1288:, has infinitely many elements, and we cannot use any natural number to give its size. It might seem natural to divide the sets into different classes: put all the sets containing one element together; all the sets containing two elements together; ...; finally, put together all infinite sets and consider them as having the same size. This view works well for countably infinite sets and was the prevailing assumption before Georg Cantor's work. For example, there are infinitely many odd integers, infinitely many even integers, and also infinitely many integers overall. We can consider all these sets to have the same "size" because we can arrange things such that, for every integer, there is a distinct even integer:
3945:
3457:
1102:
10097:
7630:
3940:{\displaystyle {\begin{array}{c|c|c }{\text{Index}}&{\text{Tuple}}&{\text{Element}}\\\hline 0&(0,0)&{\textbf {a}}_{0}\\1&(0,1)&{\textbf {a}}_{1}\\2&(1,0)&{\textbf {b}}_{0}\\3&(0,2)&{\textbf {a}}_{2}\\4&(1,1)&{\textbf {b}}_{1}\\5&(2,0)&{\textbf {c}}_{0}\\6&(0,3)&{\textbf {a}}_{3}\\7&(1,2)&{\textbf {b}}_{2}\\8&(2,1)&{\textbf {c}}_{1}\\9&(3,0)&{\textbf {d}}_{0}\\10&(0,4)&{\textbf {a}}_{4}\\\vdots &&\end{array}}}
10167:
7618:
2172:
2230:
8339:
3397:
1000:, called roster form. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ("...") is used to represent many elements between the starting element and the end element in a set, if the writer believes that the reader can easily guess what ... represents; for example,
2002:
2433:
2167:{\displaystyle {\begin{matrix}0\leftrightarrow 1,&1\leftrightarrow 2,&2\leftrightarrow 3,&3\leftrightarrow 4,&4\leftrightarrow 5,&\ldots \\0\leftrightarrow 0,&1\leftrightarrow 2,&2\leftrightarrow 4,&3\leftrightarrow 6,&4\leftrightarrow 8,&\ldots \end{matrix}}}
7215:
Cantor's discovery of uncountable sets in 1874 was one of the most unexpected events in the history of mathematics. Before 1874, infinity was not even considered a legitimate mathematical subject by most people, so the need to distinguish between countable and uncountable infinities could not have
1481:
showed that not all infinite sets are countably infinite. For example, the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers). The set of real numbers has a greater cardinality than the set of natural numbers and is said to be uncountable.
1398:
1443:
that maps between two sets such that each element of each set corresponds to a single element in the other set. This mathematical notion of "size", cardinality, is that two sets are of the same size if and only if there is a bijection between them. We call all sets that are in one-to-one
2250:
4025:
This set is the union of the length-1 sequences, the length-2 sequences, the length-3 sequences, each of which is a countable set (finite
Cartesian product). So we are talking about a countable union of countable sets, which is countable by the previous theorem.
86:
from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
5851:
2927:-tuples made by the Cartesian product of finitely many different sets, each element in each tuple has the correspondence to a natural number, so every tuple can be written in natural numbers then the same logic is applied to prove the theorem.
1291:
3377:
With the foresight of knowing that there are uncountable sets, we can wonder whether or not this last result can be pushed any further. The answer is "yes" and "no", we can extend it, but we need to assume a new axiom to do so.
1647:
6494:
3359:. Since the set of natural number pairs is one-to-one mapped (actually one-to-one correspondence or bijection) to the set of natural numbers as shown above, the positive rational number set is proved as countable.
5026:
939:
is uncountable, thus showing that not all infinite sets are countable. In 1878, he used one-to-one correspondences to define and compare cardinalities. In 1883, he extended the natural numbers with his infinite
5632:
6139:
5375:
4803:
4005:
3452:
2865:, maps to a different natural number, a difference between two n-tuples by a single element is enough to ensure the n-tuples being mapped to different natural numbers. So, an injection from the set of
3103:
are integers), then for every positive fraction, we can come up with a distinct natural number corresponding to it. This representation also includes the natural numbers, since every natural number
4709:
2542:
5674:-th root of the polynomial, where the roots are sorted by absolute value from small to big, then sorted by argument from small to big. We can define an injection (i. e. one-to-one) function
5466:
5056:
4759:
2224:
5708:
5182:
1591:
4435:
6073:
6838:
6632:
2428:{\displaystyle 0\leftrightarrow (0,0),1\leftrightarrow (1,0),2\leftrightarrow (0,1),3\leftrightarrow (2,0),4\leftrightarrow (1,1),5\leftrightarrow (0,2),6\leftrightarrow (3,0),\ldots }
784:
4864:
1286:
2174:
Every countably infinite set is countable, and every infinite countable set is countably infinite. Furthermore, any subset of the natural numbers is countable, and more generally:
6224:
1968:
1936:
1048:
6726:
6560:
6267:
4972:
4938:
1997:
1876:
510:
4386:
4309:
3151:. So we can conclude that there are exactly as many positive rational numbers as there are positive integers. This is also true for all rational numbers, as can be seen below.
6005:
4044:
The elements of any finite subset can be ordered into a finite sequence. There are only countably many finite sequences, so also there are only countably many finite subsets.
6196:
1427:
7836:
6307:
5713:
5436:
8151:
8074:
8035:
7997:
7969:
7941:
7913:
7801:
7768:
7740:
7712:
6402:
6353:
5488:
5331:
5146:
4650:
3202:
3180:
3015:
2993:
2968:
2905:
1826:
702:
654:
441:
372:
324:
275:
5289:
1691:
1473:
1204:
921:
605:
246:
5224:
4480:
can be used to show that this minimal model is countable. The fact that the notion of "uncountability" makes sense even in this model, and in particular that this model
5108:
5082:
2767:
2691:
1733:
998:
6758:
6592:
4203:
4128:
1938:, the set of even integers. We can show these sets are countably infinite by exhibiting a bijection to the natural numbers. This can be achieved using the assignments
5652:
5520:
3101:
2647:
864:
6429:
6380:
5941:
5878:
3357:
2863:
2831:
2799:
2723:
2569:
838:
811:
3325:
3149:
3055:
1596:
578:
219:
6858:
6798:
6778:
6692:
6672:
6652:
6526:
6331:
6025:
5961:
5898:
5672:
5309:
5244:
4904:
4884:
4353:
4333:
4276:
4243:
4223:
4168:
4148:
4092:
4072:
3293:
3273:
3253:
3233:
3121:
3075:
2925:
2883:
2609:
2589:
2460:
1804:
1780:
1757:
1536:
1508:
1224:
1171:
1151:
1131:
1096:
1076:
884:
725:
676:
632:
537:
463:
419:
392:
346:
302:
178:
962:, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted
8476:
6078:
148:
may also be used, e.g. referring to countable and countably infinite respectively, definitions vary and care is needed respecting the difference with
129:
Although the terms "countable" and "countably infinite" as defined here are quite common, the terminology is not universal. An alternative style uses
1393:{\displaystyle \ldots \,-\!2\!\rightarrow \!-\!4,\,-\!1\!\rightarrow \!-\!2,\,0\!\rightarrow \!0,\,1\!\rightarrow \!2,\,2\!\rightarrow \!4\,\cdots }
3454:, we first assign each element of each set a tuple, then we assign each tuple an index using a variant of the triangular enumeration we saw above:
5502:
Per definition, every algebraic number (including complex numbers) is a root of a polynomial with integer coefficients. Given an algebraic number
6434:
10624:
9151:
10774:
4977:
9234:
8375:
1058:
to list all the elements, because the number of elements in the set is finite. If we number the elements of the set 1, 2, and so on, up to
5525:
3462:
7254:
5336:
4764:
932:
3960:
3407:
102:(the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be
7662:
5110:
is countable. This result generalizes to the
Cartesian product of any finite collection of countable sets and the proof follows by
9548:
4599:
7158:
Invitation To
Algebra: A Resource Compendium For Teachers, Advanced Undergraduate Students And Graduate Students In Mathematics
6870:
9706:
7597:
7506:
7455:
7387:
7351:
7166:
7115:
7085:
7058:
7026:
6999:
6909:
8494:
10313:
10133:
9561:
8884:
7851:
7846:
6874:
10641:
9566:
9556:
9293:
9146:
8499:
7525:
4655:
17:
8490:
2469:
9702:
7567:
7546:
7498:
7473:
7405:
7208:
5441:
5031:
4734:
2199:
9044:
5677:
5151:
9799:
9543:
8368:
1544:
7806:
4405:
10619:
9104:
8797:
8224:
6030:
149:
10499:
8538:
8302:
6803:
6597:
4477:
10060:
9762:
9525:
9520:
9345:
8766:
8450:
730:
10393:
10272:
10055:
9838:
9755:
9468:
9399:
9276:
8518:
8185:
4866:
is injective. It then follows that the
Cartesian product of any two countable sets is countable, because if
4808:
4446:
4711:, it makes no difference whether one considers 0 a natural number or not. In any case, this article follows
1229:
10636:
9980:
9806:
9492:
9126:
8725:
7655:
7584:. Texts and Readings in Mathematics. Vol. 37 (Third ed.). Singapore: Springer. pp. 181â210.
6201:
1941:
1885:
1003:
10629:
10267:
10230:
9858:
9853:
9463:
9202:
9131:
8460:
8361:
7811:
6697:
6531:
4943:
4909:
1973:
6229:
1831:
472:
9787:
9377:
8771:
8739:
8430:
7608:
5846:{\displaystyle f(\alpha )=2^{k-1}\cdot 3^{a_{0}}\cdot 5^{a_{1}}\cdot 7^{a_{2}}\cdots {p_{n+2}}^{a_{n}}}
4358:
4281:
47:
10284:
5966:
10779:
10318:
10210:
10198:
10193:
10077:
10026:
9923:
9421:
9382:
8859:
8504:
8219:
8175:
6310:
6144:
3950:
3389:
1403:
91:
71:
8533:
7817:
6284:
5380:
944:, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities.
10126:
9918:
9848:
9387:
9239:
9222:
8945:
8425:
8342:
8214:
8134:
8057:
8018:
7980:
7952:
7924:
7896:
7784:
7751:
7723:
7695:
6385:
6336:
5471:
5314:
5129:
4633:
3185:
3163:
2998:
2976:
2951:
2888:
1809:
685:
637:
424:
355:
307:
258:
10738:
10656:
10531:
10483:
10297:
10220:
9750:
9727:
9688:
9574:
9515:
9161:
9081:
8925:
8869:
8482:
7648:
5249:
2234:
1652:
1451:
1176:
899:
583:
224:
7577:
5187:
10690:
10571:
10383:
10203:
10040:
9767:
9745:
9712:
9605:
9451:
9436:
9409:
9360:
9244:
9179:
9004:
8970:
8965:
8839:
8670:
8647:
7415:
5111:
5087:
5061:
4473:
2728:
2652:
1700:
1440:
965:
679:
7341:
7075:
7048:
6731:
6565:
4176:
4101:
10606:
10576:
10520:
10440:
10420:
10398:
9970:
9823:
9615:
9333:
9069:
8975:
8834:
8819:
8700:
8675:
8287:
8123:
7246:
7198:
7105:
7016:
6989:
6899:
5637:
5505:
4515:
3080:
2614:
1101:
843:
7156:
10680:
10670:
10504:
10435:
10388:
10328:
10215:
9943:
9905:
9782:
9586:
9426:
9350:
9328:
9156:
9114:
9013:
8980:
8844:
8632:
8543:
8040:
7773:
6407:
6358:
5919:
5856:
3330:
2836:
2804:
2772:
2696:
2547:
1429:(see picture). What we have done here is arrange the integers and the even integers into a
816:
789:
8:
10675:
10586:
10494:
10489:
10303:
10245:
10183:
10119:
10072:
9963:
9948:
9928:
9885:
9772:
9722:
9648:
9593:
9530:
9323:
9318:
9266:
9034:
9023:
8695:
8595:
8523:
8514:
8510:
8445:
8440:
8251:
8161:
8118:
8100:
7878:
4580:. This is the key definition that determines whether a total order is also a well order.
4504:
4389:
4254:
3302:
3126:
3032:
553:
349:
194:
4250:
These follow from the definitions of countable set as injective / surjective functions.
