6339:
3466:
6409:
1332:, which is also the ordinal number 1, and this may be confusing. A possible compromise (to take advantage of the alignment in finite arithmetic while avoiding reliance on the axiom of choice and confusion in infinite arithmetic) is to apply von Neumann assignment to the cardinal numbers of finite sets (those which can be well ordered and are not equipotent to proper subsets) and to use Scott's trick for the cardinal numbers of other sets.
4581:
27:
59:
979:. Supposing there is an innkeeper at a hotel with an infinite number of rooms. The hotel is full, and then a new guest arrives. It is possible to fit the extra guest in by asking the guest who was in room 1 to move to room 2, the guest in room 2 to move to room 3, and so on, leaving room 1 vacant. We can explicitly write a segment of this mapping:
1161:
Von
Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers. However,
809:
The intuition behind the formal definition of cardinal is the construction of a notion of the relative size or "bigness" of a set, without reference to the kind of members which it has. For finite sets this is easy; one simply counts the number of elements a set has. In order to compare the sizes
794:
More formally, a non-zero number can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. For finite sets and sequences it is easy to see that these two notions coincide, since for every number describing a position in a sequence we can
1017:
When considering these large objects, one might also want to see if the notion of counting order coincides with that of cardinal defined above for these infinite sets. It happens that it does not; by considering the above example we can see that if some object "one greater than infinity" exists,
1009:
With this assignment, we can see that the set {1,2,3,...} has the same cardinality as the set {2,3,4,...}, since a bijection between the first and the second has been shown. This motivates the definition of an infinite set being any set that has a proper subset of the same cardinality (i.e., a
1627:
operations on cardinal numbers that generalize the ordinary operations for natural numbers. It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.
1985:
293:
147:(bijection) between the elements of the two sets. In the case of finite sets, this agrees with the intuitive notion of number of elements. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to
1037:) are concerned with discovering whether there is some cardinal between some pair of other infinite cardinals. In more recent times, mathematicians have been describing the properties of larger and larger cardinals.
3348:. Both the continuum hypothesis and the generalized continuum hypothesis have been proved to be independent of the usual axioms of set theory, the Zermelo–Fraenkel axioms together with the axiom of choice (
1271:
697:
3452:
795:
construct a set that has exactly the right size. For example, 3 describes the position of 'c' in the sequence <'a','b','c','d',...>, and we can construct the set {a,b,c}, which has 3 elements.
2447:
2050:
1206:
1885:
2581:
1642:
If the axiom of choice holds, then every cardinal κ has a successor, denoted κ, where κ > κ and there are no cardinals between κ and its successor. (Without the axiom of choice, using
3273:
1022:, based on the ideas of counting and considering each number in turn, and we discover that the notions of cardinality and ordinality are divergent once we move out of the finite numbers.
1680:
172:
2185:
3696:
3092:
3051:
2978:
1304:
1806:
1613:
3191:
3215:
728:
596:
4078:
1330:
555:
4393:
4316:
4277:
4239:
4211:
4183:
4155:
4043:
4010:
3982:
3954:
3407:
3346:
3154:
1583:
1556:
1501:
755:
626:
425:
757:. This hypothesis is independent of the standard axioms of mathematical set theory, that is, it can neither be proved nor disproved from them. This was shown in 1963 by
3380:
3319:
3299:
2998:
1529:
126:
2664:
2632:
466:
may be encoded as a finite sequence of integers, which are the coefficients in the polynomial equation of which it is a solution, i.e. the ordered n-tuple (
802:, it is essential to distinguish between the two, since the two notions are in fact different for infinite sets. Considering the position aspect leads to
4718:
1018:
then it must have the same cardinality as the infinite set we started out with. It is possible to use a different formal notion for number, called
1476:
Assuming the axiom of choice, it can be proved that the
Dedekind notions correspond to the standard ones. It can also be proved that the cardinal
601:
Cantor also developed a large portion of the general theory of cardinal numbers; he proved that there is a smallest transfinite cardinal number (
6873:
5393:
1040:
Since cardinality is such a common concept in mathematics, a variety of names are in use. Sameness of cardinality is sometimes referred to as
558:", Cantor proved that there exist higher-order cardinal numbers, by showing that the set of real numbers has cardinality greater than that of
5476:
4617:
3664:
1356:
909:
159:
of an infinite set to have the same cardinality as the original set—something that cannot happen with proper subsets of finite sets.
1211:
634:
3904:
874:. The element d has no element mapping to it, but this is permitted as we only require an injective mapping, and not necessarily a
5790:
3412:
1092:. If the axiom of choice is not assumed, then a different approach is needed. The oldest definition of the cardinality of a set
976:
3732:
Robert A. McCoy and Ibula Ntantu, Topological
Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315,
1718:. If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace
372:, in 1874–1884. Cardinality can be used to compare an aspect of finite sets. For example, the sets {1,2,3} and {4,5,6} are not
19:
This article is about the mathematical concept. For number words indicating quantity ("three" apples, "four" birds, etc.), see
5948:
3848:
2402:
458:
is also denumerable, since every rational can be represented by a pair of integers. He later proved that the set of all real
151:
shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of
4736:
6562:
6375:
5803:
5126:
4093:
2004:
1682:) For finite cardinals, the successor is simply κ + 1. For infinite cardinals, the successor cardinal differs from the
4088:
1165:
6890:
5808:
5798:
5535:
5388:
4741:
1980:{\displaystyle (\kappa \leq \mu )\rightarrow ((\kappa +\nu \leq \mu +\nu ){\mbox{ and }}(\nu +\kappa \leq \nu +\mu )).}
4732:
2896:
1531:) of the set of natural numbers is the smallest infinite cardinal (i.e., any infinite set has a subset of cardinality
5944:
3820:
3632:
3505:
5286:
972:. It is however possible to discuss the relative cardinality of sets without explicitly assigning names to objects.
6041:
5785:
4610:
3784:
3278:
2515:
4048:
3228:
288:{\displaystyle 0,1,2,3,\ldots ,n,\ldots ;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\ }
6868:
6462:
5346:
5039:
4466:
3349:
1089:
961:
6748:
4780:
4544:
6302:
6004:
5767:
5762:
5587:
5008:
4692:
1649:
2122:
6642:
6521:
6297:
6080:
5997:
5710:
5641:
5518:
4760:
4427:
3875:
3757:
D. A. Vladimirov, Boolean
Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.
3495:
3064:
3023:
2794:
1030:
567:
3109:≤ 2. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of
6885:
6222:
6048:
5734:
5368:
4967:
3897:
3218:
571:
432:
314:
is true, this transfinite sequence includes every cardinal number. If the axiom of choice is not true (see
6878:
6516:
6479:
6100:
6095:
5705:
5444:
5373:
4702:
4603:
4053:
3870:
3515:
2950:
1276:
1135:
1740:
1588:
6029:
5619:
5013:
4981:
4672:
758:
349:
6533:
3159:
384:(i.e., a one-to-one correspondence) between the two sets, such as the correspondence {1→4, 2→5, 3→6}.
7023:
6567:
6452:
6440:
6435:
6319:
6268:
6165:
5663:
5624:
5101:
4746:
4461:
4417:
3865:
3792:
3196:
709:
577:
144:
4775:
4059:
1309:
6368:
6160:
6090:
5629:
5481:
5464:
5187:
4667:
4584:
4456:
2388:
Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either
4376:
4299:
4260:
4222:
4194:
4166:
4138:
4026:
3993:
3965:
3937:
964:; for this definition to make sense, it must be proved that every set has the same cardinality as
54:
demonstrates that the sets have the same cardinality, in this case equal to the cardinal number 4.
6987:
6905:
6780:
6732:
6546:
6469:
5992:
5969:
5930:
5816:
5757:
5403:
5323:
5167:
5111:
4724:
3890:
3811:
3748:, Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.
3385:
3324:
3132:
1561:
1534:
1479:
969:
733:
604:
450:, even though this might appear to run contrary to intuition. He also proved that the set of all
403:
387:
Cantor applied his concept of bijection to infinite sets (for example the set of natural numbers
6939:
6820:
6632:
6445:
6282:
6009:
5987:
5954:
5847:
5693:
5678:
5651:
5602:
5486:
5421:
5246:
5212:
5207:
5081:
4912:
4889:
2591:
3815:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
315:
6855:
6825:
6769:
6689:
6669:
6647:
6212:
6065:
5857:
5575:
5311:
5217:
5076:
5061:
4942:
4917:
4529:
4365:
3365:
3304:
3284:
2983:
1514:
1098:
1011:
341:
111:
3117:, though they lack some of the properties that logarithms of positive real numbers possess.
6929:
6919:
6753:
6684:
6637:
6577:
6457:
6185:
6147:
6024:
5828:
5668:
5592:
5570:
5398:
5356:
5255:
5222:
5086:
4874:
4785:
4282:
4015:
3644:
3126:
2637:
2605:
1111:
1034:
703:
163:
1646:, it can be shown that for any cardinal number κ, there is a minimal cardinal κ such that
1029:
is greater than that of the natural numbers just described. This can be visualized using
8:
6924:
6835:
6743:
6738:
6552:
6494:
6425:
6361:
6314:
6205:
6190:
6170:
6127:
6014:
5964:
5890:
5835:
5772:
5565:
5560:
5508:
5276:
5265:
4937:
4837:
4765:
4756:
4752:
4687:
4682:
4493:
4403:
4360:
4342:
4120:
3490:
3356:
1146:
and related systems. However, if we restrict from this class to those equinumerous with
155:
is greater than the cardinality of the set of natural numbers. It is also possible for a
1990:
Assuming the axiom of choice, addition of infinite cardinal numbers is easy. If either
6847:
6842:
6627:
6582:
6489:
6343:
6112:
6075:
6060:
6053:
6036:
5840:
5822:
5688:
5614:
5597:
5550:
5363:
5272:
5106:
5091:
5051:
5003:
4988:
4976:
4932:
4907:
4677:
4626:
4398:
4110:
3773:
3688:
3610:
3471:
3110:
2810:
1707:
1637:
852:= {a,b,c,d}, then using this notion of size, we would observe that there is a mapping:
822:
783:
beginning with 0. The counting numbers are exactly what can be defined formally as the
566:, but in an 1891 paper, he proved the same result using his ingenious and much simpler
136:
102:
31:
5296:
6704:
6541:
6504:
6474:
6398:
6338:
6278:
6085:
5895:
5885:
5777:
5658:
5493:
5469:
5250:
5234:
5139:
5116:
4993:
4962:
4927:
4822:
4657:
4556:
4519:
4483:
4422:
4408:
4103:
4083:
3844:
3816:
3780:
3692:
3680:
3628:
3614:
3465:
3114:
2113:
1683:
1415:
1151:
882:
825:
82:
3593:
Deiser, Oliver (May 2010). "On the
Development of the Notion of a Cardinal Number".
