:
7971:
7966:
7964:
7961:
7959:
7956:
7953:
7949:
7946:
7944:
7941:
7937:
7934:
7932:
7929:
7927:
7924:
7923:
7922:
7919:
7916:
7912:
7909:
7907:
7904:
7902:
7899:
7897:
7894:
7893:
7891:
7888:
7884:
7878:
7875:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7841:
7838:
7836:
7833:
7832:
7831:
7828:
7826:
7823:
7821:
7818:
7816:
7813:
7811:
7808:
7805:
7801:
7798:
7796:
7793:
7791:
7788:
7787:
7785:
7779:
7776:
7775:
7772:
7766:
7763:
7761:
7758:
7756:
7753:
7751:
7748:
7746:
7743:
7741:
7738:
7736:
7733:
7730:
7727:
7725:
7722:
7721:
7719:
7717:
7713:
7705:
7704:specification
7702:
7700:
7697:
7696:
7695:
7692:
7691:
7688:
7685:
7683:
7680:
7678:
7675:
7673:
7670:
7668:
7665:
7663:
7660:
7658:
7655:
7653:
7650:
7646:
7643:
7642:
7641:
7638:
7636:
7633:
7629:
7626:
7624:
7621:
7619:
7616:
7615:
7614:
7611:
7609:
7606:
7605:
7603:
7601:
7597:
7592:
7582:
7579:
7578:
7576:
7572:
7568:
7561:
7556:
7554:
7549:
7547:
7542:
7541:
7538:
7528:
7527:
7522:
7514:
7508:
7505:
7503:
7500:
7498:
7495:
7493:
7490:
7486:
7483:
7482:
7481:
7478:
7476:
7473:
7471:
7468:
7466:
7462:
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7438:
7436:
7432:
7426:
7423:
7421:
7418:
7416:
7415:Recursive set
7413:
7411:
7408:
7406:
7403:
7401:
7398:
7396:
7393:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7369:
7366:
7365:
7364:
7361:
7359:
7356:
7354:
7351:
7349:
7346:
7344:
7341:
7339:
7336:
7335:
7333:
7331:
7327:
7321:
7318:
7316:
7313:
7311:
7308:
7306:
7303:
7301:
7298:
7296:
7293:
7291:
7288:
7284:
7281:
7279:
7276:
7274:
7271:
7270:
7269:
7266:
7264:
7261:
7259:
7256:
7254:
7251:
7249:
7246:
7244:
7241:
7237:
7234:
7233:
7232:
7229:
7225:
7224:of arithmetic
7222:
7221:
7220:
7217:
7213:
7210:
7208:
7205:
7203:
7200:
7198:
7195:
7193:
7190:
7189:
7188:
7185:
7181:
7178:
7176:
7173:
7172:
7171:
7168:
7167:
7165:
7163:
7159:
7153:
7150:
7148:
7145:
7143:
7140:
7138:
7135:
7132:
7131:from ZFC
7128:
7125:
7123:
7120:
7114:
7111:
7110:
7109:
7106:
7104:
7101:
7099:
7096:
7095:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7069:
7066:
7064:
7061:
7060:
7058:
7056:
7052:
7042:
7041:
7037:
7036:
7031:
7030:non-Euclidean
7028:
7024:
7021:
7019:
7016:
7014:
7013:
7009:
7008:
7006:
7003:
7002:
7000:
6996:
6992:
6989:
6987:
6984:
6983:
6982:
6978:
6974:
6971:
6970:
6969:
6965:
6961:
6958:
6956:
6953:
6951:
6948:
6946:
6943:
6941:
6938:
6936:
6933:
6932:
6930:
6926:
6925:
6923:
6918:
6912:
6907:Example
6904:
6896:
6891:
6890:
6889:
6886:
6884:
6881:
6877:
6874:
6872:
6869:
6867:
6864:
6862:
6859:
6858:
6857:
6854:
6852:
6849:
6847:
6844:
6842:
6839:
6835:
6832:
6830:
6827:
6826:
6825:
6822:
6818:
6815:
6813:
6810:
6808:
6805:
6803:
6800:
6799:
6798:
6795:
6793:
6790:
6786:
6783:
6781:
6778:
6776:
6773:
6772:
6771:
6768:
6764:
6761:
6759:
6756:
6754:
6751:
6749:
6746:
6744:
6741:
6739:
6736:
6735:
6734:
6731:
6729:
6726:
6724:
6721:
6719:
6716:
6712:
6709:
6707:
6704:
6702:
6699:
6697:
6694:
6693:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6670:
6667:
6665:
6664:by definition
6662:
6661:
6660:
6657:
6653:
6650:
6649:
6648:
6645:
6643:
6640:
6638:
6635:
6633:
6630:
6628:
6625:
6624:
6621:
6618:
6616:
6612:
6607:
6601:
6597:
6587:
6584:
6582:
6579:
6577:
6574:
6572:
6569:
6567:
6564:
6562:
6559:
6557:
6554:
6552:
6551:KripkeâPlatek
6549:
6547:
6544:
6540:
6537:
6535:
6532:
6531:
6530:
6527:
6526:
6524:
6520:
6512:
6509:
6508:
6507:
6504:
6502:
6499:
6495:
6492:
6491:
6490:
6487:
6485:
6482:
6480:
6477:
6475:
6472:
6470:
6467:
6464:
6460:
6456:
6453:
6449:
6446:
6444:
6441:
6439:
6436:
6435:
6434:
6430:
6427:
6426:
6424:
6422:
6418:
6414:
6406:
6403:
6401:
6398:
6396:
6395:constructible
6393:
6392:
6391:
6388:
6386:
6383:
6381:
6378:
6376:
6373:
6371:
6368:
6366:
6363:
6361:
6358:
6356:
6353:
6351:
6348:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6327:
6325:
6323:
6318:
6310:
6307:
6305:
6302:
6300:
6297:
6295:
6292:
6290:
6287:
6285:
6282:
6281:
6279:
6275:
6272:
6270:
6267:
6266:
6265:
6262:
6260:
6257:
6255:
6252:
6250:
6247:
6245:
6241:
6237:
6235:
6232:
6228:
6225:
6224:
6223:
6220:
6219:
6216:
6213:
6211:
6207:
6197:
6194:
6192:
6189:
6187:
6184:
6182:
6179:
6177:
6174:
6172:
6169:
6165:
6162:
6161:
6160:
6157:
6153:
6148:
6147:
6146:
6143:
6142:
6140:
6138:
6134:
6126:
6123:
6121:
6118:
6116:
6113:
6112:
6111:
6108:
6106:
6103:
6101:
6098:
6096:
6093:
6091:
6088:
6086:
6083:
6081:
6078:
6077:
6075:
6073:
6072:Propositional
6069:
6063:
6060:
6058:
6055:
6053:
6050:
6048:
6045:
6043:
6040:
6038:
6035:
6031:
6028:
6027:
6026:
6023:
6021:
6018:
6016:
6013:
6011:
6008:
6006:
6003:
6001:
6000:Logical truth
5998:
5996:
5993:
5992:
5990:
5988:
5984:
5981:
5979:
5975:
5969:
5966:
5964:
5961:
5959:
5956:
5954:
5951:
5949:
5946:
5944:
5940:
5936:
5932:
5930:
5927:
5925:
5922:
5920:
5916:
5913:
5912:
5910:
5908:
5902:
5897:
5891:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5853:
5851:
5848:
5846:
5843:
5841:
5838:
5836:
5833:
5829:
5826:
5825:
5824:
5821:
5820:
5818:
5814:
5810:
5803:
5798:
5796:
5791:
5789:
5784:
5783:
5780:
5770:
5766:
5762:
5756:
5752:
5743:
5742:0-85312-563-5
5739:
5735:
5734:0-85312-612-7
5731:
5725:
5716:
5715:
5705:
5701:
5700:
5692:
5685:
5683:0-09-944068-7
5679:
5675:
5671:
5665:
5657:
5653:
5648:
5643:
5639:
5635:
5630:
5625:
5621:
5617:
5613:
5609:
5605:
5598:
5590:
5586:
5581:
5576:
5572:
5568:
5563:
5558:
5554:
5550:
5546:
5542:
5538:
5531:
5525:
5520:
5518:3-540-42224-2
5514:
5510:
5509:
5504:
5500:
5494:
5487:
5483:
5479:
5475:
5471:
5467:
5463:
5459:
5458:
5453:
5449:
5445:
5444:David Hilbert
5441:
5437:
5433:
5427:
5412:
5408:
5402:
5400:
5391:
5387:
5380:
5373:
5365:
5358:
5339:
5335:
5328:
5321:
5314:
5310:
5306:
5302:
5299:
5293:
5279:
5278:Math Timeline
5275:
5269:
5262:
5258:
5257:
5250:
5236:
5235:brilliant.org
5232:
5226:
5217:
5216:
5211:
5208:
5201:
5193:
5187:
5183:
5176:
5172:
5162:
5159:
5157:
5154:
5152:
5149:
5147:
5146:Countable set
5144:
5142:
5139:
5137:
5134:
5132:
5129:
5127:
5124:
5123:
5117:
5103:
5073:
5065:
5053:
5052:
5051:
5037:
5003:
5000:
4989:
4978:
4968:
4967:
4966:
4952:
4949:
4943:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4885:
4882:
4879:
4876:
4837:
4831:
4825:
4819:
4816:
4813:
4792:
4789:
4783:
4780:
4770:
4762:
4750:
4749:
4748:
4710:
4698:
4694:
4674:
4670:
4667:
4664:
4660:
4656:
4653:
4650:
4646:
4642:
4638:
4635:
4632:
4628:
4624:
4620:
4616:
4613:
4610:
4606:
4598:
4597:
4596:
4594:
4593:intersections
4590:
4571:
4567:
4564:
4561:
4557:
4553:
4550:
4547:
4543:
4539:
4535:
4532:
4529:
4525:
4517:
4516:
4515:
4513:
4512:disjoint sets
4509:
4505:
4499:
4471:
4468:
4465:
4454:
4450:
4447:
4444:
4440:
4436:
4432:
4428:
4424:
4420:
4416:
4413:
4409:
4405:
4401:
4397:
4393:
4389:
4385:
4382:
4378:
4374:
4370:
4366:
4363:, oranges), (
4362:
4358:
4355:| because { (
4354:
4350:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4318:
4314:
4313:
4293:
4282:
4278:
4271:
4263:
4252:
4248:
4236:
4229:
4220:
4216:
4211:
4188:
4174:
4164:
4157:
4148:
4144:
4136:
4127:
4121:
4111:
4107:
4100:
4090:
4083:
4074:
4070:
4065:
4058:
4037:
4023:
4013:
4006:
3997:
3993:
3985:
3976:
3973:
3969:
3965:
3960:
3955:
3948:
3939:
3935:
3930:
3925:
3909:
3908:
3907:
3905:
3881:
3877:
3841:
3831:
3824:
3791:
3781:
3776:
3760:
3752:
3748:
3727:
3722:
3718:
3714:
3703:
3695:
3694:
3693:
3686:
3682:
3678:
3674:
3671:
3667:
3663:
3659:
3655:
3651:
3650:
3649:
3648:
3647:
3645:
3641:
3612:
3610:
3606:
3602:
3598:
3593:
3591:
3587:
3583:
3579:
3575:
3570:
3568:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3535:
3533:
3530:
3509:
3501:
3494:
3485:
3477:
3476:
3475:
3473:
3469:
3461:
3457:
3453:
3437:
3429:
3422:
3413:
3405:
3404:
3403:
3402:) satisfies:
3401:
3383:
3379:
3375:
3368:
3359:
3355:
3340:
3336:
3318:
3281:
3275:
3265:
3249:
3236:
3232:
3228:
3224:
3223:Gottlob Frege
3220:
3216:
3215:infinite sets
3212:
3205:Infinite sets
3199:
3195:
3167:
3163:
3159:
3155:
3151:
3148:
3130:
3117:
3113:
3109:
3105:
3102:
3098:
3094:
3090:
3086:
3082:
3081:
3080:
3078:
3074:
3064:
3062:
3058:
3054:
3036:
3007:
2998:
2973:
2964:
2960:
2955:
2942:
2937:
2919:
2910:
2906:
2877:
2869:
2854:
2850:
2846:
2818:
2800:
2787:
2783:
2778:
2762:
2735:
2732:
2729:
2704:
2697:
2678:
2675:
2672:
2667:
2659:
2654:
2646:
2641:
2629:
2628:
2627:
2625:
2624:infinite sets
2621:
2618:Assuming the
2613:
2609:
2605:
2601:
2597:
2594:
2590:
2589:
2588:
2586:
2582:
2578:
2574:
2570:
2566:
2561:
2557:
2547:
2545:
2541:
2537:
2336:
2330:
2324:
2318:
2272:
2268:
2237:
2221:
2197:
2173:
2149:
2135:
2129:
2122:
2106:
2082:
2055:
2047:
2039:
2009:
2001:
1993:
1980:
1976:
1955:
1947:
1939:
1909:
1901:
1893:
1863:
1855:
1847:
1833:
1817:
1793:
1769:
1745:
1731:
1725:
1718:
1702:
1665:
1657:
1630:
1606:
1596:
1577:
1569:
1563:
1560:
1557:
1551:
1545:
1538:, defined by
1523:
1473:
1451:
1448:
1445:
1439:
1433:
1426:, defined by
1411:
1361:
1337:
1313:
1289:
1265:
1255:
1237:
1213:
1203:
1185:
1161:
1150:
1134:
1086:
1083:
1080:
1074:
1068:
1060:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1009:
996:
972:
969:
966:
963:
960:
957:
954:
951:
945:
942:
933:
931:
930:
925:
921:
917:
913:
910:that is both
895:
871:
861:
858:, that is, a
843:
819:
809:
800:
794:
787:
780:
776:
772:
768:
763:
758:
754:
750:
746:
741:
736:
732:
728:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
684:
680:
676:
672:
669:
665:
661:
654:
650:
646:
642:
638:
633:
624:
622:
618:
614:
610:
606:
594:
592:
588:
584:
580:
576:
572:
568:
567:
560:
558:
539:
525:
499:
493:
490:
469:
445:
439:
419:
411:
407:
386:
361:
353:
332:
324:
316:
289:
265:
254:
252:
248:
244:
240:
236:
232:
231:infinite sets
228:
209:
206:
203:
200:
197:
191:
188:
165:
162:
159:
156:
153:
147:
144:
136:
132:
128:
108:
105:
97:
72:
64:
48:
39:
33:
19:
8135:Georg Cantor
8130:Paul Bernays
8061:MorseâKelley
8036:
7969:
7968:Subset
7915:hereditarily
7877:Venn diagram
7835:ordered pair
7794:
7750:Intersection
7694:Axiom schema
7517:
7315:Ultraproduct
7162:Model theory
7127:Independence
7063:Formal proof
7055:Proof theory
7038:
7011:
6968:real numbers
6940:second-order
6851:Substitution
6728:Metalanguage
6669:conservative
6642:Axiom schema
6586:Constructive
6556:MorseâKelley
6522:Set theories
6501:Aleph number
6494:inaccessible
6420:
6400:Grothendieck
6284:intersection
6171:Higher-order
6159:Second-order
6105:Truth tables
6062:Venn diagram
5845:Formal proof
5834:
5755:
5724:
5713:
5703:
5697:
5691:
5673:
5664:
5611:
5607:
5597:
5544:
5540:
5530:
5507:
5493:
5461:
5455:
5426:
5415:. Retrieved
5413:. 2019-12-05
5410:
5389:
5385:
5372:
5363:
5357:
5345:. Retrieved
5338:the original
5333:
5320:
5308:
5292:
5281:. Retrieved
5277:
5268:
5254:
5249:
5238:. Retrieved
5234:
5225:
5213:
5200:
5181:
5175:
5126:Aleph number
5095:
5029:
4936:, therefore
4865:
4702:
4586:
4507:
4503:
4501:
4438:
4434:
4430:
4426:
4422:
4418:
4411:
4407:
4403:
4399:
4398:such that |
4395:
4391:
4387:
4380:
4376:
4372:
4368:
4364:
4360:
4359:, apples), (
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4328:
4324:
4320:
4316:
3756:
3691:
3684:
3680:
3676:
3669:
3665:
3661:
3656:, i.e., the
3653:
3613:
3609:such a proof
3594:
3585:
3580:between the
3571:
3566:
3562:
3558:
3554:
3536:
3526:
3465:
3338:
3334:
3277:
3219:Georg Cantor
3208:
3193:
3165:
3161:
3157:
3153:
3115:
3111:
3107:
3096:
3092:
3084:
3070:
2938:
2817:real numbers
2779:
2693:
2626:are denoted
2617:
2592:
2584:
2580:
2562:
2559:
2536:real numbers
2238:
2140:
2133:
2127:
1834:
1736:
1729:
1723:
1253:
1201:
1151:
995:even numbers
934:
929:equinumerous
927:
923:
919:
805:
798:
792:
784:
778:
774:
770:
766:
761:
756:
752:
748:
744:
739:
734:
730:
726:
722:
714:
710:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
663:
652:
648:
645:even numbers
640:
636:
605:Georg Cantor
595:
586:
582:
578:
574:
564:
561:
547:as early as
545:
406:vertical bar
351:
255:
250:
130:
124:
8160:Thomas Jech
8003:Alternative
7982:Uncountable
7936:Ultrafilter
7795:Cardinality
7699:replacement
7640:Determinacy
7425:Type theory
7373:undecidable
7305:Truth value
7192:equivalence
6871:non-logical
6484:Enumeration
6474:Isomorphism
6421:cardinality
6405:Von Neumann
6370:Ultrafilter
6335:Uncountable
6269:equivalence
6186:Quantifiers
6176:Fixed-point
6145:First-order
6025:Consistency
6010:Proposition
5987:Traditional
5958:Lindström's
5948:Compactness
5890:Type theory
5835:Cardinality
5436:Felix Klein
5131:Beth number
3561:, although
3211:finite sets
3198:uncountable
3075:holds, the
3053:independent
2784:is denoted
924:equipollent
693:), the set
647:. Although
639:to the set
601: 1880
557:mathematics
131:cardinality
127:mathematics
8200:Categories
8155:Kurt Gödel
8140:Paul Cohen
7977:Transitive
7745:Identities
7729:Complement
7716:Operations
7677:Regularity
7645:projective
7608:Adjunction
7567:Set theory
7236:elementary
6929:arithmetic
6797:Quantifier
6775:functional
6647:Expression
6365:Transitive
6309:identities
6294:complement
6227:hereditary
6210:Set theory
5670:Penrose, R
5417:2020-08-23
5283:2018-04-26
5240:2020-08-23
5167:References
5156:Ordinality
4433:|, then |
3664:, written
3588:(see also
3101:finite set
2943:says that
2786:aleph-null
2311:, because
2073:for every
920:equipotent
916:surjective
609:set theory
243:injections
239:bijections
235:arithmetic
8088:Paradoxes
8008:Axiomatic
7987:Universal
7963:Singleton