10598:
10593:
10378:
10333:
10240:
10101:
9870:
9833:
9818:
9811:
9794:
9598:
9580:
9446:
9372:
9355:
9308:
9121:
9030:
8864:
8849:
8809:
8761:
8746:
8734:
8690:
8665:
8435:
8384:
8156:
7868:
7634:
7435:
6843:
6783:
6763:
6677:
6657:
6637:
6511:
6316:
6010:
5946:
5883:
5657:
5294:
5229:
4889:
4869:
4716:
4338:
4318:
4261:
4228:
4208:
4153:
4133:
4077:
4057:
3370:
3278:
3258:
3238:
3218:
3106:
3060:
2910:
2868:
2594:
2574:
2445:
2226:
is countably infinite, as can be seen by following a path like the one in the picture:
1789:
1765:
1742:
1521:
1493:
1209:
1156:
1136:
1116:
1081:
1061:
869:
710:
661:
617:
522:
448:
404:
377:
331:
287:
281:
163:
83:
9054:
4562:
The usual order of rational numbers (Cannot be explicitly written as an ordered list!)
3299:
can easily be one-to-one mapped to the set of natural number pairs (2-tuples) because
10784:
10455:
10292:
10255:
10225:
10156:
10096:
10036:
9843:
9653:
9643:
9535:
9416:
9251:
9227:
9008:
8992:
8897:
8874:
8751:
8720:
8685:
8580:
8415:
8314:
8277:
8241:
8180:
8166:
7861:
7841:
7629:
7622:
7593:
7563:
7542:
7521:
7502:
7494:
7469:
7451:
7439:
7401:
7383:
7347:
7204:
7162:
7111:
7081:
7054:
7022:
6995:
6905:
3295:
is one-to-one mapped to the set of natural numbers. For example, the set of positive
2938:
2242:
2193:
954:
59:
10743:
10733:
10718:
10713:
10581:
10235:
10050:
10045:
9938:
9895:
9717:
9678:
9673:
9658:
9484:
9441:
9338:
9136:
9086:
8660:
8622:
8332:
8261:
8236:
8170:
8079:
8045:
7886:
7856:
7778:
7681:
7585:
7486:
7427:
4511:
3209:
7448:
Labyrinth of
Thought: A History of Set Theory and Its Role in Mathematical Thought
10612:
10550:
10368:
10188:
10031:
10021:
9975:
9958:
9913:
9875:
9777:
9697:
9504:
9431:
9404:
9392:
9298:
9212:
9186:
9141:
9109:
8910:
8712:
8655:
8605:
8570:
8528:
8209:
8113:
7745:
7431:
4604:
3296:
3026:
2971:
1539:
1153:
is any integer that can be specified. (No matter how large the specified integer
253:
114:
40:
7589:
10748:
10545:
10526:
10430:
10415:
10372:
10308:
10250:
10016:
9995:
9953:
9933:
9828:
9683:
9281:
9271:
9261:
9256:
9190:
9064:
8940:
8829:
8824:
8802:
8403:
8256:
8246:
8231:
8050:
7918:
7689:
7371:
7194:
4538:
941:
75:
6496:, rather than an injection, there is no requirement that the sets be disjoint.
10768:
10753:
10555:
10464:
9990:
9668:
9175:
8960:
8950:
8920:
8575:
8319:
8292:
8201:
7376:
10723:
1642:{\displaystyle a\leftrightarrow 1,\ b\leftrightarrow 2,\ c\leftrightarrow 3}
10703:
10698:
10516:
10445:
10403:
10262:
10166:
9890:
9737:
9638:
9630:
9510:
9458:
9367:
9303:
9286:
9217:
9076:
8935:
8637:
8420:
8282:
8084:
7555:
6489:{\displaystyle G:\mathbf {N} \times \mathbf {N} \to \bigcup _{i\in I}A_{i}}
5901:
4589:
4392:. As an immediate consequence of this and the Basic Theorem above we have:
2189:
1478:
542:
110:
10728:
10363:
10000:
9880:
9059:
9049:
8996:
8680:
8600:
8585:
8465:
8410:
8108:
7890:
7513:
7482:
4527:
4469:
4453:
4438:
3235:
to be shown as countable is one-to-one mapped (injection) to another set
3022:
936:
892:
189:
118:
99:
55:
2591:
are natural numbers, by repeatedly mapping the first two elements of an
10708:
10479:
10142:
8930:
8785:
8756:
8562:
8089:
7946:
7534:
4534:
466:
249:
67:
36:
32:
5021:{\displaystyle f\times g:\mathbb {N} \times \mathbb {N} \to A\times B}
10511:
10474:
10425:
10323:
10082:
9985:
9038:
8955:
8915:
8879:
8815:
8627:
8617:
8590:
8353:
7382:, Calculus, vol. 2 (2nd ed.), New York: John Wiley + Sons,
4761:
is countable as a consequence of the definition because the function
4712:
4627:
4312:
3018:
2439:
1783:
1515:
1435:
611:
398:
10067:
9865:
9313:
9018:
8612:
8197:
8128:
7974:
5627:{\displaystyle a_{0}x^{0}+a_{1}x^{1}+a_{2}x^{2}+\cdots +a_{n}x^{n}}
4594:
4457:
4018:
1759:
is countable. Similarly we can show all finite sets are countable.
117:, that is, sets that are not countable; for example the set of the
7501:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
2229:
9663:
8455:
7717:
7640:
6134:{\displaystyle G:I\times \mathbf {N} \to \bigcup _{i\in I}A_{i},}
5370:{\displaystyle g:\mathbb {Z} \times \mathbb {N} \to \mathbb {Q} }
4798:{\displaystyle f:\mathbb {N} \times \mathbb {N} \to \mathbb {N} }
4472:) of ZFC set theory, then there is a minimal standard model (see
2946:
1879:
1226:
elements.) For example, the set of natural numbers, denotable by
1051:
4000:{\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}},\dots }
3447:{\displaystyle {\textbf {a}},{\textbf {b}},{\textbf {c}},\dots }
3396:
10536:
10358:
7671:
7343:
Introduction to Set Theory, Third
Edition, Revised and Expanded
4037:
4559:
The usual order of integers (..., â3, â2, â1, 0, 1, 2, 3, ...)
2833:
maps to 39. Since a different 2-tuple, that is a pair such as
10408:
10175:
10111:
9207:
8553:
8398:
4503:
was seen as paradoxical in the early days of set theory; see
2463:
896:
if it is not countable, i.e. its cardinality is greater than
7520:, Dover series in mathematics and physics, New York: Dover,
7378:
Multi-Variable
Calculus and Linear Algebra with Applications
5311:
is negative, is an injective function. The rational numbers
4547:
The integers in the order (0, 1, 2, 3, ...; â1, â2, â3, ...)
2007:
31:"Countable" redirects here. For the linguistic concept, see
4463:
3392:) The union of countably many countable sets is countable.
7104:
Katzourakis, Nikolaos; Varvaruca, Eugen (2 January 2018).
4544:
The usual order of natural numbers (0, 1, 2, 3, 4, 5, ...)
3017:. But looks can be deceiving. If a pair is treated as the
2237:
assigns one natural number to each pair of natural numbers
4568:
In both examples of well orders here, any subset has a
1054:
from 1 to 100. Even in this case, however, it is still
6232:
1078:, this gives us the usual definition of "sets of size
8137:
8060:
8021:
7983:
7955:
7927:
7899:
7820:
7787:
7754:
7726:
7698:
7606:
7200:
Roads to
Infinity: The Mathematics of Truth and Proof
7155:
Dlab, Vlastimil; Williams, Kenneth S. (9 June 2020).
6846:
6806:
6786:
6766:
6734:
6700:
6680:
6660:
6640:
6600:
6568:
6534:
6514:
6437:
6410:
6388:
6361:
6339:
6319:
6287:
6204:
6147:
6081:
6033:
6013:
5969:
5949:
5922:
5886:
5859:
5716:
5680:
5660:
5640:
5528:
5508:
5474:
5444:
5383:
5339:
5317:
5297:
5252:
5232:
5190:
5154:
5132:
5090:
5064:
5034:
4980:
4946:
4912:
4892:
4872:
4811:
4767:
4737:
4658:
4636:
4408:
4361:
4341:
4321:
4284:
4264:
4231:
4211:
4179:
4156:
4136:
4104:
4080:
4060:
3963:
3460:
3410:
3333:
3305:
3281:
3261:
3241:
3221:
3188:
3166:
3129:
3109:
3083:
3063:
3035:
3001:
2979:
2954:
2913:
2891:
2871:
2839:
2807:
2775:
2731:
2699:
2655:
2617:
2597:
2577:
2550:
2472:
2448:
2253:
2202:
2005:
1976:
1944:
1888:
1834:
1812:
1792:
1768:
1745:
1739:
vice versa, this defines a bijection, and shows that
1703:
1655:
1599:
1547:
1524:
1496:
1454:
1406:
1294:
1232:
1212:
1179:
1159:
1139:
1119:
1084:
1064:
1006:
968:
902:
872:
846:
819:
792:
733:
713:
688:
664:
640:
620:
586:
556:
525:
475:
451:
427:
407:
380:
358:
334:
310:
290:
261:
227:
197:
166:
7103:
6431:. Note that since we are considering the surjection
5634:
be a polynomial with integer coefficients such that
610:
There is an injective and surjective (and therefore
133:
to mean what is here called countably infinite, and
8145:
8068:
8029:
7991:
7963:
7935:
7907:
7830:
7795:
7762:
7734:
7706:
7375:
6852:
6832:
6792:
6772:
6752:
6720:
6686:
6666:
6646:
6626:
6586:
6554:
6520:
6488:
6423:
6396:
6382:from the non-empty collection of surjections from
6374:
6347:
6325:
6301:
6261:
6218:
6190:
6133:
6067:
6019:
5999:
5955:
5935:
5892:
5872:
5845:
5702:
5666:
5646:
5626:
5514:
5482:
5460:
5430:
5369:
5325:
5303:
5283:
5238:
5218:
5176:
5140:
5102:
5076:
5050:
5020:
4966:
4932:
4898:
4878:
4858:
4797:
4753:
4703:
4644:
4456:is uncountable, and so is the set of all infinite
4429:
4380:
4347:
4327:
4303:
4270:
4237:
4217:
4197:
4162:
4142:
4122:
4086:
4066:
3999:
3939:
3446:
3400:Enumeration for countable number of countable sets
3351:
3319:
3287:
3267:
3247:
3227:
3196:
3174:
3143:
3115:
3095:
3069:
3049:
3009:
2987:
2962:
2919:
2899:
2877:
2857:
2825:
2793:
2761:
2717:
2685:
2641:
2603:
2583:
2563:
2536:
2454:
2427:
2218:
2166:
1991:
1962:
1930:
1870:
1820:
1798:
1774:
1751:
1727:
1685:
1641:
1585:
1530:
1502:
1467:
1421:
1392:
1280:
1218:
1198:
1165:
1145:
1125:
1090:
1070:
1042:
992:
915:
878:
858:
832:
805:
778:
719:
696:
670:
648:
626:
599:
572:
531:
504:
457:
435:
413:
386:
366:
340:
318:
296:
269:
240:
213:
172:
4704:{\displaystyle \mathbb {N} ^{*}=\{1,2,3,\dots \}}
3215:Sometimes more than one mapping is useful: a set
3204:(the set of all rational numbers) are countable.
1382:
1378:
1367:
1363:
1352:
1348:
1337:
1333:
1329:
1325:
1314:
1310:
1306:
1302:
10766:
4468:If there is a set that is a standard model (see
2537:{\displaystyle (a_{1},a_{2},a_{3},\dots ,a_{n})}
7420:Journal fĂŒr die Reine und Angewandte Mathematik
7107:An Illustrative Introduction to Modern Analysis
5461:{\displaystyle \mathbb {Z} \times \mathbb {N} }
5051:{\displaystyle \mathbb {N} \times \mathbb {N} }
4754:{\displaystyle \mathbb {N} \times \mathbb {N} }
2219:{\displaystyle \mathbb {N} \times \mathbb {N} }
7493:Reprinted by Springer-Verlag, New York, 1974.
7099:
7097:
5703:{\displaystyle f:\mathbb {A} \to \mathbb {Q} }
5177:{\displaystyle f:\mathbb {Z} \to \mathbb {N} }
2941:of finitely many countable sets is countable.
1105:Bijective mapping from integer to even numbers
10127:
8369:
7656:
7463:
7340:Hrbacek, Karel; Jech, Thomas (22 June 1999).
7327:
6528:is countable there is an injective function
4906:are two countable sets there are surjections
1586:{\displaystyle \mathbb {N} =\{0,1,2,\dots \}}
7154:
5994:
5976:
4698:
4674:
4510:The minimal standard model includes all the
4430:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4335:, then there is no surjective function from
2435:This mapping covers all such ordered pairs.
1925:
1895:
1865:
1841:
1722:
1704:
1680:
1662:
1580:
1556:
1275:
1233:
1037:
1007:
987:
969:
7339:
7094:
6694:is empty or there is a surjective function
6068:{\displaystyle g_{i}:\mathbb {N} \to A_{i}}
4572:; and in both examples of non-well orders,
10134:
10120:
8561:
8376:
8362:
8338:
7663:
7649:
7395:
7315:
7291:
6833:{\displaystyle g\circ h:\mathbb {N} \to T}
6627:{\displaystyle h\circ f:S\to \mathbb {N} }
4518:, as well as many other kinds of numbers.