3569:
6992:
6982:
6967:
6962:
6830:
6484:
6292:
6287:
6180:
6137:
5959:
5920:
5915:
5900:
5726:
5683:
5580:
5378:
5328:
4902:
4864:
4574:
4503:
4478:
4412:
4321:
4287:
4128:
4098:
4020:
3923:
3836:
3672:
3652:
3602:
3530:
3360:
878:
mapping. The advantage of this notion is that it can be extended to infinite sets.
563:
459:
353:
337:
20:
628:, aleph-null), and that for every cardinal number there is a next-larger cardinal
6861:
6799:
6430:
6273:
6263:
6217:
6200:
6155:
6117:
6019:
5939:
5746:
5673:
5646:
5634:
5540:
5454:
5428:
5383:
5351:
5152:
4954:
4897:
4847:
4812:
4770:
4355:
3987:
3832:
3733:
3660:
2818:
1508:
1446:
1438:
1143:
1073:
803:
787:
cardinal numbers. Infinite cardinals only occur in higher-level mathematics and
780:
775:
455:
345:
311:
140:
106:
6997:
6794:
6775:
6679:
6664:
6621:
6557:
6499:
6258:
6237:
6195:
6175:
6070:
5925:
5523:
5513:
5503:
5498:
5432:
5306:
5182:
5071:
5066:
5044:
4645:
4498:
4488:
4473:
4292:
4160:
3931:
3525:
3520:
3510:
2671:
2326:
1643:
1081:
1019:
806:, while the size aspect is generalized by the cardinal numbers described here.
307:
299:
129:
94:
3840:
3745:
3606:
7017:
7002:
6804:
6718:
6713:
6232:
5910:
5417:
5202:
5192:
5162:
5147:
4817:
4561:
4534:
4443:
3684:
3656:
1703:
1419:
1052:. It is thus said that two sets with the same cardinality are, respectively,
779:, provided that 0 is included: 0, 1, 2, .... They may be identified with the
395:
333:
156:
6972:
1158:: it works because the collection of objects with any given rank is a set).
975:
The classic example used is that of the infinite hotel paradox, also called
762:
132:) marked with subscript indicating their rank among the infinite cardinals.
6952:
6947:
6765:
6694:
6652:
6511:
6408:
6132:
5979:
5880:
5872:
5752:
5700:
5609:
5545:
5528:
5459:
5318:
5177:
4879:
4662:
4524:
4326:
3480:
2824:
All the remaining propositions in this section assume the axiom of choice:
2814:
1470:
799:
451:
428:
400:, which all share the same cardinal number. This cardinal number is called
365:
329:
303:
148:
98:
1383:|. The axiom of choice is equivalent to the statement that given two sets
6977:
6612:
6242:
6122:
5301:
5291:
5238:
4922:
4842:
4827:
4707:
4652:
4350:
4132:
3806:
3648:
3485:
3222:
2261:
1858:
1827:
1306:. On the other hand, Scott's trick implies that the cardinal number 0 is
1139:
1026:
844:. This is most easily understood by an example; suppose we have the sets
439:
321:
152:
90:
70:
3627:
Enderton, Herbert. "Elements of Set Theory", Academic Press Inc., 1977.
2805:. This proves that no largest cardinal exists (because for any cardinal
773:
In informal use, a cardinal number is what is normally referred to as a
6957:
6728:
6384:
5172:
5027:
4998:
4804:
4331:
4188:
3676:
2920:
1624:
1504:
1442:
1155:
784:
391:= {0, 1, 2, 3, ...}). Thus, he called all sets having a bijection with
369:
325:
86:
62:
6760:
6723:
6674:
6572:
6324:
6227:
5280:
5197:
5157:
5121:
5057:
4869:
4859:
4832:
4595:
2786:
1344:
1118:
is non-empty, this collection is too large to be a set. In fact, for
897:
875:
381:
454:
of natural numbers is denumerable; this implies that the set of all
6309:
6107:
5555:
5260:
4854:
4439:
4370:
4216:
3500:
3053:. However, if such a cardinal exists, it is infinite and less than
1335:
Formally, the order among cardinal numbers is defined as follows: |
1122:≠ ∅ there is an injection from the universe into by mapping a set
810:
of larger sets, it is necessary to appeal to more refined notions.
570:. The new cardinal number of the set of real numbers is called the
81:
for short, is what is commonly called the number of elements of a
5905:
4697:
3959:
3882:
1273:
in cardinal arithmetic, although the von
Neumann assignment puts
364:
The notion of cardinality, as now understood, was formulated by
6785:
6607:
3913:
2602:. It is easy to check that the right-hand side depends only on
1014:); in this case {2,3,4,...} is a proper subset of {1,2,3,...}.
1266:{\displaystyle 2^{\aleph _{0}}>\aleph _{0}=\aleph _{0}^{2}}
692:{\displaystyle (\aleph _{1},\aleph _{2},\aleph _{3},\ldots ).}
328:. It is also a tool used in branches of mathematics including
6657:
6417:
6353:
5449:
4795:
4640:
3008:
Assuming the axiom of choice and, given an infinite cardinal
2935:
Assuming the axiom of choice and, given an infinite cardinal
2457:
Assuming the axiom of choice and, given an infinite cardinal
2060:
Assuming the axiom of choice and, given an infinite cardinal
1585:, and so on. For every ordinal α, there is a cardinal number
788:
556:
On a
Property of the Collection of All Real Algebraic Numbers
302:
including zero (finite cardinals), which are followed by the
3775:
Georg Cantor: His
Mathematics and Philosophy of the Infinite
318:), there are infinite cardinals that are not aleph numbers.
26:
3447:{\displaystyle \kappa <\operatorname {cf} (2^{\kappa })}
881:
We can then extend this to an equality-style relation. Two
1138:, is a proper class. The definition does work however in
1102:) is as the class of all sets that are equinumerous with
836:. An injective mapping identifies each element of the set
380:, namely three. This is established by the existence of a
3382:, the only restrictions ZFC places on the cardinality of
3129:(CH) states that there are no cardinals strictly between
1107:
58:
2809:, we can always find a larger cardinal 2). In fact, the
516:
is the unique root of the polynomial with coefficients (
3454:, and that the exponential function is non-decreasing.
2442:{\displaystyle \kappa \cdot \mu =\max\{\kappa ,\mu \}.}
105:
have been introduced, which are often denoted with the
1938:
1615:
and this list exhausts all infinite cardinal numbers.
4379:
4302:
4263:
4225:
4197:
4169:
4141:
4062:
4029:
3996:
3968:
3940:
3415:
3388:
3368:
3327:
3307:
3287:
3231:
3199:
3162:
3135:
3067:
3026:
2986:
2953:
2640:
2608:
2518:
2405:
2125:
2007:
1888:
1743:
1652:
1591:
1564:
1537:
1517:
1482:
1312:
1279:
1214:
1168:
1033:; classic questions of cardinality (for instance the
736:
712:
637:
607:
580:
406:
175:
114:
3671:, Bd. 76 (4), Leipzig: B. G. Teubner: 438–443,
3643:
3461:
3193:
The latter cardinal number is also often denoted by
2738:
Exponentiation is non-decreasing in both arguments:
2282:
Multiplication is non-decreasing in both arguments:
2045:{\displaystyle \kappa +\mu =\max\{\kappa ,\mu \}\,.}
3016:greater than 1, there may or may not be a cardinal
1507:or aleph-0, where aleph is the first letter in the
4387:
4310:
4271:
4233:
4205:
4177:
4149:
4072:
4037:
4004:
3976:
3948:
3829:Set theory : exploring independence and truth
3772:
3446:
3401:
3374:
3340:
3313:
3293:
3267:
3209:
3185:
3148:
3086:
3045:
2992:
2972:
2817:. (This proof fails in some set theories, notably
2658:
2626:
2575:
2441:
2179:
2044:
1979:
1800:
1674:
1607:
1577:
1550:
1523:
1495:
1324:
1298:
1265:
1201:{\displaystyle 2^{\omega }=\omega <\omega ^{2}}
1200:
749:
722:
691:
620:
590:
431:. He called the cardinal numbers of infinite sets
419:
287:
120:
7015:
2418:
2020:
462:is also denumerable. Each real algebraic number
1162:they differ for infinite numbers. For example,
3281:(GCH) states that for every infinite cardinal
1879:Addition is non-decreasing in both arguments:
1096:(implicit in Cantor and explicit in Frege and
6369:
4611:
3898:
3097:The logarithm of an infinite cardinal number
2576:{\displaystyle |X|^{|Y|}=\left|X^{Y}\right|,}
1025:It can be proved that the cardinality of the
948:itself is often defined as the least ordinal
3268:{\displaystyle 2^{\aleph _{0}}=\aleph _{1}.}
2879:and at least one of them is infinite, then:
2433:
2421:
2035:
2023:
1319:
1313:
1154:, then it will work (this is a trick due to
866:which is injective, and hence conclude that
3120:
6376:
6362:
4803:
4618:
4604:
4580:
3905:
3891:
3301:, there are no cardinals strictly between
2504:
2107:
1558:). The next larger cardinal is denoted by
730:of the set of real numbers is the same as
4381:
4304:
4265:
4227:
4199:
4171:
4143:
4031:
3998:
3970:
3942:
3826:
3779:, Princeton: Princeton University Press,
3720:
2038:
1675:{\displaystyle \kappa ^{+}\nleq \kappa .}
1084:α such that there is a bijection between
870:has cardinality greater than or equal to
3101:is defined as the least cardinal number
2836:are both finite and greater than 1, and
2396:is infinite and both are non-zero, then
2180:{\displaystyle |X|\cdot |Y|=|X\times Y|}
2112:The product of cardinals comes from the
1088:and α. This definition is known as the
706:is the proposition that the cardinality
57:
25:
3621:
3087:{\displaystyle \nu ^{\lambda }=\kappa }
3046:{\displaystyle \mu ^{\lambda }=\kappa }
324:is studied for its own sake as part of
7016:
4625:
3770:
3592:
3555:
1618:
1441:if such a subset does not exist. The
6357:
4599:
3886:
3803:. New York: Simon and Schuster, 1956.