7958:Recursive
7901:Countable
7896:Amorphous
7755:Power set
7672:Power set
7623:dependent
7618:countable
7507:Supertask
7410:Recursion
7368:decidable
7202:saturated
7180:of models
7103:deductive
7098:axiomatic
7018:Hilbert's
7005:Euclidean
6986:canonical
6909:axiomatic
6841:Signature
6770:Predicate
6659:Extension
6581:Ackermann
6506:Operation
6385:Universal
6375:Recursive
6350:Singleton
6345:Inhabited
6330:Countable
6320:Types of
6304:power set
6274:partition
6191:Predicate
6137:Predicate
6052:Syllogism
6042:Soundness
6015:Inference
6005:Tautology
5907:paradoxes
5486:121598654
5478:0025-5831
5215:MathWorld
5007:→
4982:↦
4947:∅
4944:⋂
4918:∈
4906:∈
4900:∀
4893:⟺
4883:⋂
4880:∈
4826:α
4823:→
4808:∃
4793:∈
4790:α
4784:⋂
4781:∩
4636:∩
4614:∪
4533:∪
4268:ℵ
4264:×
4226:ℵ
4154:ℵ
4133:ℵ
4128:×
4118:ℵ
4097:ℵ
4080:ℵ
4055:ℵ
4003:ℵ
3982:ℵ
3977:×
3945:ℵ
3821:ℵ
3719:ℶ
3658:power set
3605:hypercube
3506:ℵ
3491:ℵ
3434:ℵ
3419:ℵ
3380:ℶ
3365:ℵ
3315:ℵ
3246:ℵ
3160:| > |
3127:ℵ
3095:| < |
3033:ℵ
3004:ℵ
2970:ℵ
2952:ℵ
2916:ℵ
2874:ℵ
2797:ℵ
2763:α
2759:ℵ
2730:α
2726:ℵ
2705:α
2694:For each
2676:…
2664:ℵ
2651:ℵ
2638:ℵ
2579:of a set
2271:power set
2048:≤
2002:≤
1902:≤
1856:≤
1564:−
912:injective
808:bijection
668:power set
613:bijection
523:#
494:
404:, with a
8092:Problems
7996:Theories
7972:Superset
7948:Infinite
7777:Concepts
7657:Infinity
7574:Overview
7492:Logicism
7485:timeline
7461:Concrete
7320:Validity
7290:T-schema
7283:Kripke's
7278:Tarski's
7273:semantic
7263:Strength
7212:submodel
7207:spectrum
7175:function
7023:Tarski's
7012:Elements
6999:geometry
6955:Robinson
6876:variable
6861:function
6834:spectrum
6824:Sentence
6780:variable
6723:Language
6676:Relation
6637:Automata
6627:Alphabet
6611:language
6465:-jection
6443:codomain
6429:Function
6390:Universe
6360:Infinite
6264:Relation
6047:Validity
6037:Argument
5935:theorem,
5769:geometry
5759:Such as
5706:: 81â125
5672:(2005),
5656:16591132
5589:16578557
5501:(2002),
5450:(eds.),
5434:(1915),
5392:: 75â78.
5301:Archived
5151:Counting
5120:See also
4425:| and |
3751:Beth two
3675:the set
3582:interval
3400:Beth one
3152:Any set
3106:Any set
3083:Any set
860:function
725:, hence
566:Elements
227:elements
41:The set
8030:General
8025:Zermelo
7931:subbase
7913: (
7852:Forcing
7830:Element
7802: (
7780:Methods
7667:Pairing
7434:Related
7231:Diagram
7129: (
7108:Hilbert
7093:Systems
7088:Theorem
6966:of the
6911:systems
6691:Formula
6686:Grammar
6602: (
6546:General
6259:Forcing
6244:Element
6164:Monadic
5939:paradox
5880:Theorem
5816:General
5616:Bibcode
5549:Bibcode
5347:Nov 15,
4514:, then
3644:theorem
3543:segment
3192:> |
3091:, or |
3071:If the
3063:below.
2990:, i.e.