4495:but uncountable from the point of view of
2184:A subset of a countable set is countable.
8139:
8062:
8023:
7985:
7957:
7929:
7901:
7789:
7756:
7728:
7700:
7464:Fletcher, Peter; Patty, C. Wayne (1988),
7445:
7193:
7014:
6875:Finite intersection property#Applications
6820:
6708:
6620:
6548:
6390:
6341:
6295:
6212:
6048:
5696:
5688:
5476:
5454:
5446:
5363:
5355:
5347:
5319:
5170:
5162:
5134:
5044:
5036:
5002:
4994:
4954:
4920:
4791:
4783:
4775:
4747:
4739:
4661:
4638:
4420:
3190:
3168:
3003:
2981:
2956:
2893:
2611:-tuple to a natural number. For example,
2212:
2204:
1814:
1593:. For example, define the correspondence
1549:
1386:
1374:
1359:
1344:
1321:
1298:
779:{\displaystyle a_{0},a_{1},a_{2},\ldots }
690:
642:
516:All of these definitions are equivalent.
429:
360:
312:
263:
7416:"Ein Beitrag zur Mannigfaltigkeitslehre"
7042:
7040:
7038:
5114:on the number of sets in the collection.
4464:Minimal model of set theory is countable
3395:
2228:
1762:As for the case of infinite sets, a set
1100:
727:can be arranged in an infinite sequence
124:
7468:, Boston: PWS-KENT Publishing Company,
7370:
7236:FerreirĂłs 2007, pp. 268, 272–273.
6987:
6975:
6897:
5438:is a surjection from the countable set
5028:is a surjection from the countable set
4859:{\displaystyle f(m,n)=2^{m}\cdot 3^{n}}
4622:
4620:
137:to mean what is here called countable.
27:Mathematical set that can be enumerated
14:
10767:
8383:
7481:
7413:
7279:
7267:
7130:
6969:
6959:
6957:
6654:is countable. For (2) observe that if
4445:For an elaboration of this result see
2995:may intuitively seem much bigger than
2885:-tuples to the set of natural numbers
1281:{\displaystyle \{0,1,2,3,4,5,\dots \}}
90:In more technical terms, assuming the
10775:Basic concepts in infinite set theory
10115:
8357:
7644:
7554:
7541:, Berlin, New York: Springer-Verlag,
7512:
7303:
7142:
7073:
7067:
7047:Yaqub, Aladdin M. (24 October 2014).
7046:
7035:
6994:. BoD - Books on Demand. p. 24.
6948:
6924:
4040:of the natural numbers is countable.
7533:
7450:(2nd revised ed.), BirkhÀuser,
7398:Foundations for Advanced Mathematics
6963:
6918:
6219:{\displaystyle I\times \mathbb {N} }
4719:, which takes 0 as a natural number.
4617:
4600:Hilbert's paradox of the Grand Hotel
1963:{\displaystyle n\leftrightarrow n+1}
1931:{\displaystyle B=\{0,2,4,6,\dots \}}
1782:is countably infinite if there is a
1043:{\displaystyle \{1,2,3,\dots ,100\}}
7852:Set-theoretically definable numbers
7575:
7560:Principles of Mathematical Analysis
7181:
7021:. Academic Publishers. p. 22.
6954:
6936:
6871:Cantor's first uncountability proof
6800:are both empty, or the composition
6721:{\displaystyle h:\mathbb {N} \to S}
6555:{\displaystyle h:T\to \mathbb {N} }
6262:{\textstyle \bigcup _{i\in I}A_{i}}
5333:are countable because the function
5148:are countable because the function
4967:{\displaystyle g:\mathbb {N} \to B}
4933:{\displaystyle f:\mathbb {N} \to A}
3986:
3976:
3966:
3913:
3873:
3833:
3793:
3753:
3713:
3673:
3633:
3593:
3553:
3513:
3433:
3423:
3413:
1992:{\displaystyle n\leftrightarrow 2n}
35:. For the statistical concept, see
24:
7823:
7670:
7074:Singh, Tej Bahadur (17 May 2019).
4411:
4388:. A proof is given in the article
4364:
4287:
3404:For example, given countable sets
1871:{\displaystyle A=\{1,2,3,\dots \}}
1485:
1456:
904:
588:
505:{\displaystyle |S|<\aleph _{0}}
493:
229:
25:
10796:
7466:Foundations of Higher Mathematics
4381:{\displaystyle {\mathcal {P}}(A)}
4315:, i.e. the set of all subsets of
4304:{\displaystyle {\mathcal {P}}(A)}
4021:of natural numbers is countable.
1828:. As examples, consider the sets
1444:correspondence with the integers
252:), the cardinality of the set of
10165:
10095:
8337:
7628:
7616:
7257:from the original on 2020-09-18.
7247:"What Are Sets and Roster Form?"
6988:Thierry, Vialar (4 April 2017).
6453:
6445:
6095:
6000:{\displaystyle I=\{1,\dots ,n\}}
3373:of countable sets is countable.
3208:In a similar manner, the set of
2438:This form of triangular mapping
2196:of two sets of natural numbers,
935:, Cantor proved that the set of
7400:, Scott, Foresman and Company,
7333:
7321:
7309:
7297:
7285:
7273:
7261:
7239:
7230:
7221:
7187:
7175:
7161:. World Scientific. p. 8.
7148:
7136:
7124:
7015:Mukherjee, Subir Kumar (2009).
6898:Manetti, Marco (19 June 2015).
6863:
6499:
6272:
6191:{\displaystyle G(i,m)=g_{i}(m)}
6027:there is a surjective function
5907:
5493:
5117:
4722:
4715:and the standard convention in
4521:
4514:and all effectively computable
1422:{\displaystyle n\rightarrow 2n}
1206:, infinite sets have more than
947:
10141:
7831:{\displaystyle {\mathcal {P}}}
7491:, D. Van Nostrand Company, Inc
7008:
6981:
6942:
6930:
6891:
6840:is surjective. In either case
6824:
6744:
6712:
6616:
6578:
6544:
6457:
6302:{\displaystyle I=\mathbb {N} }
6185:
6179:
6163:
6151:
6099:
6052:
5726:
5720:
5692:
5431:{\displaystyle g(m,n)=m/(n+1)}
5425:
5413:
5399:
5387:
5359:
5262:
5256:
5200:
5194:
5166:
5006:
4958:
4924:
4827:
4815:
4787:
4530:in various ways, for example:
4424:
4416:
4375:
4369:
4298:
4292:
4189:
4114:
3905:
3893:
3865:
3853:
3825:
3813:
3785:
3773:
3745:
3733:
3705:
3693:
3665:
3653:
3625:
3613:
3585:
3573:
3545:
3533:
3505:
3493:
3346:
3334:
3182:(the set of all integers) and
2852:
2840:
2820:
2808:
2788:
2776:
2756:
2747:
2735:
2732:
2712:
2700:
2680:
2671:
2659:
2656:
2636:
2618:
2531:
2473:
2416:
2404:
2401:
2392:
2380:
2377:
2368:
2356:
2353:
2344:
2332:
2329:
2320:
2308:
2305:
2296:
2284:
2281:
2272:
2260:
2257:
2146:
2132:
2118:
2104:
2090:
2069:
2055:
2041:
2027:
2013:
1980:
1948:
1633:
1618:
1603:
1448:and say they have cardinality
1410:
1379:
1364:
1349:
1330:
1307:
1050:presumably denotes the set of
566:
558:
485:
477:
207:
199:
113:, who proved the existence of
13:
1:
10056:History of mathematical logic
8186:Plane-based geometric algebra
7364:
7018:First Course in Real Analysis
6594:is injective the composition
6281:: As in the finite case, but
4437:is not countable; i.e. it is
4017:The set of all finite-length
155:
109:The concept is attributed to
48:(recursively) enumerable sets
9981:Primitive recursive function
8146:{\displaystyle \mathbb {S} }
8069:{\displaystyle \mathbb {C} }
8030:{\displaystyle \mathbb {R} }
7992:{\displaystyle \mathbb {O} }
7964:{\displaystyle \mathbb {H} }
7936:{\displaystyle \mathbb {C} }
7908:{\displaystyle \mathbb {R} }
7796:{\displaystyle \mathbb {A} }
7763:{\displaystyle \mathbb {Q} }
7735:{\displaystyle \mathbb {Z} }
7707:{\displaystyle \mathbb {N} }
7539:Real and Functional Analysis
7432:10.1515/crelle-1878-18788413
7050:An Introduction to Metalogic
6884:
6397:{\displaystyle \mathbb {N} }
6348:{\displaystyle \mathbb {N} }
5943:is a countable set for each
5483:{\displaystyle \mathbb {Q} }
5326:{\displaystyle \mathbb {Q} }
5141:{\displaystyle \mathbb {Z} }
4645:{\displaystyle \mathbb {N} }
4484:contains elements that are:
3197:{\displaystyle \mathbb {Q} }
3175:{\displaystyle \mathbb {Z} }
3010:{\displaystyle \mathbb {N} }
2988:{\displaystyle \mathbb {Q} }
2963:{\displaystyle \mathbb {Z} }
2900:{\displaystyle \mathbb {N} }
1821:{\displaystyle \mathbb {N} }
1113:; these sets have more than
933:his first set theory article
697:{\displaystyle \mathbb {N} }
649:{\displaystyle \mathbb {N} }
436:{\displaystyle \mathbb {N} }
367:{\displaystyle \mathbb {N} }
319:{\displaystyle \mathbb {N} }
270:{\displaystyle \mathbb {N} }
7:
7590:10.1007/978-981-10-1789-6_8
5284:{\displaystyle f(n)=3^{-n}}
4583:
3029:(a fraction in the form of
1686:{\displaystyle S=\{a,b,c\}}
1468:{\displaystyle \aleph _{0}}
1199:{\displaystyle n=10^{1000}}
916:{\displaystyle \aleph _{0}}
600:{\displaystyle \aleph _{0}}
348:is empty or there exists a
241:{\displaystyle \aleph _{0}}
10:
10801:
10625:von NeumannâBernaysâGödel
9045:SchröderâBernstein theorem
8772:Monadic predicate calculus
8431:Foundations of mathematics
7396:Avelsgaard, Carol (1990),
7346:. CRC Press. p. 141.
6978:, p. 23, Chapter 1.14
6508:: For (1) observe that if
5219:{\displaystyle f(n)=2^{n}}
5084:and the Corollary implies
4626:Since there is an obvious
4447:Cantor's diagonal argument
3275:is proved as countable if
2907:is proved. For the set of
2466:of natural numbers, i.e.,
926:
29:
10689:
10652:
10564:
10454:
10426:One-to-one correspondence
10342:
10283:
10174:
10163:
10149:
10091:
10078:Philosophy of mathematics
10027:Automated theorem proving
10009:
9904:
9736:
9629:
9481:
9198:
9174:
9152:Von NeumannâBernaysâGödel
9097:
8991:
8895:
8793:
8784:
8711:
8646:
8552:
8474:
8391:
8328:
8270:
8196:
8176:Algebra of physical space
8098:
8006:
7877:
7679:
7562:, New York: McGraw-Hill,
7328:Fletcher & Patty 1988
7227:Cantor 1878, p. 242.
7203:, CRC Press, p. 10,
7080:. Springer. p. 422.
6311:axiom of countable choice
5103:{\displaystyle A\times B}
5077:{\displaystyle A\times B}
3951:axiom of countable choice
3390:axiom of countable choice
2762:{\displaystyle ((0,2),3)}
2686:{\displaystyle ((0,2),3)}
1728:{\displaystyle \{1,2,3\}}
1431:one-to-one correspondence
993:{\displaystyle \{3,4,5\}}
680:one-to-one correspondence
221:is less than or equal to
92:axiom of countable choice
78:. Equivalently, a set is
72:one to one correspondence
8232:Extended complex numbers
8215:Extended natural numbers
7446:Ferreirós, José (2007),
7077:Introduction to Topology
6904:. Springer. p. 26.
6877:for a topological proof.
6753:{\displaystyle g:S\to T}
6587:{\displaystyle f:S\to T}
6226:is countable, the union
6198:is a surjection. Since
4610:
4478:LöwenheimâSkolem theorem
4198:{\displaystyle g:S\to T}
4123:{\displaystyle f:S\to T}
2192:of natural numbers (the
512:) or countably infinite.