3567:
2911:< cf(2) for any infinite cardinal
1631:
397:denumerable (countably infinite) sets
139:. Two sets have the same cardinality
3716:
3714:
3712:
3491:The paradox of the greatest cardinal
1689:
1067:
977:Hilbert's paradox of the Grand Hotel
912:, this is equivalent to there being
4094:Set-theoretically definable numbers
3202:
2973:{\displaystyle \nu ^{\mu }=\kappa }
1299:{\displaystyle \aleph _{0}=\omega }
715:
583:
498:together with a pair of rationals (
316:Axiom of choice § Independence
306:. The aleph numbers are indexed by
135:Cardinality is defined in terms of
13:
4065:
3912:
3799:, Part IX, Chapter 2, Volume 3 of
3665:"Über das Problem der Wohlordnung"
3253:
3238:
3169:
3137:
2670:κ = 1 (in particular 0 = 1), see
2489:. It will be unique (and equal to
2092:. It will be unique (and equal to
1801:{\displaystyle |X|+|Y|=|X\cup Y|.}
1608:{\displaystyle \aleph _{\alpha },}
1593:
1566:
1539:
1518:
1484:
1357:Cantor–Bernstein–Schroeder theorem
1316:
1281:
1249:
1236:
1221:
738:
668:
655:
642:
609:
562:. His proof used an argument with
408:
264:
245:
232:
219:
115:
14:
7035:
3858:
3709:
3061:greater than 1 will also satisfy
2218:One is a multiplicative identity
840:with a unique element of the set
6407:
6337:
4579:
3464:
3279:generalized continuum hypothesis
3186:{\displaystyle 2^{\aleph _{0}}.}
761:, complementing earlier work by
65:, the smallest infinite cardinal
3827:Schindler, Ralf-Dieter (2014).
3699:from the original on 2016-04-16
3595:History and Philosophy of Logic
3210:{\displaystyle {\mathfrak {c}}}
2233:Multiplication is associative (
1090:von Neumann cardinal assignment
968:ordinal; this statement is the
962:von Neumann cardinal assignment
723:{\displaystyle {\mathfrak {c}}}
591:{\displaystyle {\mathfrak {c}}}
97:. For dealing with the case of
16:Size of a possibly infinite set
6383:
4073:{\displaystyle {\mathcal {P}}}
3771:Dauben, Joseph Warren (1990),
3751:
3739:
3726:
3637:
3586:
3561:
3549:
3441:
3428:
2651:
2643:
2619:
2611:
2543:
2535:
2529:
2520:
2173:
2159:
2151:
2143:
2135:
2127:
2055:
1971:
1968:
1944:
1934:
1910:
1907:
1904:
1901:
1889:
1791:
1777:
1769:
1761:
1753:
1745:
1325:{\displaystyle \{\emptyset \}}
683:
638:
348:, the cardinal numbers form a
298:This sequence starts with the
1:
6298:History of mathematical logic
4428:Plane-based geometric algebra
3537:
3506:Inclusion–exclusion principle
3496:Cardinal number (linguistics)
3057:, and any finite cardinality
3003:
2943:greater than 0, the cardinal
2859:is finite and non-zero, then
1811:Zero is an additive identity
1343:| means that there exists an
768:
537:) that lies in the interval (
6223:Primitive recursive function
4388:{\displaystyle \mathbb {S} }
4311:{\displaystyle \mathbb {C} }
4272:{\displaystyle \mathbb {R} }
4234:{\displaystyle \mathbb {O} }
4206:{\displaystyle \mathbb {H} }
4178:{\displaystyle \mathbb {C} }
4150:{\displaystyle \mathbb {R} }
4038:{\displaystyle \mathbb {A} }
4005:{\displaystyle \mathbb {Q} }
3977:{\displaystyle \mathbb {Z} }
3949:{\displaystyle \mathbb {N} }
3219:cardinality of the continuum
2785:2 is the cardinality of the
1208:in ordinal arithmetic while
1110:or other related systems of
817:is at least as big as a set
572:cardinality of the continuum
446:has the same cardinality as
433:transfinite cardinal numbers
7:
3871:Encyclopedia of Mathematics
3516:Names of numbers in English
3457:
3402:{\displaystyle 2^{\kappa }}
3341:{\displaystyle 2^{\kappa }}
3149:{\displaystyle \aleph _{0}}
2509:Exponentiation is given by
2452:
1706:, addition is given by the
1578:{\displaystyle \aleph _{1}}
1551:{\displaystyle \aleph _{0}}
1496:{\displaystyle \aleph _{0}}
1136:axiom of limitation of size
1076:, the cardinality of a set
910:Schroeder–Bernstein theorem
798:However, when dealing with
750:{\displaystyle \aleph _{1}}
621:{\displaystyle \aleph _{0}}
574:and Cantor used the symbol
420:{\displaystyle \aleph _{0}}
10:
7040:
6874:von Neumann–Bernays–Gödel
5287:Schröder–Bernstein theorem
5014:Monadic predicate calculus
4673:Foundations of mathematics
3823:(Springer-Verlag edition).
2795:Cantor's diagonal argument
2465:, there exists a cardinal
2068:, there exists a cardinal
1635:
1453:is finite if and only if |
1449:, in the sense that a set
1031:Cantor's diagonal argument
944:|. The cardinal number of
928:an injective mapping from
916:an injective mapping from
892:are said to have the same
359:
89:, its cardinal number, or
18:
6938:
6901:
6813:
6703:
6675:One-to-one correspondence
6591:
6532:
6416:
6405:
6391:
6333:
6320:Philosophy of mathematics
6269:Automated theorem proving
6251:
6146:
5978:
5871:
5723:
5440:
5416:
5394:Von Neumann–Bernays–Gödel
5339:
5233:
5137:
5035:
5026:
4953:
4888:
4794:
4716:
4633:
4570:
4512:
4438:
4418:Algebra of physical space
4340:
4248:
4119:
3921:
3841:10.1007/978-3-319-06725-4
3607:10.1080/01445340903545904
145:one-to-one correspondence
103:infinite cardinal numbers
4474:Extended complex numbers
4457:Extended natural numbers
3801:The World of Mathematics
3121:The continuum hypothesis
2930:
2461:and a non-zero cardinal
1465:for some natural number
1106:. This does not work in
5970:Self-verifying theories
5791:Tarski's axiomatization
4742:Tarski's undefinability
4737:incompleteness theorems
3375:{\displaystyle \kappa }
3314:{\displaystyle \kappa }
3294:{\displaystyle \kappa }
2993:{\displaystyle \kappa }
2505:Cardinal exponentiation
2108:Cardinal multiplication
1524:{\displaystyle \aleph }
1445:cardinals are just the
1072:Formally, assuming the
970:well-ordering principle
438:Cantor proved that any
121:{\displaystyle \aleph }
6633:Constructible universe
6453:Constructibility (V=L)
6344:Mathematics portal
5955:Proof of impossibility
5603:propositional variable
4913:Propositional calculus
4530:Transcendental numbers
4389:
4366:Hyperbolic quaternions
4312:
4273:
4235:
4207:
4179:
4151:
4074:
4039:
4006:
3978:
3950:
3831:. Universitext. Cham:
3448:
3403:
3376:
3342:
3315:
3295:
3269:
3211:
3187:
3150:
3088:
3047:
3012:and a finite cardinal
2994:
2974:
2939:and a finite cardinal
2660:
2628:
2577:
2443:
2181:
2046:
1981:
1802:
1676:
1609:
1579:
1552:
1525:
1497:
1326:
1300:
1267:
1202:
960:|. This is called the
751:
724:
693:
622:
592:
421:
289:
122:
66:
55:
6856:Principia Mathematica
6690:Transfinite induction
6549:(i.e. set difference)
6213:Kolmogorov complexity
6166:Computably enumerable
6066:Model complete theory
5858:Principia Mathematica
4918:Propositional formula
4747:Banach–Tarski paradox
4462:Extended real numbers
4390:
4313:
4283:Split-complex numbers
4274:
4236:
4208:
4180:
4152:
4075:
4040:
4016:Constructible numbers
4007:
3979:
3951:
3574:mathworld.wolfram.com
3449:
3404:
3377:
3343:
3316:
3296:
3270:
3212:
3188:
3151:
3089:
3048:
2995:
2975:
2661:
2659:{\displaystyle {|Y|}}
2629:
2627:{\displaystyle {|X|}}
2578:
2444:
2182:
2047:
1982:
1803:
1677:
1636:Further information:
1610:
1580:
1553:
1526:
1498:
1327:
1301:
1268:
1203:
1099:Principia Mathematica
1012:Dedekind-infinite set
828:from the elements of
752:
725:
694:
623:
593:
422:
342:mathematical analysis
290:
166:of cardinal numbers:
123:
61:
29:
6930:Burali-Forti paradox
6685:Set-builder notation
6638:Continuum hypothesis
6578:Symmetric difference
6161:Church–Turing thesis
6148:Computability theory
5357:continuum hypothesis
4875:Square of opposition
4733:Gödel's completeness
4494:Supernatural numbers
4404:Multicomplex numbers
4377:
4361:Dual-complex numbers
4300:
4261:
4223:
4195:
4167:
4139:
4121:Composition algebras
4089:Arithmetical numbers
4060:
4027:
3994:
3966:
3938:
3645:Friedrich M. Hartogs
3413:
3386:
3366:
3325:
3305:
3285:
3229:
3197:
3160:
3133:
3127:continuum hypothesis
3065:
3024:
2984:
2951:
2638:
2606:
2516:
2403:
2123:
2005:
1886:
1741:
1650:
1589:
1562:
1535:
1515:
1480:
1469:. Any other set is
1310:
1277:
1212:
1166:
1150:that have the least
1112:axiomatic set theory
1035:continuum hypothesis
734:
710:
704:continuum hypothesis
635:
605:
578:
404:
368:, the originator of
173:
164:transfinite sequence
112:
6891:Tarski–Grothendieck
6315:Mathematical object
6206:P versus NP problem
6171:Computable function
5965:Reverse mathematics
5891:Logical consequence
5768:primitive recursive
5763:elementary function
5536:Free/bound variable
5389:Tarski–Grothendieck
4908:Logical connectives
4838:Logical equivalence
4688:Logical consequence
4399:Split-biquaternions
4111:Eisenstein integers
4049:Closed-form numbers
3568:Weisstein, Eric W.