2855:, that
2696:ordinal
2571:on the
2534:of all
2532:
2510:
2506:
2484:
2480:
2445:
2441:
2419:
2415:
2380:
2376:
2354:
2309:
2274:
2265:of all
2263:
2241:
2234:
2214:
2210:
2190:
2186:
2166:
2162:
2142:
2119:
2099:
2095:
2075:
1977:). The
1927:, then
1830:
1810:
1806:
1786:
1782:
1762:
1758:
1738:
1715:
1695:
1643:
1623:
1619:
1599:
1536:
1516:
1512:
1490:
1486:
1466:
1424:
1404:
1400:
1378:
1374:
1354:
1350:
1330:
1326:
1306:
1302:
1282:
1278:
1258:
1252:, then
1250:
1230:
1226:
1206:
1198:
1178:
1174:
1154:
1147:
1127:
1123:
1101:
908:
888:
884:
864:
856:
836:
832:
812:
542:History
302:
282:
278:
258:
61:of all
7921:Filter
7911:Finite
7847:Family
7790:Almost
7628:global
7613:Choice
7600:Axioms
7197:finite
6960:Skolem
6913:
6888:Theory
6856:Symbol
6846:String
6829:atomic
6706:ground
6701:closed
6696:atomic
6652:ground
6615:syntax
6511:binary
6438:domain
6355:Finite
6120:finite
5978:Logics
5937:
5885:Theory
5761:length
5740:
5732:
5680:
5654:
5647:300611
5644:
5636:
5587:
5580:221287
5577:
5569:
5515:
5484:
5476:
5261:Quanta
5188:
4589:unions
4437:| = |
4429:| †|
4421:| †|
4406:| and
4402:| = |
4390:| †|
4351:| = |
4343:, and
4331:} and
3672:) or 2
3114:| = |
2137:|
1733:|
1593:(see:
802:|
8013:Naive
7943:Fuzzy
7906:Empty
7889:types
7840:tuple
7810:Class
7804:large
7765:Union
7682:Union
7187:Model
6935:Peano
6792:Proof
6632:Arity
6561:Naive
6448:image
6380:Fuzzy
6340:Empty
6289:union
6234:Class
5875:Model
5865:Lemma
5823:Axiom
5638:72252
5634:JSTOR
5571:71858
5567:JSTOR
5482:S2CID
5382:(PDF)
5341:(PDF)
5330:(PDF)
4703:Here
4591:and
4417:If |
4386:If |
4383:is 3.
3749:(see
3454:(see
3398:(see
2573:class
1808:into
1488:from
1376:from
1200:, if
926:, or
862:from
719:range
681:from
515:, or
7926:base
7310:Type
7113:list
6917:list
6894:list
6883:Term
6817:rank
6711:open
6605:list
6417:Maps
6322:sets
6181:Free
6151:list
5901:list
5828:list
5765:area
5763:and
5738:ISBN
5730:ISBN
5678:ISBN
5652:PMID
5585:PMID
5513:ISBN
5474:ISSN
5349:2019
5186:ISBN
4510:are
4506:and
4379:and
4371:and
3854:and
3757:The
3728:>
3642:and
3549:and
3466:The
3430:>
3168:| =
3149:set.
3118:| =
2939:The
2870:>
2673:<
2660:<
2647:<
2097:and
1881:and
1621:nor
1328:and
1254:each
1202:some
1176:and
914:and
762:blue
585:and
577:and
491:card
280:and
251:size
241:and
181:and
135:sets
7887:Set
6997:of
6979:of
6927:of
6459:Sur
6433:Map
6240:Ur-
6222:Set
5767:in
5642:PMC
5624:doi
5575:PMC
5557:doi
5466:doi
5079:Ord
5012:Ord
4798:Ord
4776:Ord
4733:Ord
4695:or
4693:NBG
4502:If
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932:.
886:to
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740:red
721:of
697:= {
685:to
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551:000
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6463:Bi
6455:In
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3792:,
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2060:|
2056:A
2052:|
2044:|
2040:B
2036:|
2014:|
2010:B
2006:|
1998:|
1994:A
1990:|
1960:|
1956:B
1952:|
1948:=
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1936:|
1914:|
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1906:|
1898:|
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201:,
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166:3
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