46:Not to be confused with
9728:Self-verifying theories
9549:Tarski's axiomatization
8500:Tarski's undefinability
8495:incompleteness theorems
6991:Handbook of Mathematics
6075:and hence the function
5647:{\displaystyle \alpha }
5515:{\displaystyle \alpha }
3096:{\displaystyle b\neq 0}
2642:{\displaystyle (0,2,3)}
2235:Cantor pairing function
1649:Since every element of
859:{\displaystyle i\neq j}
39:. For the company, see
10384:Constructible universe
10211:Constructibility (V=L)
10102:Mathematics portal
9713:Proof of impossibility
9361:propositional variable
8671:Propositional calculus
8288:Transcendental numbers
8147:
8124:Hyperbolic quaternions
8070:
8031:
7993:
7965:
7937:
7909:
7832:
7797:
7764:
7736:
7708:
7414:Cantor, Georg (1878),
6854:
6834:
6794:
6774:
6760:is surjective, either
6754:
6722:
6688:
6668:
6648:
6628:
6588:
6556:
6522:
6490:
6425:
6398:
6376:
6349:
6327:
6303:
6263:
6220:
6192:
6135:
6069:
6021:
6001:
5957:
5937:
5894:
5874:
5847:
5704:
5668:
5648:
5628:
5516:
5484:
5462:
5432:
5371:
5327:
5305:
5285:
5240:
5220:
5178:
5142:
5104:
5078:
5052:
5022:
4968:
4934:
4900:
4880:
4860:
4799:
4755:
4705:
4646:
4576:subsets do not have a
4526:Countable sets can be
4516:transcendental numbers
4474:Constructible universe
4431:
4382:
4349:
4329:
4305:
4272:
4239:
4219:
4199:
4164:
4144:
4124:
4088:
4068:
4036:The set of all finite
4001:
3941:
3448:
3401:
3353:
3321:
3289:
3269:
3249:
3229:
3198:
3176:
3145:
3117:
3097:
3071:
3051:
3011:
2989:
2964:
2921:
2901:
2879:
2859:
2827:
2795:
2763:
2719:
2687:
2643:
2605:
2585:
2565:
2538:
2456:
2429:
2238:
2220:
2168:
1993:
1964:
1932:
1878:, the set of positive
1872:
1822:
1800:
1776:
1753:
1729:
1687:
1643:
1587:
1532:
1504:
1469:
1423:
1394:
1282:
1220:
1200:
1167:
1147:
1127:
1106:
1092:
1072:
1044:
994:
917:
880:
860:
834:
807:
780:
721:
698:
672:
650:
628:
601:
574:
533:
506:
459:
437:
415:
388:
368:
342:
320:
298:
271:
242:
215:
174:
150:recursively enumerable
125:A note on terminology
10607:Principia Mathematica
10441:Transfinite induction
10300:(i.e. set difference)
9971:Kolmogorov complexity
9924:Computably enumerable
9824:Model complete theory
9616:Principia Mathematica
8676:Propositional formula
8505:BanachâTarski paradox
8220:Extended real numbers
8148:
8071:
8041:Split-complex numbers
8032:
7994:
7966:
7938:
7910:
7833:
7798:
7774:Constructible numbers
7765:
7737:
7709:
7576:Tao, Terence (2016).
6855:
6835:
6795:
6775:
6755:
6723:
6689:
6674:is countable, either
6669:
6649:
6629:
6589:
6557:
6523:
6491:
6426:
6424:{\displaystyle A_{i}}
6399:
6377:
6375:{\displaystyle g_{i}}
6350:
6328:
6304:
6264:
6221:
6193:
6136:
6070:
6022:
6002:
5958:
5938:
5936:{\displaystyle A_{i}}
5895:
5875:
5873:{\displaystyle p_{n}}
5848:
5705:
5669:
5649:
5629:
5517:
5485:
5463:
5433:
5372:
5328:
5306:
5286:
5241:
5221:
5179:
5143:
5105:
5079:
5053:
5023:
4969:
4935:
4901:
4881:
4861:
4800:
4756:
4706:
4647:
4432:
4383:
4350:
4330:
4306:
4273:
4240:
4220:
4200:
4165:
4145:
4125:
4089:
4069:
4002:
3942:
3449:
3399:
3354:
3352:{\displaystyle (p,q)}
3322:
3290:
3270:
3250:
3230:
3199:
3177:
3146:
3118:
3098:
3072:
3052:
3012:
2990:
2965:
2922:
2902:
2880:
2860:
2858:{\displaystyle (a,b)}
2828:
2826:{\displaystyle (5,3)}
2796:
2794:{\displaystyle (5,3)}
2764:
2720:
2718:{\displaystyle (0,2)}
2688:
2644:
2606:
2586:
2566:
2564:{\displaystyle a_{i}}
2539:
2457:
2430:
2245:proceeds as follows:
2232:
2221:
2169:
1994:
1965:
1933:
1873:
1823:
1801:
1777:
1754:
1730:
1688:
1644:
1588:
1533:
1505:
1490:By definition, a set
1470:
1424:
1395:
1283:
1221:
1201:
1168:
1148:
1128:
1104:
1093:
1073:
1045:
995:
918:
881:
866:and every element of
861:
835:
833:{\displaystyle a_{j}}
808:
806:{\displaystyle a_{i}}
781:
722:
699:
673:
651:
629:
602:
575:
534:
507:
460:
438:
416:
389:
369:
343:
321:
299:
272:
243:
216:
175:
70:or it can be made in
10681:Burali-Forti paradox
10436:Set-builder notation
10389:Continuum hypothesis
10329:Symmetric difference
9919:ChurchâTuring thesis
9906:Computability theory
9115:continuum hypothesis
8633:Square of opposition
8491:Gödel's completeness
8252:Supernatural numbers
8162:Multicomplex numbers
8135:
8119:Dual-complex numbers
8058:
8019:
7981:
7953:
7925:
7897:
7879:Composition algebras
7847:Arithmetical numbers
7818:
7785:
7752:
7724:
7696:
7509:(Paperback edition).
6844:
6804:
6784:
6764:
6732:
6698:
6678:
6658:
6638:
6598:
6566:
6532:
6512:
6435:
6408:
6386:
6359:
6337:
6317:
6285:
6230:
6202:
6145:
6079:
6031:
6011:
5967:
5947:
5920:
5884:
5857:
5714:
5678:
5658:
5638:
5526:
5506:
5472:
5442:
5381:
5337:
5315:
5295:
5250:
5246:is non-negative and
5230:
5188:
5152:
5130:
5088:
5062:
5032:
4978:
4944:
4910:
4890:
4870:
4809:
4765:
4735:
4656:
4634:
4460:of natural numbers.
4406:
4359:
4339:
4319:
4282:
4262:
4229:
4209:
4177:
4154:
4134:
4102:
4078:
4058:
3961:
3458:
3408:
3331:
3303:
3279:
3259:
3239:
3219:
3186:
3164:
3127:
3107:
3081:
3061:
3033:
2999:
2977:
2952:
2911:
2889:
2869:
2837:
2805:
2773:
2729:
2697:
2653:
2615:
2595:
2575:
2548:
2470:
2446:
2251:
2200:
2003:
1974:
1942:
1886:
1832:
1810:
1790:
1766:
1743:
1701:
1653:
1597:
1545:
1538:and a subset of the
1522:
1494:
1452:
1404:
1400:or, more generally,
1292:
1230:
1210:
1177:
1157:
1137:
1117:
1082:
1062:
1004:
966:
900:
870:
844:
817:
790:
731:
711:
686:
662:
638:
618:
584:
554:
523:
473:
449:
425:
405:
378:
356:
332:
308:
288:
259:
225:
195:
164:
10642:TarskiâGrothendieck
10073:Mathematical object
9964:P versus NP problem
9929:Computable function
9723:Reverse mathematics
9649:Logical consequence
9526:primitive recursive
9521:elementary function
9294:Free/bound variable
9147:TarskiâGrothendieck
8666:Logical connectives
8596:Logical equivalence
8446:Logical consequence
8157:Split-biquaternions
7869:Eisenstein integers
7807:Closed-form numbers
7053:. Broadview Press.
4400: —
4052: —
4034: —
4015: —
3386: —
3367: —
3320:{\displaystyle p/q}
3159: —
3144:{\displaystyle n/1}
3123:is also a fraction
3050:{\displaystyle a/b}
2970:and the set of all
2935: —
2182: —
958:is a collection of
573:{\displaystyle |S|}
350:surjective function
214:{\displaystyle |S|}
82:if there exists an
10231:Limitation of size
9871:Transfer principle
9834:Semantics of logic
9819:Categorical theory
9795:Non-standard model
9309:Logical connective
8436:Information theory
8385:Mathematical logic
8315:Profinite integers
8278:Irrational numbers
8143:
8066:
8027:
7989:
7961:
7933:
7905:
7862:Gaussian rationals
7842:Computable numbers
7828:
7793:
7760:
7732:
7704:
7195:Stillwell, John C.
6850:
6830:
6790:
6770:
6750:
6718:
6684:
6664:
6644:
6624:
6584:
6552:
6518:
6486:
6475:
6421:
6394:
6372:
6345:
6323:
6299:
6259:
6248:
6216:
6188:
6131:
6117:
6065:
6017:
5997:
5953:
5933:
5890:
5870:
5843:
5700:
5664:
5644:
5624:
5512:
5480:
5458:
5428:
5367:
5323:
5301:
5281:
5236:
5216:
5174:
5138:
5100:
5074:
5048:
5018:
4964:
4930:
4896:
4876:
4856:
4795:
4751:
4717:mathematical logic
4701:
4642:
4492:, hence countable,
4427:
4398:
4378:
4345:
4325:
4301:
4268:
4235:
4225:is countable then
4215:
4205:is surjective and
4195:
4160:
4150:is countable then
4140:
4120:
4084:
4064:
4050:
4032:
4013:
3997:
3937:
3935:
3444:
3402:
3384:
3365:
3349:
3317:
3285:
3265:
3245:
3225:
3194:
3172:
3157:
3141:
3113:
3093:
3067:
3047:
3007:
2985:
2960:
2933:
2917:
2897:
2875:
2855:
2823:
2791:
2759:
2715:
2683:
2649:can be written as
2639:
2601:
2581:
2561:
2534:
2452:
2425:
2239:
2216:
2180:
2164:
2162:
1989:
1960:
1928:
1868:
1818:
1796:
1772:
1749:
1725:
1683:
1639:
1583:
1528:
1514:if there exists a
1500:
1465:
1446:countably infinite
1419:
1390:
1278:
1216:
1196:
1163:
1143:
1123:
1107:
1088:
1068:
1040:
990:
913:
876:
856:
830:
803:
776:
717:
694:
668:
646:
624:
614:) mapping between
597:
570:
529:
502:
455:
433:
411:
384:
364:
338:
316:
294:
282:injective function
267:
238:
211:
170:
104:countably infinite
84:injective function
18:Countably infinite
10762:
10761:
10671:Russell's paradox
10620:ZermeloâFraenkel
10521:Dedekind-infinite
10394:Diagonal argument
10293:Cartesian product
10157:Set (mathematics)
10109:
10108:
10041:Abstract category
9844:Theories of truth
9654:Rule of inference
9644:Natural deduction
9625:
9624:
9170:
9169:
8875:Cartesian product
8780:
8779:
8686:Many-valued logic
8661:Boolean functions
8544:Russell's paradox
8519:diagonal argument
8416:First-order logic
8351:
8350:
8262:Superreal numbers
8242:Levi-Civita field
8237:Hyperreal numbers
8181:Spacetime algebra
8167:Geometric algebra
8080:Bicomplex numbers
8046:Split-quaternions
7887:Division algebras
7857:Gaussian integers
7779:Algebraic numbers
7682:definable numbers
7599:978-981-10-1789-6
7507:978-1-61427-131-4
7457:978-3-7643-8349-7
7389:978-0-471-00007-5
7353:978-0-8247-7915-3
7168:978-981-12-1999-3
7117:978-1-351-76532-9
7087:978-981-13-6954-4
7060:978-1-4604-0244-3
7028:978-81-89781-90-3
7001:978-2-9551990-1-5
6966:, §2 of Chapter I
6911:978-3-319-16958-3
6853:{\displaystyle T}
6793:{\displaystyle T}
6773:{\displaystyle S}
6687:{\displaystyle S}
6667:{\displaystyle S}
6647:{\displaystyle S}
6634:is injective, so
6521:{\displaystyle T}
6460:
6326:{\displaystyle i}
6313:to pick for each
6233:
6102:
6020:{\displaystyle i}
5956:{\displaystyle i}
5893:{\displaystyle n}
5667:{\displaystyle k}
5468:to the rationals
5304:{\displaystyle n}
5239:{\displaystyle n}
4899:{\displaystyle B}
4879:{\displaystyle A}
4512:algebraic numbers
4396:
4348:{\displaystyle A}
4328:{\displaystyle A}
4271:{\displaystyle A}
4238:{\displaystyle T}
4218:{\displaystyle S}
4163:{\displaystyle S}
4143:{\displaystyle T}
4130:is injective and
4087:{\displaystyle T}
4067:{\displaystyle S}
4048:
4030:
4011:
3988:
3978:
3968:
3915:
3875:
3835:
3795:
3755:
3715:
3675:
3635:
3595:
3555:
3515:
3482:
3475:
3468:
3435:
3425:
3415:
3382:
3363:
3288:{\displaystyle B}
3268:{\displaystyle A}
3248:{\displaystyle B}
3228:{\displaystyle A}
3210:algebraic numbers
3155:
3116:{\displaystyle n}
3070:{\displaystyle a}
2939:Cartesian product
2931:
2920:{\displaystyle n}
2878:{\displaystyle n}
2604:{\displaystyle n}
2584:{\displaystyle n}
2455:{\displaystyle n}
2194:Cartesian product
2178:
1799:{\displaystyle S}
1775:{\displaystyle S}
1752:{\displaystyle S}
1629:
1614:
1531:{\displaystyle S}
1503:{\displaystyle S}
1219:{\displaystyle n}
1166:{\displaystyle n}
1146:{\displaystyle n}
1126:{\displaystyle n}
1091:{\displaystyle n}
1071:{\displaystyle n}
879:{\displaystyle S}
813:is distinct from
720:{\displaystyle S}
671:{\displaystyle S}
627:{\displaystyle S}
532:{\displaystyle S}
458:{\displaystyle S}
414:{\displaystyle S}
387:{\displaystyle S}
341:{\displaystyle S}
297:{\displaystyle S}
173:{\displaystyle S}
135:at most countable
16:(Redirected from
10792:
10780:Cardinal numbers
10744:Bertrand Russell
10734:John von Neumann
10719:Abraham Fraenkel
10714:Richard Dedekind
10676:Suslin's problem
10587:Cantor's theorem
10304:De Morgan's laws
10169:
10136:
10129:
10122:
10113:
10112:
10100:
10099:
10051:History of logic
10046:Category of sets
9939:Decision problem
9718:Ordinal analysis
9659:Sequent calculus
9557:Boolean algebras
9497:
9496:
9471:
9442:logical/constant
9196:
9195:
9182:
9105:ZermeloâFraenkel
8856:Set operations:
8791:
8790:
8728:
8559:
8558:
8539:LöwenheimâSkolem
8426:Formal semantics
8378:
8371:
8364:
8355:
8354:
8341:
8340:
8308:
8298:
8210:Cardinal numbers
8171:Clifford algebra
8152:
8150:
8149:
8144:
8142:
8114:Dual quaternions
8075:
8073:
8072:
8067:
8065:
8036:
8034:
8033:
8028:
8026:
7998:
7996:
7995:
7990:
7988:
7970:
7968:
7967:
7962:
7960:
7942:
7940:
7939:
7934:
7932:
7914:
7912:
7911:
7906:
7904:
7837:
7835:
7834:
7829:
7827:
7826:
7802:
7800:
7799:
7794:
7792:
7769:
7767:
7766:
7761:
7759:
7746:Rational numbers
7741:
7739:
7738:
7733:
7731:
7713:
7711:
7710:
7705:
7703:
7665:
7658:
7651:
7642:
7641:
7633:
7632:
7621:
7620:
7619:
7612:
7603:
7572:
7551:
7530:
7492:
7488:Naive Set Theory
7478:
7460:
7442:
7410:
7392:
7381:
7358:
7357:
7337:
7331:
7325:
7319:
7313:
7307:
7301:
7295:
7289:
7283:
7277:
7271:
7265:
7259:
7258:
7243:
7237:
7234:
7228:
7225:
7219:
7218:
7191:
7185:
7179:
7173:
7172:
7152:
7146:
7140:
7134:
7128:
7122:
7121:
7101:
7092:
7091:
7071:
7065:
7064:
7044:
7033:
7032:
7012:
7006:
7005:
6985:
6979:
6973:
6967:
6961:
6952:
6946:
6940:
6934:
6928:
6922:
6916:
6915:
6895:
6878:
6867:
6861:
6859:
6857:
6856:
6851:
6839:
6837:
6836:
6831:
6823:
6799:
6797:
6796:
6791:
6779:
6777:
6776:
6771:
6759:
6757:
6756:
6751:
6727:
6725:
6724:
6719:
6711:
6693:
6691:
6690:
6685:
6673:
6671:
6670:
6665:
6653:
6651:
6650:
6645:
6633:
6631:
6630:
6625:
6623:
6593:
6591:
6590:
6585:
6561:
6559:
6558:
6553:
6551:
6527:
6525:
6524:
6519:
6503:
6497:
6495:
6493:
6492:
6487:
6485:
6484:
6474:
6456:
6448:
6430:
6428:
6427:
6422:
6420:
6419:
6403:
6401:
6400:
6395:
6393:
6381:
6379:
6378:
6373:
6371:
6370:
6354:
6352:
6351:
6346:
6344:
6332:
6330:
6329:
6324:
6308:
6306:
6305:
6300:
6298:
6276:
6270:
6268:
6266:
6265:
6260:
6258:
6257:
6247:
6225:
6223:
6222:
6217:
6215:
6197:
6195:
6194:
6189:
6178:
6177:
6140:
6138:
6137:
6132:
6127:
6126:
6116:
6098:
6074:
6072:
6071:
6066:
6064:
6063:
6051:
6043:
6042:
6026:
6024:
6023:
6018:
6007:, then for each
6006:
6004:
6003:
5998:
5962:
5960:
5959:
5954:
5942:
5940:
5939:
5934:
5932:
5931:
5911:
5905:
5899:
5897:
5896:
5891:
5879:
5877:
5876:
5871:
5869:
5868:
5852:
5850:
5849:
5844:
5842:
5841:
5840:
5839:
5829:
5828:
5827:
5807:
5806:
5805:
5804:
5787:
5786:
5785:
5784:
5767:
5766:
5765:
5764:
5747:
5746:
5709:
5707:
5706:
5701:
5699:
5691:
5673:
5671:
5670:
5665:
5653:
5651:
5650:
5645:
5633:
5631:
5630:
5625:
5623:
5622:
5613:
5612:
5594:
5593:
5584:
5583:
5571:
5570:
5561:
5560:
5548:
5547:
5538:
5537:
5521:
5519:
5518:
5513:
5497:
5491:
5489:
5487:
5486:
5481:
5479:
5467:
5465:
5464:
5459:
5457:
5449:
5437:
5435:
5434:
5429:
5412:
5376:
5374:
5373:
5368:
5366:
5358:
5350:
5332:
5330:
5329:
5324:
5322:
5310:
5308:
5307:
5302:
5290:
5288:
5287:
5282:
5280:
5279:
5245:
5243:
5242:
5237:
5225:
5223:
5222:
5217:
5215:
5214:
5183:
5181:
5180:
5175:
5173:
5165:
5147:
5145:
5144:
5139:
5137:
5121:
5115:
5109:
5107:
5106:
5101:
5083:
5081:
5080:
5075:
5057:
5055:
5054:
5049:
5047:
5039:
5027:
5025:
5024:
5019:
5005:
4997:
4973:
4971:
4970:
4965:
4957:
4939:
4937:
4936:
4931:
4923:
4905:
4903:
4902:
4897:
4885:
4883:
4882:
4877:
4865:
4863:
4862:
4857:
4855:
4854:
4842:
4841:
4804:
4802:
4801:
4796:
4794:
4786:
4778:
4760:
4758:
4757:
4752:
4750:
4742:
4726:
4720:
4710:
4708:
4707:
4702:
4670:
4669:
4664:
4651:
4649:
4648:
4643:
4641:
4624:
4505:Skolem's paradox
4436:
4434:
4433:
4428:
4423:
4415:
4414:
4401:
4390:Cantor's theorem
4387:
4385:
4384:
4379:
4368:
4367:
4354:
4352:
4351:
4346:
4334:
4332:
4331:
4326:
4310:
4308:
4307:
4302:
4291:
4290:
4277:
4275:
4274:
4269:
4258:asserts that if
4255:Cantor's theorem
4244:
4242:
4241:
4236:
4224:
4222:
4221:
4216:
4204:
4202:
4201:
4196:
4173:If the function
4169:
4167:
4166:
4161:
4149:
4147:
4146:
4141:
4129:
4127:
4126:
4121:
4098:If the function
4093:
4091:
4090:
4085:
4073:
4071:
4070:
4065:
4053:
4035:
4016:
4007:simultaneously.
4006:
4004:
4003:
3998:
3990:
3989:
3980:
3979:
3970:
3969:
3946:
3944:
3943:
3938:
3936:
3933:
3932:
3923:
3922:
3917:
3916:
3883:
3882:
3877:
3876:
3843:
3842:
3837:
3836:
3803:
3802:
3797:
3796:
3763:
3762:
3757:
3756:
3723:
3722:
3717:
3716:
3683:
3682:
3677:
3676:
3643:
3642:
3637:
3636:
3603:
3602:
3597:
3596:
3563:
3562:
3557:
3556:
3523:
3522:
3517:
3516:
3483:
3480:
3476:
3473:
3469:
3466:
3453:
3451:
3450:
3445:
3437:
3436:
3427:
3426:
3417:
3416:
3387:
3368:
3358:
3356:
3355:
3350:
3326:
3324:
3323:
3318:
3313:
3297:rational numbers
3294:
3292:
3291:
3286:
3274:
3272:
3271:
3266:
3254:
3252:
3251:
3246:
3234:
3232:
3231:
3226:
3203:
3201:
3200:
3195:
3193:
3181:
3179:
3178:
3173:
3171:
3160:
3150:
3148:
3147:
3142:
3137:
3122:
3120:
3119:
3114:
3102:
3100:
3099:
3094:
3076:
3074:
3073:
3068:
3056:
3054:
3053:
3048:
3043:
3016:
3014:
3013:
3008:
3006:
2994:
2992:
2991:
2986:
2984:
2972:rational numbers
2969:
2967:
2966:
2961:
2959:
2936:
2926:
2924:
2923:
2918:
2906:
2904:
2903:
2898:
2896:
2884:
2882:
2881:
2876:
2864:
2862:
2861:
2856:
2832:
2830:
2829:
2824:
2800:
2798:
2797:
2792:
2768:
2766:
2765:
2760:
2724:
2722:
2721:
2716:
2692:
2690:
2689:
2684:
2648:
2646:
2645:
2640:
2610:
2608:
2607:
2602:
2590:
2588:
2587:
2582:
2570:
2568:
2567:
2562:
2560:
2559:
2543:
2541:
2540:
2535:
2530:
2529:
2511:
2510:
2498:
2497:
2485:
2484:
2461:
2459:
2458:
2453:
2434:
2432:
2431:
2426:
2225:
2223:
2222:
2217:
2215:
2207:
2183:
2173:
2171:
2170:
2165:
2163:
1998:
1996:
1995:
1990:
1969:
1967:
1966:
1961:
1937:
1935:
1934:
1929:
1877:
1875:
1874:
1869:
1827:
1825:
1824:
1819:
1817:
1805:
1803:
1802:
1797:
1781:
1779:
1778:
1773:
1758:
1756:
1755:
1750:
1734:
1732:
1731:
1726:
1692:
1690:
1689:
1684:
1648:
1646:
1645:
1640:
1627:
1612:
1592:
1590:
1589:
1584:
1552:
1537:
1535:
1534:
1529:
1509:
1507:
1506:
1501:
1474:
1472:
1471:
1466:
1464:
1463:
1428:
1426:
1425:
1420:
1399:
1397:
1396:
1391:
1287:
1285:
1284:
1279:
1225:
1223:
1222:
1217:
1205:
1203:
1202:
1197:
1195:
1194:
1172:
1170:
1169:
1164:
1152:
1150:
1149:
1144:
1132:
1130:
1129:
1124:
1097:
1095:
1094:
1089:
1077:
1075:
1074:
1069:
1049:
1047:
1046:
1041:
999:
997:
996:
991:
922:
920:
919:
914:
912:
911:
885:
883:
882:
877:
865:
863:
862:
857:
839:
837:
836:
831:
829:
828:
812:
810:
809:
804:
802:
801:
785:
783:
782:
777:
769:
768:
756:
755:
743:
742:
726:
724:
723:
718:
707:The elements of
703:
701:
700:
695:
693:
677:
675:
674:
669:
655:
653:
652:
647:
645:
633:
631:
630:
625:
606:
604:
603:
598:
596:
595:
579:
577:
576:
571:
569:
561:
550:Its cardinality
538:
536:
535:
530:
511:
509:
508:
503:
501:
500:
488:
480:
464:
462:
461:
456:
442:
440:
439:
434:
432:
421:and a subset of
420:
418:
417:
412:
401:mapping between
393:
391:
390:
385:
373:
371:
370:
365:
363:
347:
345:
344:
339:
325:
323:
322:
317:
315:
303:
301:
300:
295:
280:There exists an
276:
274:
273:
268:
266:
247:
245:
244:
239:
237:
236:
220:
218:
217:
212:
210:
202:
179:
177:
176:
171:
115:uncountable sets
74:with the set of
66:if either it is
51:
44:
21:
10800:
10799:
10795:
10794:
10793:
10791:
10790:
10789:
10765:
10764:
10763:
10758:
10685:
10664:
10648:
10613:New Foundations
10560:
10450:
10369:Cardinal number
10352:
10338:
10279:
10170:
10161:
10145:
10140:
10110:
10105:
10094:
10087:
10032:Category theory
10022:Algebraic logic
10005:
9976:Lambda calculus
9914:Church encoding
9900:
9876:Truth predicate
9732:
9698:Complete theory
9621:
9490:
9486:
9482:
9477:
9469:
9189: and
9185:
9180:
9166:
9142:New Foundations
9110:axiom of choice
9093:
9055:Gödel numbering
8995: and
8987:
8891:
8776:
8726:
8707:
8656:Boolean algebra
8642:
8606:Equiconsistency
8571:Classical logic
8548:
8529:Halting problem
8517: and
8493: and
8481: and
8480:
8475:Theorems (
8470:
8387:
8382:
8352:
8347:
8324:
8303:
8293:
8266:
8257:Surreal numbers
8247:Ordinal numbers
8192:
8138:
8136:
8133:
8132:
8094:
8061:
8059:
8056:
8055:
8053:
8051:Split-octonions
8022:
8020:
8017:
8016:
8008:
8002:
7984:
7982:
7979:
7978:
7956:
7954:
7951:
7950:
7928:
7926:
7923:
7922:
7919:Complex numbers
7900:
7898:
7895:
7894:
7873:
7822:
7821:
7819:
7816:
7815:
7788:
7786:
7783:
7782:
7755:
7753:
7750:
7749:
7727:
7725:
7722:
7721:
7699:
7697:
7694:
7693:
7690:Natural numbers
7675:
7669:
7639:
7627:
7617:
7615:
7607:
7600:
7578:"Infinite sets"
7570:
7549:
7528:
7483:Halmos, Paul R.