3111:cardinal invariants
2797:shows that 2 > |
1619:Cardinal arithmetic
1262:
832:to the elements of
554:In his 1874 paper "
137:bijective functions
85:. In the case of a
6480:Limitation of size
6113:Transfer principle
6076:Semantics of logic
6061:Categorical theory
6037:Non-standard model
5551:Logical connective
4678:Information theory
4627:Mathematical logic
4557:Profinite integers
4520:Irrational numbers
4385:
4308:
4269:
4231:
4203:
4175:
4147:
4104:Gaussian rationals
4084:Computable numbers
4070:
4035:
4002:
3974:
3946:
3677:10.1007/bf01458215
3472:Mathematics portal
3444:
3399:
3372:
3338:
3311:
3291:
3265:
3207:
3183:
3146:
3115:topological spaces
3084:
3043:
2990:
2970:
2840:is infinite, then
2813:of cardinals is a
2656:
2624:
2590:is the set of all
2573:
2439:
2260:Multiplication is
2177:
2042:
1998:is infinite, then
1977:
1942:
1798:
1672:
1638:Successor cardinal
1632:Successor cardinal
1605:
1575:
1548:
1521:
1493:
1418:if there exists a
1322:
1296:
1263:
1248:
1198:
896:if there exists a
747:
720:
689:
618:
588:
417:
285:
118:
67:
56:
32:bijective function
7011:
7010:
6920:Russell's paradox
6869:Zermelo–Fraenkel
6770:Dedekind-infinite
6643:Diagonal argument
6542:Cartesian product
6399:Set (mathematics)
6351:
6350:
6283:Abstract category
6086:Theories of truth
5896:Rule of inference
5886:Natural deduction
5867:
5866:
5412:
5411:
5117:Cartesian product
5022:
5021:
4928:Many-valued logic
4903:Boolean functions
4786:Russell's paradox
4761:diagonal argument
4658:First-order logic
4593:
4592:
4504:Superreal numbers
4484:Levi-Civita field
4479:Hyperreal numbers
4423:Spacetime algebra
4409:Geometric algebra
4322:Bicomplex numbers
4288:Split-quaternions
4129:Division algebras
4099:Gaussian integers
4021:Algebraic numbers
3924:definable numbers
3866:"Cardinal number"
3850:978-3-319-06725-4
3570:"Cardinal Number"
3361:regular cardinals
2493:) if and only if
2114:Cartesian product
2096:) if and only if
1941:
1690:Cardinal addition
1684:successor ordinal
1416:Dedekind-infinite
1068:Formal definition
936:. We then write |
568:diagonal argument
460:algebraic numbers
284:
7031:
7024:Cardinal numbers
6993:Bertrand Russell
6983:John von Neumann
6968:Abraham Fraenkel
6963:Richard Dedekind
6925:Suslin's problem
6836:Cantor's theorem
6553:De Morgan's laws
6411:
6378:
6371:
6364:
6355:
6354:
6342:
6341:
6293:History of logic
6288:Category of sets
6181:Decision problem
5960:Ordinal analysis
5901:Sequent calculus
5799:Boolean algebras
5739:
5738:
5713:
5684:logical/constant
5438:
5437:
5424:
5347:Zermelo–Fraenkel
5098:Set operations:
5033:
5032:
4970:
4801:
4800:
4781:Löwenheim–Skolem
4668:Formal semantics
4620:
4613:
4606:
4597:
4596:
4583:
4582:
4550:
4540:
4452:Cardinal numbers
4413:Clifford algebra
4394:
4392:
4391:
4386:
4384:
4356:Dual quaternions
4317:
4315:
4314:
4309:
4307:
4278:
4276:
4275:
4270:
4268:
4240:
4238:
4237:
4232:
4230:
4212:
4210:
4209:
4204:
4202:
4184:
4182:
4181:
4176:
4174:
4156:
4154:
4153:
4148:
4146:
4079:
4077:
4076:
4071:
4069:
4068:
4044:
4042:
4041:
4036:
4034:
4011:
4009:
4008:
4003:
4001:
3988:Rational numbers
3983:
3981:
3980:
3975:
3973:
3955:
3953:
3952:
3947:
3945:
3907:
3900:
3893:
3884:
3883:
3879:
3854:
3812:Naive set theory
3789:
3778:
3758:
3755:
3749:
3743:
3737:
3730:
3724:
3718:
3707:
3706:
3705:
3704:
3653:Walther von Dyck
3641:
3635:
3625:
3619:
3618:
3590:
3584:
3583:
3581:
3580:
3565:
3559:
3553:
3531:Regular cardinal
3474:
3469:
3468:
3453:
3451:
3450:
3445:
3440:
3439:
3408:
3406:
3405:
3400:
3398:
3397:
3381:
3379:
3378:
3373:
3359:shows that, for
3357:Easton's theorem
3347:
3345:
3344:
3339:
3337:
3336:
3320:
3318:
3317:
3312:
3300:
3298:
3297:
3292:
3274:
3272:
3271:
3266:
3261:
3260:
3248:
3247:
3246:
3245:
3225:). In this case
3216:
3214:
3213:
3208:
3206:
3205:
3192:
3190:
3189:
3184:
3179:
3178:
3177:
3176:
3155:
3153:
3152:
3147:
3145:
3144:
3093:
3091:
3090:
3085:
3077:
3076:
3052:
3050:
3049:
3044:
3036:
3035:
2999:
2997:
2996:
2991:
2979:
2977:
2976:
2971:
2963:
2962:
2899:, one can prove
2855:is infinite and
2665:
2663:
2662:
2657:
2655:
2654:
2646:
2633:
2631:
2630:
2625:
2623:
2622:
2614:
2582:
2580:
2579:
2574:
2569:
2565:
2564:
2548:
2547:
2546:
2538:
2532:
2523:
2448:
2446:
2445:
2440:
2186:
2184:
2183:
2178:
2176:
2162:
2154:
2146:
2138:
2130:
2051:
2049:
2048:
2043:
1986:
1984:
1983:
1978:
1943:
1939:
1807:
1805:
1804:
1799:
1794:
1780:
1772:
1764:
1756:
1748:
1681:
1679:
1678:
1673:
1662:
1661:
1644:Hartogs' theorem
1614:
1612:
1611:
1606:
1601:
1600:
1584:
1582:
1581:
1576:
1574:
1573:
1557:
1555:
1554:
1549:
1547:
1546:
1530:
1528:
1527:
1522:
1502:
1500:
1499:
1494:
1492:
1491:
1359:states that if |
1331:
1329:
1328:
1323:
1305:
1303:
1302:
1297:
1289:
1288:
1272:
1270:
1269:
1264:
1261:
1256:
1244:
1243:
1231:
1230:
1229:
1228:
1207:
1205:
1204:
1199:
1197:
1196:
1178:
1177:
1134:, and so by the
756:
754:
753:
748:
746:
745:
729:
727:
726:
721:
719:
718:
698:
696:
695:
690:
676:
675:
663:
662:
650:
649:
627:
625:
624:
619:
617:
616:
597:
595:
594:
589:
587:
586:
564:nested intervals
456:rational numbers
440:unbounded subset
426:
424:
423:
418:
416:
415:
378:same cardinality
354:category of sets
338:abstract algebra
294:
292:
291:
286:
282:
272:
271:
253:
252:
240:
239:
227:
226:
127:
125:
124:
119:
21:Cardinal numeral
7039:
7038:
7034:
7033:
7032:
7030:
7029:
7028:
7014:
7013:
7012:
7007:
6934:
6913:
6897:
6862:New Foundations
6809:
6699:
6618:Cardinal number
6601:
6587:
6528:
6412:
6403:
6387:
6382:
6352:
6347:
6336:
6329:
6274:Category theory
6264:Algebraic logic
6247:
6218:Lambda calculus
6156:Church encoding
6142:
6118:Truth predicate
5974:
5940:Complete theory
5863:
5732:
5728:
5724:
5719:
5711:
5431: and
5427:
5422:
5408:
5384:New Foundations
5352:axiom of choice
5335:
5297:Gödel numbering
5237: and
5229:
5133:
5018:
4968:
4949:
4898:Boolean algebra
4884:
4848:Equiconsistency
4813:Classical logic
4790:
4771:Halting problem
4759: and
4735: and
4723: and
4722:
4717:Theorems (
4712:
4629:
4624:
4594:
4589:
4566:
4545:
4535:
4508:
4499:Surreal numbers
4489:Ordinal numbers
4434:
4380:
4378:
4375:
4374:
4336:
4303:
4301:
4298:
4297:
4295:
4293:Split-octonions
4264:
4262:
4259:
4258:
4250:
4244:
4226:
4224:
4221:
4220:
4198:
4196:
4193:
4192:
4170:
4168:
4165:
4164:
4161:Complex numbers
4142:
4140:
4137:
4136:
4115:
4064:
4063:
4061:
4058:
4057:
4030:
4028:
4025:
4024:
3997:
3995:
3992:
3991:
3969:
3967:
3964:
3963:
3941:
3939:
3936:
3935:
3932:Natural numbers
3917:
3911:
3864:
3861:
3851:
3833:Springer-Verlag
3787:
3762:
3761:
3756:
3752:
3744:
3740:
3734:Springer-Verlag
3731:
3727:
3719:
3710:
3702:
3700:
3661:Otto Blumenthal