7476:
7458:
7426:(84): 242â248,
7408:
7390:
7372:Apostol, Tom M.
7367:
7362:
7361:
7354:
7338:
7334:
7326:
7322:
7316:Avelsgaard 1990
7314:
7310:
7302:
7298:
7292:Avelsgaard 1990
7290:
7286:
7278:
7274:
7266:
7262:
7245:
7244:
7240:
7235:
7231:
7226:
7222:
7211:
7192:
7188:
7180:
7176:
7169:
7153:
7149:
7141:
7137:
7129:
7125:
7118:
7102:
7095:
7088:
7072:
7068:
7061:
7045:
7036:
7029:
7013:
7009:
7002:
6986:
6982:
6974:
6970:
6962:
6955:
6947:
6943:
6935:
6931:
6923:
6919:
6912:
6896:
6892:
6887:
6882:
6881:
6868:
6864:
6845:
6842:
6841:
6819:
6805:
6802:
6801:
6785:
6782:
6781:
6765:
6762:
6761:
6733:
6730:
6729:
6707:
6699:
6696:
6695:
6679:
6676:
6675:
6659:
6656:
6655:
6639:
6636:
6635:
6619:
6599:
6596:
6595:
6567:
6564:
6563:
6547:
6533:
6530:
6529:
6513:
6510:
6509:
6504:
6500:
6480:
6476:
6464:
6452:
6444:
6436:
6433:
6432:
6415:
6411:
6409:
6406:
6405:
6389:
6387:
6384:
6383:
6366:
6362:
6360:
6357:
6356:
6340:
6338:
6335:
6334:
6318:
6315:
6314:
6309:and we use the
6294:
6286:
6283:
6282:
6277:
6273:
6253:
6249:
6237:
6231:
6228:
6227:
6211:
6203:
6200:
6199:
6173:
6169:
6146:
6143:
6142:
6122:
6118:
6106:
6094:
6080:
6077:
6076:
6059:
6055:
6047:
6038:
6034:
6032:
6029:
6028:
6012:
6009:
6008:
5968:
5965:
5964:
5948:
5945:
5944:
5927:
5923:
5921:
5918:
5917:
5912:
5908:
5885:
5882:
5881:
5864:
5860:
5858:
5855:
5854:
5835:
5831:
5830:
5817:
5813:
5812:
5811:
5800:
5796:
5795:
5791:
5780:
5776:
5775:
5771:
5760:
5756:
5755:
5751:
5736:
5732:
5715:
5712:
5711:
5695:
5687:
5679:
5676:
5675:
5659:
5656:
5655:
5639:
5636:
5635:
5618:
5614:
5608:
5604:
5589:
5585:
5579:
5575:
5566:
5562:
5556:
5552:
5543:
5539:
5533:
5529:
5527:
5524:
5523:
5507:
5504:
5503:
5498:
5494:
5475:
5473:
5470:
5469:
5453:
5445:
5443:
5440:
5439:
5408:
5382:
5379:
5378:
5362:
5354:
5346:
5338:
5335:
5334:
5318:
5316:
5313:
5312:
5296:
5293:
5292:
5272:
5268:
5251:
5248:
5247:
5231:
5228:
5227:
5210:
5206:
5189:
5186:
5185:
5169:
5161:
5153:
5150:
5149:
5133:
5131:
5128:
5127:
5122:
5118:
5089:
5086:
5085:
5063:
5060:
5059:
5043:
5035:
5033:
5030:
5029:
5001:
4993:
4979:
4976:
4975:
4953:
4945:
4942:
4941:
4919:
4911:
4908:
4907:
4891:
4888:
4887:
4871:
4868:
4867:
4850:
4846:
4837:
4833:
4810:
4807:
4806:
4790:
4782:
4774:
4766:
4763:
4762:
4746:
4738:
4736:
4733:
4732:
4727:
4723:
4665:
4660:
4659:
4657:
4654:
4653:
4637:
4635:
4632:
4631:
4625:
4618:
4613:
4605:Uncountable set
4586:
4528:totally ordered
4524:
4466:
4443:
4419:
4410:
4409:
4407:
4404:
4403:
4399:
4363:
4362:
4360:
4357:
4356:
4340:
4337:
4336:
4320:
4317:
4316:
4286:
4285:
4283:
4280:
4279:
4263:
4260:
4259:
4248:
4230:
4227:
4226:
4210:
4207:
4206:
4178:
4175:
4174:
4155:
4152:
4151:
4135:
4132:
4131:
4103:
4100:
4099:
4079:
4076:
4075:
4059:
4056:
4055:
4051:
4042:
4033:
4023:
4014:
3985:
3984:
3975:
3974:
3965:
3964:
3962:
3959:
3958:
3934:
3931:
3925:
3924:
3918:
3912:
3911:
3910:
3908:
3891:
3885:
3884:
3878:
3872:
3871:
3870:
3868:
3851:
3845:
3844:
3838:
3832:
3831:
3830:
3828:
3811:
3805:
3804:
3798:
3792:
3791:
3790:
3788:
3771:
3765:
3764:
3758:
3752:
3751:
3750:
3748:
3731:
3725:
3724:
3718:
3712:
3711:
3710:
3708:
3691:
3685:
3684:
3678:
3672:
3671:
3670:
3668:
3651:
3645:
3644:
3638:
3632:
3631:
3630:
3628:
3611:
3605:
3604:
3598:
3592:
3591:
3590:
3588:
3571:
3565:
3564:
3558:
3552:
3551:
3550:
3548:
3531:
3525:
3524:
3518:
3512:
3511:
3510:
3508:
3491:
3485:
3484:
3479:
3477:
3472:
3470:
3465:
3461:
3459:
3456:
3455:
3432:
3431:
3422:
3421:
3412:
3411:
3409:
3406:
3405:
3394:
3385:
3375:
3366:
3332:
3329:
3328:
3309:
3304:
3301:
3300:
3280:
3277:
3276:
3260:
3257:
3256:
3240:
3237:
3236:
3220:
3217:
3216:
3206:
3189:
3187:
3184:
3183:
3167:
3165:
3162:
3161:
3158:
3133:
3128:
3125:
3124:
3108:
3105:
3104:
3082:
3079:
3078:
3062:
3059:
3058:
3039:
3034:
3031:
3030:
3027:vulgar fraction
3002:
3000:
2997:
2996:
2980:
2978:
2975:
2974:
2955:
2953:
2950:
2949:
2945:The set of all
2943:
2934:
2912:
2909:
2908:
2892:
2890:
2887:
2886:
2870:
2867:
2866:
2838:
2835:
2834:
2806:
2803:
2802:
2774:
2771:
2770:
2730:
2727:
2726:
2698:
2695:
2694:
2654:
2651:
2650:
2616:
2613:
2612:
2596:
2593:
2592:
2576:
2573:
2572:
2555:
2551:
2549:
2546:
2545:
2525:
2521:
2506:
2502:
2493:
2489:
2480:
2476:
2471:
2468:
2467:
2447:
2444:
2443:
2442:generalizes to
2252:
2249:
2248:
2211:
2203:
2201:
2198:
2197:
2188:The set of all
2186:
2181:
2161:
2160:
2155:
2141:
2127:
2113:
2099:
2084:
2083:
2078:
2064:
2050:
2036:
2022:
2006:
2004:
2001:
2000:
1975:
1972:
1971:
1943:
1940:
1939:
1887:
1884:
1883:
1833:
1830:
1829:
1813:
1811:
1808:
1807:
1791:
1788:
1787:
1767:
1764:
1763:
1744:
1741:
1740:
1702:
1699:
1698:
1693:is paired with
1654:
1651:
1650:
1598:
1595:
1594:
1548:
1546:
1543:
1542:
1540:natural numbers
1523:
1520:
1519:
1495:
1492:
1491:
1488:
1486:Formal overview
1459:
1455:
1453:
1450:
1449:
1405:
1402:
1401:
1293:
1290:
1289:
1231:
1228:
1227:
1211:
1208:
1207:
1190:
1186:
1178:
1175:
1174:
1158:
1155:
1154:
1138:
1135:
1134:
1133:elements where
1118:
1115:
1114:
1083:
1080:
1079:
1063:
1060:
1059:
1005:
1002:
1001:
967:
964:
963:
950:
929:
907:
903:
901:
898:
897:
871:
868:
867:
845:
842:
841:
824:
820:
818:
815:
814:
797:
793:
791:
788:
787:
764:
760:
751:
747:
738:
734:
732:
729:
728:
712:
709:
708:
689:
687:
684:
683:
663:
660:
659:
641:
639:
636:
635:
619:
616:
615:
591:
587:
585:
582:
581:
565:
557:
555:
552:
551:
524:
521:
520:
496:
492:
484:
476:
474:
471:
470:
450:
447:
446:
428:
426:
423:
422:
406:
403:
402:
397:There exists a
379:
376:
375:
359:
357:
354:
353:
333:
330:
329:
311:
309:
306:
305:
289:
286:
285:
262:
260:
257:
256:
254:natural numbers
232:
228:
226:
223:
222:
206:
198:
196:
193:
192:
165:
162:
161:
158:
127:
76:natural numbers
52:
45:
41:Countable (app)
30:
28:
23:
22:
15:
12:
11:
5:
10798:
10788:
10787:
10782:
10777:
10760:
10759:
10757:
10756:
10751:
10749:Thoralf Skolem
10746:
10741:
10736:
10731:
10726:
10721:
10716:
10711:
10706:
10701:
10695:
10693:
10687:
10686:
10684:
10683:
10678:
10673:
10667:
10665:
10663:
10662:
10659:
10653:
10650:
10649:
10647:
10646:
10645:
10644:
10639:
10634:
10633:
10632:
10617:
10616:
10615:
10603:
10602:
10601:
10590:
10589:
10584:
10579:
10574:
10568:
10566:
10562:
10561:
10559:
10558:
10553:
10548:
10543:
10534:
10529:
10524:
10514:
10509:
10508:
10507:
10502:
10497:
10487:
10477:
10472:
10467:
10461:
10459:
10452:
10451:
10449:
10448:
10443:
10438:
10433:
10431:Ordinal number
10428:
10423:
10418:
10413:
10412:
10411:
10406:
10396:
10391:
10386:
10381:
10376:
10366:
10361:
10355:
10353:
10351:
10350:
10347:
10343:
10340:
10339:
10337:
10336:
10331:
10326:
10321:
10316:
10311:
10309:Disjoint union
10306:
10301:
10295:
10289:
10287:
10281:
10280:
10278:
10277:
10276:
10275:
10270:
10259:
10258:
10256:Martin's axiom
10253:
10248:
10243:
10238:
10233:
10228:
10223:
10221:Extensionality
10218:
10213:
10208:
10207:
10206:
10201:
10196:
10186:
10180:
10178:
10172:
10171:
10164:
10162:
10160:
10159:
10153:
10151:
10147:
10146:
10139:
10138:
10131:
10124:
10116:
10107:
10106:
10092:
10089:
10088:
10086:
10085:
10080:
10075:
10070:
10065:
10064:
10063:
10053:
10048:
10043:
10034:
10029:
10024:
10019:
10017:Abstract logic
10013:
10011:
10007:
10006:
10004:
10003:
9998:
9996:Turing machine
9993:
9988:
9983:
9978:
9973:
9968:
9967:
9966:
9961:
9956:
9951:
9946:
9936:
9934:Computable set
9931:
9926:
9921:
9916:
9910:
9908:
9902:
9901:
9899:
9898:
9893:
9888:
9883:
9878:
9873:
9868:
9863:
9862:
9861:
9856:
9851:
9841:
9836:
9831:
9829:Satisfiability
9826:
9821:
9816:
9815:
9814:
9804:
9803:
9802:
9792:
9791:
9790:
9785:
9780:
9775:
9770:
9760:
9759:
9758:
9753:
9746:Interpretation
9742:
9740:
9734:
9733:
9731:
9730:
9725:
9720:
9715:
9710:
9700:
9695:
9694:
9693:
9692:
9691:
9681:
9676:
9666:
9661:
9656:
9651:
9646:
9641:
9635:
9633:
9627:
9626:
9623:
9622:
9620:
9619:
9611:
9610:
9609:
9608:
9603:
9602:
9601:
9596:
9591:
9571:
9570:
9569:
9567:minimal axioms
9564:
9553:
9552:
9551:
9540:
9539:
9538:
9533:
9528:
9523:
9518:
9513:
9500:
9498:
9479:
9478:
9476:
9475:
9474:
9473:
9461:
9456:
9455:
9454:
9449:
9444:
9439:
9429:
9424:
9419:
9414:
9413:
9412:
9407:
9397:
9396:
9395:
9390:
9385:
9380:
9370:
9365:
9364:
9363:
9358:
9353:
9343:
9342:
9341:
9336:
9331:
9326:
9321:
9316:
9306:
9301:
9296:
9291:
9290:
9289:
9284:
9279:
9274:
9264:
9259:
9257:Formation rule
9254:
9249:
9248:
9247:
9242:
9232:
9231:
9230:
9220:
9215:
9210:
9205:
9199:
9193:
9176:Formal systems
9172:
9171:
9168:
9167:
9165:
9164:
9159:
9154:
9149:
9144:
9139:
9134:
9129:
9124:
9119:
9118:
9117:
9112:
9101:
9099:
9095:
9094:
9092:
9091:
9090:
9089:
9079:
9074:
9073:
9072:
9065:Large cardinal
9062:
9057:
9052:
9047:
9042:
9028:
9027:
9026:
9021:
9016:
9001:
8999:
8989:
8988:
8986:
8985:
8984:
8983:
8978:
8973:
8963:
8958:
8953:
8948:
8943:
8938:
8933:
8928:
8923:
8918:
8913:
8908:
8902:
8900:
8893:
8892:
8890:
8889:
8888:
8887:
8882:
8877:
8872:
8867:
8862:
8854:
8853:
8852:
8847:
8837:
8832:
8830:Extensionality
8827:
8825:Ordinal number
8822:
8812:
8807:
8806:
8805:
8794:
8788:
8782:
8781:
8778:
8777:
8775:
8774:
8769:
8764:
8759:
8754:
8749:
8744:
8743:
8742:
8732:
8731:
8730:
8717:
8715:
8709:
8708:
8706:
8705:
8704:
8703:
8698:
8693:
8683:
8678:
8673:
8668:
8663:
8658:
8652:
8650:
8644:
8643:
8641:
8640:
8635:
8630:
8625:
8620:
8615:
8610:
8609:
8608:
8598:
8593:
8588:
8583:
8578:
8573:
8567:
8565:
8556:
8550:
8549:
8547:
8546:
8541:
8536:
8531:
8526:
8521:
8509:Cantor's
8507:
8502:
8497:
8487:
8485:
8472:
8471:
8469:
8468:
8463:
8458:
8453:
8448:
8443:
8438:
8433:
8428:
8423:
8418:
8413:
8408:
8407:
8406:
8395:
8393:
8389:
8388:
8381:
8380:
8373:
8366:
8358:
8349:
8348:
8346:
8345:
8335:
8333:Classification
8329:
8326:
8325:
8323:
8322:
8320:Normal numbers
8317:
8312:
8290:
8285:
8280:
8274:
8272:
8268:
8267:
8265:
8264:
8259:
8254:
8249:
8244:
8239:
8234:
8229:
8228:
8227:
8217:
8212:
8206:
8204:
8202:infinitesimals
8194:
8193:
8191:
8190:
8189:
8188:
8183:
8178:
8164:
8159:
8154:
8141:
8126:
8121:
8116:
8111:
8105:
8103:
8096:
8095:
8093:
8092:
8087:
8082:
8077:
8064:
8048:
8043:
8038:
8025:
8012:
8010:
8004:
8003:
8001:
8000:
7987:
7972:
7959:
7944:
7931:
7916:
7903:
7883:
7881:
7875:
7874:
7872:
7871:
7866:
7865:
7864:
7854:
7849:
7844:
7839:
7825:
7809:
7804:
7791:
7776:
7771:
7758:
7743:
7730:
7715:
7702:
7686:
7684:
7677:
7676:
7668:
7667:
7660:
7653:
7645:
7638:
7637:
7625:
7605:
7604:
7598:
7573:
7568:
7552:
7547:
7531:
7527:978-0486601410
7526:
7518:Theory of Sets
7510:
7479:
7474:
7461:
7456:
7443:
7411:
7406:
7393:
7388:
7366:
7363:
7360:
7359:
7352:
7332:
7320:
7308:
7306:, pp. 3â4
7296:
7284:
7272:
7260:
7253:. 2021-05-09.
7238:
7229:
7220:
7216:been imagined.
7209:
7186:
7174:
7167:
7147:
7135:
7123:
7116:
7093:
7086:
7066:
7059:
7034:
7027:
7007:
7000:
6980:
6968:
6953:
6941:
6929:
6917:
6910:
6889:
6888:
6886:
6883:
6880:
6879:
6862:
6849:
6829:
6826:
6822:
6818:
6815:
6812:
6809:
6789:
6769:
6749:
6746:
6743:
6740:
6737:
6717:
6714:
6710:
6706:
6703:
6683:
6663:
6643:
6622:
6618:
6615:
6612:
6609:
6606:
6603:
6583:
6580:
6577:
6574:
6571:
6550:
6546:
6543:
6540:
6537:
6517:
6498:
6483:
6479:
6473:
6470:
6467:
6463:
6459:
6455:
6451:
6447:
6443:
6440:
6418:
6414:
6392:
6369:
6365:
6343:
6322:
6297:
6293:
6290:
6271:
6256:
6252:
6246:
6243:
6240:
6236:
6214:
6210:
6207:
6187:
6184:
6181:
6176:
6172:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6130:
6125:
6121:
6115:
6112:
6109:
6105:
6101:
6097:
6093:
6090:
6087:
6084:
6062:
6058:
6054:
6050:
6046:
6041:
6037:
6016:
5996:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5952:
5930:
5926:
5906:
5889:
5867:
5863:
5838:
5834:
5826:
5823:
5820:
5816:
5810:
5803:
5799:
5794:
5790:
5783:
5779:
5774:
5770:
5763:
5759:
5754:
5750:
5745:
5742:
5739:
5735:
5731:
5728:
5725:
5722:
5719:
5698:
5694:
5690:
5686:
5683:
5663:
5643:
5621:
5617:
5611:
5607:
5603:
5600:
5597:
5592:
5588:
5582:
5578:
5574:
5569:
5565:
5559:
5555:
5551:
5546:
5542:
5536:
5532:
5511:
5492:
5478:
5456:
5452:
5448:
5427:
5424:
5421:
5418:
5415:
5411:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5365:
5361:
5357:
5353:
5349:
5345:
5342:
5321:
5300:
5278:
5275:
5271:
5267:
5264:
5261:
5258:
5255:
5235:
5213:
5209:
5205:
5202:
5199:
5196:
5193:
5172:
5168:
5164:
5160:
5157:
5136:
5116:
5099:
5096:
5093:
5073:
5070:
5067:
5046:
5042:
5038:
5017:
5014:
5011:
5008:
5004:
5000:
4996:
4992:
4989:
4986:
4983:
4963:
4960:
4956:
4952:
4949:
4929:
4926:
4922:
4918:
4915:
4895:
4875:
4853:
4849:
4845:
4840:
4836:
4832:
4829:
4826:
4823:
4820:
4817:
4814:
4793:
4789:
4785:
4781:
4777:
4773:
4770:
4749:
4745:
4741:
4721:
4700:
4697:
4694:
4691:
4688:
4685:
4682:
4679:
4676:
4673:
4668:
4663:
4640:
4615:
4614:
4612:
4609:
4608:
4607:
4602:
4597:
4592:
4585:
4582:
4566:
4565:
4564:
4563:
4560:
4556:well orders):
4550:
4549:
4548:
4545:
4539:ordinal number
4523:
4520:
4501:
4500:
4493:
4465:
4462:
4426:
4422:
4418:
4413:
4394:
4377:
4374:
4371:
4366:
4344:
4324:
4300:
4297:
4294:
4289:
4267:
4247:
4246:
4234:
4214:
4194:
4191:
4188:
4185:
4182:
4171:
4159:
4139:
4119:
4116:
4113:
4110:
4107:
4083:
4063:
4046:
4028:
4009:
3996:
3993:
3983:
3973:
3930:
3927:
3926:
3921:
3909:
3907:
3904:
3901:
3898:
3895:
3892:
3890:
3887:
3886:
3881:
3869:
3867:
3864:
3861:
3858:
3855:
3852:
3850:
3847:
3846:
3841:
3829:
3827:
3824:
3821:
3818:
3815:
3812:
3810:
3807:
3806:
3801:
3789:
3787:
3784:
3781:
3778:
3775:
3772:
3770:
3767:
3766:
3761:
3749:
3747:
3744:
3741:
3738:
3735:
3732:
3730:
3727:
3726:
3721:
3709:
3707:
3704:
3701:
3698:
3695:
3692:
3690:
3687:
3686:
3681:
3669:
3667:
3664:
3661:
3658:
3655:
3652:
3650:
3647:
3646:
3641:
3629:
3627:
3624:
3621:
3618:
3615:
3612:
3610:
3607:
3606:
3601:
3589:
3587:
3584:
3581:
3578:
3575:
3572:
3570:
3567:
3566:
3561:
3549:
3547:
3544:
3541:
3538:
3535:
3532:
3530:
3527:
3526:
3521:
3509:
3507:
3504:
3501:
3498:
3495:
3492:
3490:
3487:
3486:
3478:
3471:
3464:
3463:
3443:
3440:
3430:
3420:
3388:(Assuming the
3380:
3361:
3348:
3345:
3342:
3339:
3336:
3316:
3312:
3308:
3284:
3264:
3244:
3224:
3212:is countable.