3642:
3638:
3626:
3622:
3591:
3587:
3578:
3576:
3566:
3562:
3554:
3550:
3540:
3535:
3470:
3463:
3460:
3435:
3431:
3414:
3411:
3410:
3393:
3389:
3387:
3384:
3383:
3367:
3364:
3363:
3332:
3328:
3326:
3323:
3322:
3306:
3303:
3302:
3286:
3283:
3282:
3277:Similarly, the
3256:
3252:
3241:
3237:
3236:
3232:
3230:
3227:
3226:
3201:
3200:
3198:
3195:
3194:
3172:
3168:
3167:
3163:
3161:
3158:
3157:
3140:
3136:
3134:
3131:
3130:
3123:
3072:
3068:
3066:
3063:
3062:
3031:
3027:
3025:
3022:
3021:
3006:
2985:
2982:
2981:
2958:
2954:
2952:
2949:
2948:
2933:
2897:König's theorem
2819:New Foundations
2650:
2642:
2641:
2639:
2636:
2635:
2618:
2610:
2609:
2607:
2604:
2603:
2560:
2556:
2552:
2542:
2534:
2533:
2528:
2527:
2519:
2517:
2514:
2513:
2507:
2481:if and only if
2455:
2404:
2401:
2400:
2329:over addition:
2325:Multiplication
2172:
2158:
2150:
2142:
2134:
2126:
2124:
2121:
2120:
2110:
2084:if and only if
2064:and a cardinal
2058:
2006:
2003:
2002:
1940: and
1937:
1887:
1884:
1883:
1790:
1776:
1768:
1760:
1752:
1744:
1742:
1739:
1738:
1692:
1657:
1653:
1651:
1648:
1647:
1640:
1634:
1621:
1596:
1592:
1590:
1587:
1586:
1569:
1565:
1563:
1560:
1559:
1542:
1538:
1536:
1533:
1532:
1516:
1513:
1512:
1509:Hebrew alphabet
1487:
1483:
1481:
1478:
1477:
1447:natural numbers
1439:Dedekind-finite
1311:
1308:
1307:
1284:
1280:
1278:
1275:
1274:
1257:
1252:
1239:
1235:
1224:
1220:
1219:
1215:
1213:
1210:
1209:
1192:
1188:
1173:
1169:
1167:
1164:
1163:
1144:New Foundations
1074:axiom of choice
1070:
821:if there is an
804:ordinal numbers
781:natural numbers
776:counting number
771:
741:
737:
735:
732:
731:
714:
713:
711:
708:
707:
671:
667:
658:
654:
645:
641:
636:
633:
632:
612:
608:
606:
603:
602:
582:
581:
579:
576:
575:
550:
543:
535:
529:
522:
511:
504:
492:
485:
479:
472:
411:
407:
405:
402:
401:
376:, but have the
362:
346:category theory
312:axiom of choice
308:ordinal numbers
300:natural numbers
267:
263:
248:
244:
235:
231:
222:
218:
174:
171:
170:
141:if, and only if
113:
110:
109:
93:is therefore a
75:cardinal number
24:
17:
12:
11:
5:
7037:
7027:
7026:
7009:
7008:
7006:
7005:
7000:
6998:Thoralf Skolem
6995:
6990:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6944:
6942:
6936:
6935:
6933:
6932:
6927:
6922:
6916:
6914:
6912:
6911:
6908:
6902:
6899:
6898:
6896:
6895:
6894:
6893:
6888:
6883:
6882:
6881:
6866:
6865:
6864:
6852:
6851:
6850:
6839:
6838:
6833:
6828:
6823:
6817:
6815:
6811:
6810:
6808:
6807:
6802:
6797:
6792:
6783:
6778:
6773:
6763:
6758:
6757:
6756:
6751:
6746:
6736:
6726:
6721:
6716:
6710:
6708:
6701:
6700:
6698:
6697:
6692:
6687:
6682:
6680:Ordinal number
6677:
6672:
6667:
6662:
6661:
6660:
6655:
6645:
6640:
6635:
6630:
6625:
6615:
6610:
6604:
6602:
6600:
6599:
6596:
6592:
6589:
6588:
6586:
6585:
6580:
6575:
6570:
6565:
6560:
6558:Disjoint union
6555:
6550:
6544:
6538:
6536:
6530:
6529:
6527:
6526:
6525:
6524:
6519:
6508:
6507:
6505:Martin's axiom
6502:
6497:
6492:
6487:
6482:
6477:
6472:
6470:Extensionality
6467:
6466:
6465:
6455:
6450:
6449:
6448:
6443:
6438:
6428:
6422:
6420:
6414:
6413:
6406:
6404:
6402:
6401:
6395:
6393:
6389:
6388:
6381:
6380:
6373:
6366:
6358:
6349:
6348:
6334:
6331:
6330:
6328:
6327:
6322:
6317:
6312:
6307:
6306:
6305:
6295:
6290:
6285:
6276:
6271:
6266:
6261:
6259:Abstract logic
6255:
6253:
6249:
6248:
6246:
6245:
6240:
6238:Turing machine
6235:
6230:
6225:
6220:
6215:
6210:
6209:
6208:
6203:
6198:
6193:
6188:
6178:
6176:Computable set
6173:
6168:
6163:
6158:
6152:
6150:
6144:
6143:
6141:
6140:
6135:
6130:
6125:
6120:
6115:
6110:
6105:
6104:
6103:
6098:
6093:
6083:
6078:
6073:
6071:Satisfiability
6068:
6063:
6058:
6057:
6056:
6046:
6045:
6044:
6034:
6033:
6032:
6027:
6022:
6017:
6012:
6002:
6001:
6000:
5995:
5988:Interpretation
5984:
5982:
5976:
5975:
5973:
5972:
5967:
5962:
5957:
5952:
5942:
5937:
5936:
5935:
5934:
5933:
5923:
5918:
5908:
5903:
5898:
5893:
5888:
5883:
5877:
5875:
5869:
5868:
5865:
5864:
5862:
5861:
5853:
5852:
5851:
5850:
5845:
5844:
5843:
5838:
5833:
5813:
5812:
5811:
5809:minimal axioms
5806:
5795:
5794:
5793:
5782:
5781:
5780:
5775:
5770:
5765:
5760:
5755:
5742:
5740:
5721:
5720:
5718:
5717:
5716:
5715:
5703:
5698:
5697:
5696:
5691:
5686:
5681:
5671:
5666:
5661:
5656:
5655:
5654:
5649:
5639:
5638:
5637:
5632:
5627:
5622:
5612:
5607:
5606:
5605:
5600:
5595:
5585:
5584:
5583:
5578:
5573:
5568:
5563:
5558:
5548:
5543:
5538:
5533:
5532:
5531:
5526:
5521:
5516:
5506:
5501:
5499:Formation rule
5496:
5491:
5490:
5489:
5484:
5474:
5473:
5472:
5462:
5457:
5452:
5447:
5441:
5435:
5418:Formal systems
5414:
5413:
5410:
5409:
5407:
5406:
5401:
5396:
5391:
5386:
5381:
5376:
5371:
5366:
5361:
5360:
5359:
5354:
5343:
5341:
5337:
5336:
5334:
5333:
5332:
5331:
5321:
5316:
5315:
5314:
5307:Large cardinal
5304:
5299:
5294:
5289:
5284:
5270:
5269:
5268:
5263:
5258:
5243:
5241:
5231:
5230:
5228:
5227:
5226:
5225:
5220:
5215:
5205:
5200:
5195:
5190:
5185:
5180:
5175:
5170:
5165:
5160:
5155:
5150:
5144:
5142:
5135:
5134:
5132:
5131:
5130:
5129:
5124:
5119:
5114:
5109:
5104:
5096:
5095:
5094:
5089:
5079:
5074:
5072:Extensionality
5069:
5067:Ordinal number
5064:
5054:
5049:
5048:
5047:
5036:
5030:
5024:
5023:
5020:
5019:
5017:
5016:
5011:
5006:
5001:
4996:
4991:
4986:
4985:
4984:
4974:
4973:
4972:
4959:
4957:
4951:
4950:
4948:
4947:
4946:
4945:
4940:
4935:
4925:
4920:
4915:
4910:
4905:
4900:
4894:
4892:
4886:
4885:
4883:
4882:
4877:
4872:
4867:
4862:
4857:
4852:
4851:
4850:
4840:
4835:
4830:
4825:
4820:
4815:
4809:
4807:
4798:
4792:
4791:
4789:
4788:
4783:
4778:
4773:
4768:
4763:
4751:Cantor's
4749:
4744:
4739:
4729:
4727:
4714:
4713:
4711:
4710:
4705:
4700:
4695:
4690:
4685:
4680:
4675:
4670:
4665:
4660:
4655:
4650:
4649:
4648:
4637:
4635:
4631:
4630:
4623:
4622:
4615:
4608:
4600:
4591:
4590:
4588:
4587:
4577:
4575:Classification
4571:
4568:
4567:
4565:
4564:
4562:Normal numbers
4559:
4554:
4532:
4527:
4522:
4516:
4514:
4510:
4509:
4507:
4506:
4501:
4496:
4491:
4486:
4481:
4476:
4471:
4470:
4469:
4459:
4454:
4448:
4446:
4444:infinitesimals
4436:
4435:
4433:
4432:
4431:
4430:
4425:
4420:
4406:
4401:
4396:
4383:
4368:
4363:
4358:
4353:
4347:
4345:
4338:
4337:
4335:
4334:
4329:
4324:
4319:
4306:
4290:
4285:
4280:
4267:
4254:
4252:
4246:
4245:
4243:
4242:
4229:
4214:
4201:
4186:
4173:
4158:
4145:
4125:
4123:
4117:
4116:
4114:
4113:
4108:
4107:
4106:
4096:
4091:
4086:
4081:
4067:
4051:
4046:
4033:
4018:
4013:
4000:
3985:
3972:
3957:
3944:
3928:
3926:
3919:
3918:
3910:
3909:
3902:
3895:
3887:
3881:
3880:
3860:
3859:External links
3857:
3856:
3855:
3849:
3824:
3804:
3790:
3785:
3760:
3759:
3750:
3738:
3725:
3721:Schindler 2014
3708:
3636:
3620:
3601:(2): 123–143.
3585:
3560:
3547:
3546:
3539:
3536:
3534:
3533:
3528:
3526:Ordinal number
3523:
3521:Nominal number
3518:
3513:
3511:Large cardinal
3508:
3503:
3498:
3493:
3488:
3483:
3477:
3476:
3475:
3459:
3456:
3443:
3438:
3434:
3430:
3427:
3424:
3421:
3418:
3396:
3392:
3371:
3335:
3331:
3310:
3290:
3264:
3259:
3255:
3251:
3244:
3240:
3235:
3204:
3182:
3175:
3171:
3166:
3143:
3139:
3122:
3119:
3083:
3080:
3075:
3071:
3042:
3039:
3034:
3030:
3005:
3002:
2989:
2969:
2966:
2961:
2957:
2932:
2929:
2893:
2892:
2869:
2868:
2849:
2801:| for any set
2783:
2782:
2763:
2736:
2735:
2716:
2709:
2695:
2685:
2682:
2675:
2672:empty function
2653:
2649:
2645:
2621:
2617:
2613:
2584:
2583:
2572:
2568:
2563:
2559:
2555:
2551:
2545:
2541:
2537:
2531:
2526:
2522:
2506:
2503:
2454:
2451:
2450:
2449:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2188:
2187:
2175:
2171:
2168:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2109:
2106:
2057:
2054:
2053:
2052:
2041:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
1988:
1987:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1936:
1933:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1809:
1808:
1797:
1793:
1789:
1786:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1691:
1688:
1671:
1668:
1665:
1660:
1656:
1633:
1630:
1623:We can define
1620:
1617:
1604:
1599:
1595:
1572:
1568:
1545:
1541:
1520:
1511:, represented
1490:
1486:
1347:function from
1321:
1318:
1315:
1295:
1292:
1287:
1283:
1260:
1255:
1251:
1247:
1242:
1238:
1234:
1227:
1223:
1218:
1195:
1191:
1187:
1184:
1181:
1176:
1172:
1082:ordinal number
1069:
1066:
1050:equinumerosity
1007:
1006:
1003:
993:
990:
987:
984:
864:
863:
860:
857:
848:= {1,2,3} and
770:
767:
744:
740:
717:
700:
699:
688:
685:
682:
679:
674:
670:
666:
661:
657:
653:
648:
644:
640:
615:
611:
585:
548:
541:
533:
527:
520:
509:
502:
490:
483:
477:
470:
414:
410:
361:
358:
296:
295:
281:
278:
275:
270:
266:
262:
259:
256:
251:
247:
243:
238:
234:
230:
225:
221:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
117:
95:natural number
15:
9:
6:
4:
3:
2:
7036:
7025:
7022:
7021:
7019:
7004:
7003:Ernst Zermelo
7001:
6999:
6996:
6994:
6991:
6989:
6988:Willard Quine
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6945:
6943:
6941:
6940:Set theorists
6937:
6931:
6928:
6926:
6923:
6921:
6918:
6917:
6915:
6909:
6907:
6904:
6903:
6900:
6892:
6889:
6887:
6886:Kripke–Platek
6884:
6880:
6877:
6876:
6875:
6872:
6871:
6870:
6867:
6863:
6860:
6859:
6858:
6857:
6853:
6849:
6846:
6845:
6844:
6841:
6840:
6837:
6834:
6832:
6829:
6827:
6824:
6822:
6819:
6818:
6816:
6812:
6806:
6803:
6801:
6798:
6796:
6793:
6791:
6789:
6784:
6782:
6779:
6777:
6774:
6771:
6767:
6764:
6762:
6759:
6755:
6752:
6750:
6747:
6745:
6742:
6741:
6740:
6737:
6734:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6711:
6709:
6706:
6702:
6696:
6693:
6691:
6688:
6686:
6683:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6659:
6656:
6654:
6651:
6650:
6649:
6646:
6644:
6641:
6639:
6636:
6634:
6631:
6629:
6626:
6623:
6619:
6616:
6614:
6611:
6609:
6606:
6605:
6603:
6597:
6594:
6593:
6590:
6584:
6581:
6579:
6576:
6574:
6571:
6569:
6566:
6564:
6561:
6559:
6556:
6554:
6551:
6548:
6545:
6543:
6540:
6539:
6537:
6535:
6531:
6523:
6522:specification
6520:
6518:
6515:
6514:
6513:
6510:
6509:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6486:
6483:
6481:
6478:
6476:
6473:
6471:
6468:
6464:
6461:
6460:
6459:
6456:
6454:
6451:
6447:
6444:
6442:
6439:
6437:
6434:
6433:
6432:
6429:
6427:
6424:
6423:
6421:
6419:
6415:
6410:
6400:
6397:
6396:
6394:
6390:
6386:
6379:
6374:
6372:
6367:
6365:
6360:
6359:
6356:
6346:
6345:
6340:
6332:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6304:
6301:
6300:
6299:
6296:
6294:
6291:
6289:
6286:
6284:
6280:
6277:
6275:
6272:
6270:
6267:
6265:
6262:
6260:
6257:
6256:
6254:
6250:
6244:
6241:
6239:
6236:
6234:
6233:Recursive set
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6207:
6204:
6202:
6199:
6197:
6194:
6192:
6189:
6187:
6184:
6183:
6182:
6179:
6177:
6174:
6172:
6169:
6167:
6164:
6162:
6159:
6157:
6154:
6153:
6151:
6149:
6145:
6139:
6136:
6134:
6131:
6129:
6126:
6124:
6121:
6119:
6116:
6114:
6111:
6109:
6106:
6102:
6099:
6097:
6094:
6092:
6089:
6088:
6087:
6084:
6082:
6079:
6077:
6074:
6072:
6069:
6067:
6064:
6062:
6059:
6055:
6052:
6051:
6050:
6047:
6043:
6042:of arithmetic
6040:
6039:
6038:
6035:
6031:
6028:
6026:
6023:
6021:
6018:
6016:
6013:
6011:
6008:
6007:
6006:
6003:
5999:
5996:
5994:
5991:
5990:
5989:
5986:
5985:
5983:
5981:
5977:
5971:
5968:
5966:
5963:
5961:
5958:
5956:
5953:
5950:
5949:from ZFC
5946:
5943:
5941:
5938:
5932:
5929:
5928:
5927:
5924:
5922:
5919:
5917:
5914:
5913:
5912:
5909:
5907:
5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5884:
5882:
5879:
5878:
5876:
5874:
5870:
5860:
5859:
5855:
5854:
5849:
5848:non-Euclidean
5846:
5842:
5839:
5837:
5834:
5832:
5831:
5827:
5826:
5824:
5821:
5820:
5818:
5814:
5810:
5807:
5805:
5802:
5801:
5800:
5796:
5792:
5789:
5788:
5787:
5783:
5779:
5776:
5774:
5771:
5769:
5766:
5764:
5761:
5759:
5756:
5754:
5751:
5750:
5748:
5744:
5743:
5741:
5736:
5730:
5725:Example
5722:
5714:
5709:
5708:
5707:
5704:
5702:
5699:
5695:
5692:
5690:
5687:
5685:
5682:
5680:
5677:
5676:
5675:
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5653:
5650:
5648:
5645:
5644:
5643:
5640:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5618:
5617:
5616:
5613:
5611:
5608:
5604:
5601:
5599:
5596:
5594:
5591:
5590:
5589:
5586:
5582:
5579:
5577:
5574:
5572:
5569:
5567:
5564:
5562:
5559:
5557:
5554:
5553:
5552:
5549:
5547:
5544:
5542:
5539:
5537:
5534:
5530:
5527:
5525:
5522:
5520:
5517:
5515:
5512:
5511:
5510:
5507:
5505:
5502:
5500:
5497:
5495:
5492:
5488:
5485:
5483:
5482:by definition
5480:
5479:
5478:
5475:
5471:
5468:
5467:
5466:
5463:
5461:
5458:
5456:
5453:
5451:
5448:
5446:
5443:
5442:
5439:
5436:
5434:
5430:
5425:
5419:
5415:
5405:
5402:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5370:
5369:Kripke–Platek
5367:
5365:
5362:
5358:
5355:
5353:
5350:
5349:
5348:
5345:
5344:
5342:
5338:
5330:
5327:
5326:
5325:
5322:
5320:
5317:
5313:
5310:
5309:
5308:
5305:
5303:
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5282:
5278:
5274:
5271:
5267:
5264:
5262:
5259:
5257:
5254:
5253:
5252:
5248:
5245:
5244:
5242:
5240:
5236:
5232:
5224:
5221:
5219:
5216:
5214:
5213:constructible
5211:
5210:
5209:
5206:
5204:
5201:
5199:
5196:
5194:
5191:
5189:
5186:
5184:
5181:
5179:
5176:
5174:
5171:
5169:
5166:
5164:
5161:
5159:
5156:
5154:
5151:
5149:
5146:
5145:
5143:
5141:
5136:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5099:
5097:
5093:
5090:
5088:
5085:
5084:
5083:
5080:
5078:
5075:
5073:
5070:
5068:
5065:
5063:
5059:
5055:
5053:
5050:
5046:
5043:
5042:
5041:
5038:
5037:
5034:
5031:
5029:
5025:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4995:
4992:
4990:
4987:
4983:
4980:
4979:
4978:
4975:
4971:
4966:
4965:
4964:
4961:
4960:
4958:
4956:
4952:
4944:
4941:
4939:
4936:
4934:
4931:
4930:
4929:
4926:
4924:
4921:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4893:
4891:
4890:Propositional
4887:
4881:
4878:
4876:
4873:
4871:
4868:
4866:
4863:
4861:
4858:
4856:
4853:
4849:
4846:
4845:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4818:Logical truth
4816:
4814:
4811:
4810:
4808:
4806:
4802:
4799:
4797:
4793:
4787:
4784:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4758:
4754:
4750:
4748:
4745:
4743:
4740:
4738:
4734:
4731:
4730:
4728:
4726:
4720:
4715:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4654:
4651:
4647:
4644:
4643:
4642:
4639:
4638:
4636:
4632:
4628:
4621:
4616:
4614:
4609:
4607:
4602:
4601:
4598:
4586:
4578:
4576:
4573:
4572:
4569:
4563:
4560:
4558:
4555:
4552:
4548:
4542:
4538:
4533:
4531:
4528:
4526:
4525:Fuzzy numbers
4523:
4521:
4518:
4517:
4515:
4511:
4505:
4502:
4500:
4497:
4495:
4492:
4490:
4487:
4485:
4482:
4480:
4477:
4475:
4472:
4468:
4465:
4464:
4463:
4460:
4458:
4455:
4453:
4450:
4449:
4447:
4445:
4441:
4437:
4429:
4426:
4424:
4421:
4419:
4416:
4415:
4414:
4410:
4407:
4405:
4402:
4400:
4397:
4372:
4369:
4367:
4364:
4362:
4359:
4357:
4354:
4352:
4349:
4348:
4346:
4344:
4339:
4333:
4330:
4328:
4327:Biquaternions
4325:
4323:
4320:
4294:
4291:
4289:
4286:
4284:
4281:
4256:
4255:
4253:
4247:
4218:
4215:
4190:
4187:
4162:
4159:
4134:
4130:
4127:
4126:
4124:
4122:
4118:
4112:
4109:
4105:
4102:
4101:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4055:
4052:
4050:
4047:
4022:
4019:
4017:
4014:
3989:
3986:
3961:
3958:
3933:
3930:
3929:
3927:
3925:
3920:
3915:
3908:
3903:
3901:
3896:
3894:
3889:
3888:
3885:
3877:
3873:
3872:
3867:
3863:
3862:
3852:
3846:
3842:
3838:
3834:
3830:
3825:
3822:
3821:0-387-90092-6
3818:
3814:
3813:
3808:
3805:
3802:
3798:
3794:
3791:
3788:
3782:
3777:
3776:
3769:
3768:
3767:
3766:
3754:
3747:
3742:
3735:
3729:
3722:
3717:
3715:
3713:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3670:
3666:
3662:
3658:
3657:David Hilbert
3654:
3650:
3646:
3640:
3634:
3633:0-12-238440-7
3630:
3624:
3616:
3612:
3608:
3604:
3600:
3596:
3589:
3575:
3571:
3564:
3557:
3552:
3548:
3545:
3544:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3478:
3473:
3467:
3462:
3455:
3436:
3432:
3425:
3422:
3419:
3416:
3394:
3390:
3369:
3362:
3358:
3353:
3351:
3333:
3329:
3308:
3288:
3280:
3275:
3262:
3257:
3249:
3242:
3233:
3224:
3220:
3180:
3173:
3164:
3141:
3128:
3118:
3116:
3112:
3108:
3104:
3100:
3095:
3081:
3078:
3073:
3069:
3060:
3056:
3040:
3037:
3032:
3028:
3019:
3015:
3011:
3001:
2987:
2967:
2964:
2959:
2955:
2946:
2942:
2938:
2928:
2926:
2922:
2918:
2914:
2910:
2906:
2902:
2898:
2891:≤ Max (2, 2).
2890:
2886:
2882:
2881:
2880:
2878:
2874:
2866:
2862:
2858:
2854:
2850:
2847:
2843:
2839:
2835:
2831:
2827:
2826:
2825:
2822:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2780:
2776:
2772:
2768:
2764:
2761:
2757:
2753:
2749:
2745:
2741:
2740:
2739:
2733:
2729:
2725:
2721:
2717:
2714:
2710:
2707:
2703:
2699:
2696:
2693:
2689:
2686:
2683:
2681:, then 0 = 0.
2680:
2676:
2673:
2669:
2668:
2667:
2647:
2615:
2601:
2597:
2593:
2589:
2570:
2566:
2561:
2557:
2553:
2549:
2539:
2524:
2512:
2511:
2510:
2502:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2464:
2460:
2436:
2430:
2427:
2424:
2415:
2412:
2409:
2406:
2399:
2398:
2397:
2395:
2391:
2386:
2384:
2380:
2376:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2336:
2332:
2328:
2323:
2321:
2317:
2313:
2309:
2305:
2301:
2297:
2293:
2289:
2285:
2280:
2278:
2274:
2270:
2266:
2263:
2258:
2256:
2252:
2248:
2244:
2240:
2236:
2231:
2229:
2225:
2221:
2216:
2214:
2210:
2206:
2202:
2198:
2196:
2192:
2169:
2166:
2163:
2155:
2147:
2139:
2131:
2119:
2118:
2117:
2115:
2105:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2039:
2032:
2029:
2026:
2017:
2014:
2011:
2008:
2001:
2000:
1999:
1997:
1993:
1974:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1898:
1895:
1892:
1882:
1881:
1880:
1877:
1875:
1871:
1867:
1863:
1860:
1855:
1853:
1849:
1845:
1841:
1837:
1833:
1829:
1824:
1822:
1818:
1814:
1795:
1787:
1784:
1781:
1773:
1765:
1757:
1749:
1737:
1736:
1735:
1733:
1729:
1725:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1687:
1685:
1669:
1666:
1663:
1658:
1654:
1645:
1639:
1629:
1626:
1616:
1602:
1597:
1570:
1543:
1510:
1506:
1488:
1474:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1428:
1424:
1421:
1420:proper subset
1417:
1413:
1408:
1406:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1354:
1350:
1346:
1342:
1338:
1333:
1293:
1290:
1285:
1258:
1253:
1245:
1240:
1232:
1225:
1216:
1193:
1189:
1185:
1182:
1179:
1174:
1170:
1159:
1157:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1100:
1095:
1091:
1087:
1083:
1080:is the least
1079:
1075:
1065:
1063:
1059:
1055:
1051:
1047:
1043:
1038:
1036:
1032:
1028:
1023:
1021:
1015:
1013:
1004:
1001:
997:
994:
991:
988:
985:
982:
981:
980:
978:
973:
971:
967:
963:
959:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
903:
899:
895:
891:
887:
884:
879:
877:
873:
869:
861:
858:
855:
854:
853:
851:
847:
843:
839:
835:
831:
827:
824:
820:
816:
811:
807:
805:
801:
800:infinite sets
796:
792:
790:
786:
782:
778:
777:
766:
764:
760:
742:
705:
686:
680:
677:
672:
664:
659:
651:
646:
631:
630:
629:
613:
599:
573:
569:
565:
561:
557:
552:
547:
540:
536:
526:
519:
515:
508:
501:
497:
493:
486:
476:
469:
465:
461:
457:
453:
452:ordered pairs
449:
445:
441:
436:
434:
430:
412:
399:
398:
394:
390:
385:
383:
379:
375:
371:
367:
357:
355:
351:
347:
343:
339:
335:
334:combinatorics
331:
327:
323:
319:
317:
313:
309:
305:
304:aleph numbers
301:
279:
276:
273:
268:
260:
257:
254:
249:
241:
236:
228:
223:
215:
212:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
169:
168:
167:
165:
160:
158:
157:proper subset
154:
150:
146:
143:, there is a
142:
138:
133:
131:
108:
107:Hebrew letter
104:
100:
99:infinite sets
96:
92:
88:
84:
80:
76:
72:
64:
60:
53:
49:
45:
41:
37:
33:
28:
22:
6953:Georg Cantor
6948:Paul Bernays
6879:Morse–Kelley
6854:
6787:
6786:Subset
6733:hereditarily
6695:Venn diagram
6653:ordered pair
6617:
6568:Intersection
6512:Axiom schema
6335:
6133:Ultraproduct
5980:Model theory
5945:Independence
5881:Formal proof
5873:Proof theory
5856:
5829:
5786:real numbers
5758:second-order
5669:Substitution
5546:Metalanguage
5487:conservative
5460:Axiom schema
5404:Constructive
5374:Morse–Kelley
5340:Set theories
5319:Aleph number
5312:inaccessible
5218:Grothendieck
5102:intersection
4989:Higher-order
4977:Second-order
4923:Truth tables
4880:Venn diagram
4663:Formal proof
4546:
4536:
4451:
4351:Dual numbers
4343:hypercomplex
4133:Real numbers
3869:
3828:
3810:
3807:Halmos, Paul
3800:
3796:
3786:0691-02447-2
3774:
3765:Bibliography
3764:
3763:
3753:
3741:
3728:
3701:, retrieved
3668:
3639:
3623:
3598:
3594:
3588:
3577:. Retrieved
3573:
3563:
3551:
3542:
3541:
3481:Aleph number
3354:
3276:
3223:real numbers
3221:(the set of
3217:; it is the
3124:
3106:
3102:
3098:
3096:
3058:
3054:
3017:
3013:
3009:
3007:
2944:
2940:
2936:
2934:
2924:
2916:
2912:
2908:
2904:
2900:
2894:
2888:
2884:
2876:
2872:
2870:
2864:
2860:
2856:
2852:
2845:
2841:
2837:
2833:
2829:
2823:
2815:proper class
2806:
2802:
2798:
2790:
2784:
2778:
2774:
2770:
2766:
2759:
2755:
2751:
2747:
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2737:
2731:
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2324:
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2291:
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2283:
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2272:
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2259:
2254:
2250:
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2234:
2232:
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2208:
2204:
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2194:
2190:
2189:
2111:
2101:
2097:
2093:
2089:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2059:
1995:
1991:
1989:
1878:
1873:
1869:
1865:
1861:
1857:Addition is
1856:
1851:
1847:
1843:
1839:
1835:
1831:
1826:Addition is
1825:
1820:
1816:
1812:
1810:
1731:
1727:
1723:
1719:
1715:
1711:
1699:
1695:
1693:
1641:
1622:
1475:
1466:
1462:
1458:
1454:
1450:
1434:
1430:
1426:
1422:
1411:
1409:
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1400:
1396:
1392:
1388:
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1380:
1376:
1372:
1368:
1364:
1360:
1352:
1348:
1340:
1336:
1334:
1160:
1147:
1131:
1127:
1123:
1119:
1115:
1103:
1097:
1093:
1085:
1077:
1071:
1062:equinumerous
1061:
1057:
1053:
1049:
1046:equipollence
1045:
1041:
1039:
1027:real numbers
1024:
1016:
1008:
999:
995:
974:
965:
957:
953:
949:
945:
941:
937:
933:
929:
925:
921:
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901:
893:
889:
885:
880:
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867:
865:
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837:
833:
829:
818:
814:
812:
808:
797:
793:
774:
772:
701:
600:
559:
553:
545:
538:
531:
524:
517:
513:
512:) such that
506:
499:
495:
488:
481:
474:
467:
463:
447:
443:
437:
396:
392:
388:
386:
377:
373:
366:Georg Cantor
363:
330:model theory
320:
297:
161:
153:real numbers
149:Georg Cantor
134:
78:
74:
68:
51:
47:
43:
39:
35:
6978:Thomas Jech
6821:Alternative
6800:Uncountable
6754:Ultrafilter
6613:Cardinality
6517:replacement
6458:Determinacy
6243:Type theory
6191:undecidable
6123:Truth value
6010:equivalence
5689:non-logical
5302:Enumeration
5292:Isomorphism
5239:cardinality
5223:Von Neumann
5188:Ultrafilter
5153:Uncountable
5087:equivalence
5004:Quantifiers
4994:Fixed-point
4963:First-order
4843:Consistency
4828:Proposition
4805:Traditional
4776:Lindström's
4766:Compactness
4708:Type theory
4653:Cardinality
4513:Other types
4332:Bioctonions
4189:Quaternions
3746:Eduard Čech
3649:Felix Klein
3556:Dauben 1990
3486:Beth number
3020:satisfying
2947:satisfying
2915:, where cf(
2789:of the set
2327:distributes
2262:commutative
2056:Subtraction
1859:commutative
1828:associative
1140:type theory
1114:because if
1058:equipollent
1042:equipotence
894:cardinality
322:Cardinality
162:There is a
91:cardinality
71:mathematics
46:, from set
6973:Kurt Gödel
6958:Paul Cohen
6795:Transitive
6563:Identities
6547:Complement
6534:Operations
6495:Regularity
6463:projective
6426:Adjunction
6385:Set theory
6054:elementary
5747:arithmetic
5615:Quantifier
5593:functional
5465:Expression
5183:Transitive
5127:identities
5112:complement
5045:hereditary
5028:Set theory
4467:Projective
4440:Infinities
3793:Hahn, Hans
3703:2014-02-02
3669:Math. Ann.
3579:2020-09-06
3538:References
3105:such that
3004:Logarithms
2921:cofinality
2469:such that
2072:such that
1815:+ 0 = 0 +
1625:arithmetic
1505:aleph null
1391:, either |
1156:Dana Scott
1054:equipotent
908:. By the
769:Motivation
763:Kurt Gödel
759:Paul Cohen
429:aleph-null
370:set theory
326:set theory
87:finite set
63:Aleph-null
6906:Paradoxes
6826:Axiomatic
6805:Universal
6781:Singleton
6776:Recursive
6719:Countable
6714:Amorphous
6573:Power set
6490:Power set
6441:dependent
6436:countable
6325:Supertask
6228:Recursion
6186:decidable
6020:saturated
5998:of models
5921:deductive
5916:axiomatic
5836:Hilbert's
5823:Euclidean
5804:canonical
5727:axiomatic
5659:Signature
5588:Predicate
5477:Extension
5399:Ackermann
5324:Operation
5203:Universal
5193:Recursive
5168:Singleton
5163:Inhabited
5148:Countable
5138:Types of
5122:power set
5092:partition
5009:Predicate
4955:Predicate
4870:Syllogism
4860:Soundness
4833:Inference
4823:Tautology
4725:paradoxes
4551:solenoids
4371:Sedenions
4217:Octonions
3876:EMS Press
3693:121598654
3685:0025-5831
3615:171037224
3437:κ
3426:
3417:κ
3409:are that
3395:κ
3370:κ
3334:κ
3309:κ
3289:κ
3254:ℵ
3239:ℵ
3170:ℵ
3138:ℵ
3082:κ
3074:λ
3070:ν
3041:κ
3033:λ
3029:μ
2988:κ
2968:κ
2960:μ
2956:ν
2919:) is the
2787:power set
2592:functions
2431:μ
2425:κ
2413:μ
2410:⋅
2407:κ
2167:×
2140:⋅
2033:μ
2027:κ
2015:μ
2009:κ
1966:μ
1960:ν
1957:≤
1954:κ
1948:ν
1932:ν
1926:μ
1923:≤
1920:ν
1914:κ
1905:→
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1896:≤
1893:κ
1785:∪
1726:×{0} and
1667:κ
1664:≰
1655:κ
1598:α
1594:ℵ
1567:ℵ
1540:ℵ
1519:ℵ
1485:ℵ
1345:injective
1317:∅
1294:ω
1282:ℵ
1250:ℵ
1237:ℵ
1222:ℵ
1190:ω
1183:ω
1175:ω
898:bijection
876:bijective
823:injective
765:in 1940.
739:ℵ
681:…
669:ℵ
656:ℵ
643:ℵ
610:ℵ
409:ℵ
382:bijection
310:. If the
277:…
269:α
265:ℵ
258:…
246:ℵ
233:ℵ
220:ℵ
213:…
201:…
116:ℵ
7018:Category
6910:Problems
6814:Theories
6790:Superset
6766:Infinite
6595:Concepts
6475:Infinity
6392:Overview
6310:Logicism
6303:timeline
6279:Concrete
6138:Validity
6108:T-schema
6101:Kripke's
6096:Tarski's
6091:semantic
6081:Strength
6030:submodel
6025:spectrum
5993:function
5841:Tarski's
5830:Elements
5817:geometry
5773:Robinson
5694:variable
5679:function
5652:spectrum
5642:Sentence
5598:variable
5541:Language
5494:Relation
5455:Automata
5445:Alphabet
5429:language
5283:-jection
5261:codomain
5247:Function
5208:Universe
5178:Infinite
5082:Relation
4865:Validity
4855:Argument
4753:theorem,
3960:Integers
3922:Sets of
3797:Infinity
3723:, pg. 34
3697:archived
3663:(eds.),
3647:(1915),
3558:, pg. 54
3501:Counting
3458:See also
3355:Indeed,
2980:will be
2875:and 1 ≤
2453:Division
1704:disjoint
1471:infinite
1375:| then |
1020:ordinals
900:between
598:for it.
350:skeleton
79:cardinal
6848:General
6843:Zermelo
6749:subbase
6731: (
6670:Forcing
6648:Element
6620: (
6598:Methods
6485:Pairing
6252:Related
6049:Diagram
5947: (
5926:Hilbert
5911:Systems
5906:Theorem
5784:of the
5729:systems
5509:Formula
5504:Grammar
5420: (
5364:General
5077:Forcing
5062:Element
4982:Monadic
4757:paradox
4698:Theorem
4634:General
4541:numbers
4373: (
4219: (
4191: (
4163: (
4135: (
4056: (
4054:Periods
4023: (
3990: (
3962: (
3934: (
3916:systems
3878:, 2001
2887:, 2) ≤
2871:If 2 ≤
2677:If 1 ≤
2222:·1 = 1·
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2207:= 0 → (
2193:·0 = 0·
1734:×{1}).
1437:|, and
1367:| and |
1142:and in
826:mapping
530:, ...,
480:, ...,
360:History
352:of the
50:to set
6739:Filter
6729:Finite
6665:Family
6608:Almost
6446:global
6431:Choice
6418:Axioms
6015:finite
5778:Skolem
5731:
5706:Theory
5674:Symbol
5664:String
5647:atomic
5524:ground
5519:closed
5514:atomic
5470:ground
5433:syntax
5329:binary
5256:domain
5173:Finite
4938:finite
4796:Logics
4755:
4703:Theory
4341:Other
3914:Number
3847:
3819:
3783:
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2586:where
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1443:finite
1429:with |
1410:A set
1399:| or |
1355:. The
952:with |
813:A set
785:finite
283:
101:, the
6831:Naive
6761:Fuzzy
6724:Empty
6707:types
6658:tuple
6628:Class
6622:large
6583:Union
6500:Union
6005:Model
5753:Peano
5610:Proof
5450:Arity
5379:Naive
5266:image
5198:Fuzzy
5158:Empty
5107:union
5052:Class
4693:Model
4683:Lemma
4641:Axiom
4549:-adic
4539:-adic
4296:Over
4257:Over
4251:types
4249:Split
3689:S2CID
3611:S2CID
3543:Notes
2931:Roots
2903:<
2883:Max (
2811:class
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2762:) and
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2742:(1 ≤
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2197:= 0.
2100:<
1708:union
1457:| = |
1433:| = |
1403:| ≤ |
1395:| ≤ |
1379:| = |
1371:| ≤ |
1363:| ≤ |
1339:| ≤ |
1060:, or
1048:, or
989:3 → 4
986:2 → 3
983:1 → 2
956:| = |
940:| = |
862:3 → c
859:2 → b
856:1 → a
789:logic
374:equal
344:. In
130:aleph
77:, or
6744:base
6128:Type
5931:list
5735:list
5712:list
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5635:rank
5529:open
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5235:Maps
5140:sets
4999:Free
4969:list
4719:list
4646:list
4585:List
4442:and
3845:ISBN
3817:ISBN
3781:ISBN
3681:ISSN
3629:ISBN
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3321:and
3156:and
3125:The
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652:,
647:1
639:(
614:0
584:c
560:N
549:1
546:b
542:0
539:b
534:n
532:a
528:1
525:a
521:0
518:a
514:z
510:1
507:b
503:0
500:b
496:Z
491:i
489:a
484:n
482:a
478:1
475:a
471:0
468:a
464:z
448:N
444:N
413:0
393:N
389:N
280:.
274:,
261:,
255:,
250:2
242:,
237:1
229:,
224:0
216:;
210:,
207:n
204:,
198:,
195:3
192:,
189:2
186:,
183:1
180:,
177:0
128:(
48:X
44:Y
40:X
36:f
23:.
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