3192:
3170:
3153:
3140:
3136:
3132:
3112:
3092:
3089:
3086:
3066:
3046:
3042:
3038:
3005:
2983:
2958:
2929:
2916:
2895:
2874:
2854:
2851:
2848:
2845:
2842:
2822:
2819:
2816:
2813:
2810:
2790:
2787:
2784:
2781:
2778:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2714:
2711:
2708:
2705:
2702:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2600:
2580:
2558:
2554:
2533:
2528:
2524:
2520:
2517:
2514:
2509:
2505:
2501:
2496:
2492:
2488:
2483:
2479:
2475:
2451:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2241:The resulting
2214:
2210:
2206:
2176:
2159:
2156:
2154:
2151:
2148:
2145:
2142:
2140:
2137:
2134:
2131:
2128:
2126:
2123:
2120:
2117:
2114:
2112:
2109:
2106:
2103:
2100:
2098:
2095:
2092:
2089:
2086:
2085:
2082:
2079:
2077:
2074:
2071:
2068:
2065:
2063:
2060:
2057:
2054:
2051:
2049:
2046:
2043:
2040:
2037:
2035:
2032:
2029:
2026:
2023:
2021:
2018:
2015:
2012:
2009:
2008:
1988:
1985:
1982:
1979:
1959:
1956:
1953:
1950:
1947:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1816:
1795:
1771:
1748:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1638:
1635:
1632:
1626:
1623:
1620:
1617:
1611:
1608:
1605:
1602:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1551:
1527:
1499:
1487:
1484:
1462:
1458:
1439:), which is a
1418:
1415:
1412:
1409:
1389:
1385:
1381:
1377:
1373:
1370:
1366:
1362:
1358:
1355:
1351:
1347:
1343:
1340:
1336:
1332:
1328:
1324:
1320:
1317:
1313:
1309:
1305:
1301:
1297:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1215:
1193:
1189:
1185:
1182:
1162:
1142:
1122:
1109:Some sets are
1087:
1067:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
989:
986:
983:
980:
977:
974:
971:
949:
946:
928:
925:
910:
906:
888:
887:
875:
855:
852:
849:
827:
823:
800:
796:
775:
772:
767:
763:
759:
754:
750:
746:
741:
737:
716:
705:
692:
667:
657:
644:
623:
608:
594:
590:
568:
564:
560:
528:
514:
513:
499:
495:
491:
487:
483:
479:
454:
444:
431:
410:
395:
383:
362:
337:
327:
314:
293:
278:
265:
235:
231:
209:
205:
201:
169:
157:
154:
126:
123:
26:
9:
6:
4:
3:
2:
10797:
10786:
10783:
10781:
10778:
10776:
10773:
10772:
10770:
10755:
10754:Ernst Zermelo
10752:
10750:
10747:
10745:
10742:
10740:
10739:Willard Quine
10737:
10735:
10732:
10730:
10727:
10725:
10722:
10720:
10717:
10715:
10712:
10710:
10707:
10705:
10702:
10700:
10697:
10696:
10694:
10692:
10691:Set theorists
10688:
10682:
10679:
10677:
10674:
10672:
10669:
10668:
10666:
10660:
10658:
10655:
10654:
10651:
10643:
10640:
10638:
10637:KripkeâPlatek
10635:
10631:
10628:
10627:
10626:
10623:
10622:
10621:
10618:
10614:
10611:
10610:
10609:
10608:
10604:
10600:
10597:
10596:
10595:
10592:
10591:
10588:
10585:
10583:
10580:
10578:
10575:
10573:
10570:
10569:
10567:
10563:
10557:
10554:
10552:
10549:
10547:
10544:
10542:
10540:
10535:
10533:
10530:
10528:
10525:
10522:
10518:
10515:
10513:
10510:
10506:
10503:
10501:
10498:
10496:
10493:
10492:
10491:
10488:
10485:
10481:
10478:
10476:
10473:
10471:
10468:
10466:
10463:
10462:
10460:
10457:
10453:
10447:
10444:
10442:
10439:
10437:
10434:
10432:
10429:
10427:
10424:
10422:
10419:
10417:
10414:
10410:
10407:
10405:
10402:
10401:
10400:
10397:
10395:
10392:
10390:
10387:
10385:
10382:
10380:
10377:
10374:
10370:
10367:
10365:
10362:
10360:
10357:
10356:
10354:
10348:
10345:
10344:
10341:
10335:
10332:
10330:
10327:
10325:
10322:
10320:
10317:
10315:
10312:
10310:
10307:
10305:
10302:
10299:
10296:
10294:
10291:
10290:
10288:
10286:
10282:
10274:
10273:specification
10271:
10269:
10266:
10265:
10264:
10261:
10260:
10257:
10254:
10252:
10249:
10247:
10244:
10242:
10239:
10237:
10234:
10232:
10229:
10227:
10224:
10222:
10219:
10217:
10214:
10212:
10209:
10205:
10202:
10200:
10197:
10195:
10192:
10191:
10190:
10187:
10185:
10182:
10181:
10179:
10177:
10173:
10168:
10158:
10155:
10154:
10152:
10148:
10144:
10137:
10132:
10130:
10125:
10123:
10118:
10117:
10114:
10104:
10103:
10098:
10090:
10084:
10081:
10079:
10076:
10074:
10071:
10069:
10066:
10062:
10059:
10058:
10057:
10054:
10052:
10049:
10047:
10044:
10042:
10038:
10035:
10033:
10030:
10028:
10025:
10023:
10020:
10018:
10015:
10014:
10012:
10008:
10002:
9999:
9997:
9994:
9992:
9991:Recursive set
9989:
9987:
9984:
9982:
9979:
9977:
9974:
9972:
9969:
9965:
9962:
9960:
9957:
9955:
9952:
9950:
9947:
9945:
9942:
9941:
9940:
9937:
9935:
9932:
9930:
9927:
9925:
9922:
9920:
9917:
9915:
9912:
9911:
9909:
9907:
9903:
9897:
9894:
9892:
9889:
9887:
9884:
9882:
9879:
9877:
9874:
9872:
9869:
9867:
9864:
9860:
9857:
9855:
9852:
9850:
9847:
9846:
9845:
9842:
9840:
9837:
9835:
9832:
9830:
9827:
9825:
9822:
9820:
9817:
9813:
9810:
9809:
9808:
9805:
9801:
9800:of arithmetic
9798:
9797:
9796:
9793:
9789:
9786:
9784:
9781:
9779:
9776:
9774:
9771:
9769:
9766:
9765:
9764:
9761:
9757:
9754:
9752:
9749:
9748:
9747:
9744:
9743:
9741:
9739:
9735:
9729:
9726:
9724:
9721:
9719:
9716:
9714:
9711:
9708:
9707:from ZFC
9704:
9701:
9699:
9696:
9690:
9687:
9686:
9685:
9682:
9680:
9677:
9675:
9672:
9671:
9670:
9667:
9665:
9662:
9660:
9657:
9655:
9652:
9650:
9647:
9645:
9642:
9640:
9637:
9636:
9634:
9632:
9628:
9618:
9617:
9613:
9612:
9607:
9606:non-Euclidean
9604:
9600:
9597:
9595:
9592:
9590:
9589:
9585:
9584:
9582:
9579:
9578:
9576:
9572:
9568:
9565:
9563:
9560:
9559:
9558:
9554:
9550:
9547:
9546:
9545:
9541:
9537:
9534:
9532:
9529:
9527:
9524:
9522:
9519:
9517:
9514:
9512:
9509:
9508:
9506:
9502:
9501:
9499:
9494:
9488:
9483:Example
9480:
9472:
9467:
9466:
9465:
9462:
9460:
9457:
9453:
9450:
9448:
9445:
9443:
9440:
9438:
9435:
9434:
9433:
9430:
9428:
9425:
9423:
9420:
9418:
9415:
9411:
9408:
9406:
9403:
9402:
9401:
9398:
9394:
9391:
9389:
9386:
9384:
9381:
9379:
9376:
9375:
9374:
9371:
9369:
9366:
9362:
9359:
9357:
9354:
9352:
9349:
9348:
9347:
9344:
9340:
9337:
9335:
9332:
9330:
9327:
9325:
9322:
9320:
9317:
9315:
9312:
9311:
9310:
9307:
9305:
9302:
9300:
9297:
9295:
9292:
9288:
9285:
9283:
9280:
9278:
9275:
9273:
9270:
9269:
9268:
9265:
9263:
9260:
9258:
9255:
9253:
9250:
9246:
9243:
9241:
9240:by definition
9238:
9237:
9236:
9233:
9229:
9226:
9225:
9224:
9221:
9219:
9216:
9214:
9211:
9209:
9206:
9204:
9201:
9200:
9197:
9194:
9192:
9188:
9183:
9177:
9173:
9163:
9160:
9158:
9155:
9153:
9150:
9148:
9145:
9143:
9140:
9138:
9135:
9133:
9130:
9128:
9127:KripkeâPlatek
9125:
9123:
9120:
9116:
9113:
9111:
9108:
9107:
9106:
9103:
9102:
9100:
9096:
9088:
9085:
9084:
9083:
9080:
9078:
9075:
9071:
9068:
9067:
9066:
9063:
9061:
9058:
9056:
9053:
9051:
9048:
9046:
9043:
9040:
9036:
9032:
9029:
9025:
9022:
9020:
9017:
9015:
9012:
9011:
9010:
9006:
9003:
9002:
9000:
8998:
8994:
8990:
8982:
8979:
8977:
8974:
8972:
8971:constructible
8969:
8968:
8967:
8964:
8962:
8959:
8957:
8954:
8952:
8949:
8947:
8944:
8942:
8939:
8937:
8934:
8932:
8929:
8927:
8924:
8922:
8919:
8917:
8914:
8912:
8909:
8907:
8904:
8903:
8901:
8899:
8894:
8886:
8883:
8881:
8878:
8876:
8873:
8871:
8868:
8866:
8863:
8861:
8858:
8857:
8855:
8851:
8848:
8846:
8843:
8842:
8841:
8838:
8836:
8833:
8831:
8828:
8826:
8823:
8821:
8817:
8813:
8811:
8808:
8804:
8801:
8800:
8799:
8796:
8795:
8792:
8789:
8787:
8783:
8773:
8770:
8768:
8765:
8763:
8760:
8758:
8755:
8753:
8750:
8748:
8745:
8741:
8738:
8737:
8736:
8733:
8729:
8724:
8723:
8722:
8719:
8718:
8716:
8714:
8710:
8702:
8699:
8697:
8694:
8692:
8689:
8688:
8687:
8684:
8682:
8679:
8677:
8674:
8672:
8669:
8667:
8664:
8662:
8659:
8657:
8654:
8653:
8651:
8649:
8648:Propositional
8645:
8639:
8636:
8634:
8631:
8629:
8626:
8624:
8621:
8619:
8616:
8614:
8611:
8607:
8604:
8603:
8602:
8599:
8597:
8594:
8592:
8589:
8587:
8584:
8582:
8579:
8577:
8576:Logical truth
8574:
8572:
8569:
8568:
8566:
8564:
8560:
8557:
8555:
8551:
8545:
8542:
8540:
8537:
8535:
8532:
8530:
8527:
8525:
8522:
8520:
8516:
8512:
8508:
8506:
8503:
8501:
8498:
8496:
8492:
8489:
8488:
8486:
8484:
8478:
8473:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8447:
8444:
8442:
8439:
8437:
8434:
8432:
8429:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8405:
8402:
8401:
8400:
8397:
8396:
8394:
8390:
8386:
8379:
8374:
8372:
8367:
8365:
8360:
8359:
8356:
8344:
8336:
8334:
8331:
8330:
8327:
8321:
8318:
8316:
8313:
8310:
8306:
8300:
8296:
8291:
8289:
8286:
8284:
8283:Fuzzy numbers
8281:
8279:
8276:
8275:
8273:
8269:
8263:
8260:
8258:
8255:
8253:
8250:
8248:
8245:
8243:
8240:
8238:
8235:
8233:
8230:
8226:
8223:
8222:
8221:
8218:
8216:
8213:
8211:
8208:
8207:
8205:
8203:
8199:
8195:
8187:
8184:
8182:
8179:
8177:
8174:
8173:
8172:
8168:
8165:
8163:
8160:
8158:
8155:
8130:
8127:
8125:
8122:
8120:
8117:
8115:
8112:
8110:
8107:
8106:
8104:
8102:
8097:
8091:
8088:
8086:
8085:Biquaternions
8083:
8081:
8078:
8052:
8049:
8047:
8044:
8042:
8039:
8014:
8013:
8011:
8005:
7976:
7973:
7948:
7945:
7920:
7917:
7892:
7888:
7885:
7884:
7882:
7880:
7876:
7870:
7867:
7863:
7860:
7859:
7858:
7855:
7853:
7850:
7848:
7845:
7843:
7840:
7813:
7810:
7808:
7805:
7780:
7777:
7775:
7772:
7747:
7744:
7719:
7716:
7691:
7688:
7687:
7685:
7683:
7678:
7673:
7666:
7661:
7659:
7654:
7652:
7647:
7646:
7643:
7636:
7631:
7626:
7624:
7614:
7613:
7610:
7601:
7595:
7591:
7587:
7583:
7579:
7574:
7571:
7569:0-07-054235-X
7565:
7561:
7557:
7556:Rudin, Walter
7553:
7550:
7548:0-387-94001-4
7544:
7540:
7536:
7532:
7529:
7523:
7519:
7515:
7511:
7508:
7504:
7500:
7499:0-387-90092-6
7496:
7490:
7489:
7484:
7480:
7477:
7475:0-87150-164-3
7471:
7467:
7462:
7459:
7453:
7449:
7444:
7441:
7437:
7433:
7429:
7425:
7421:
7417:
7412:
7409:
7407:0-673-38152-8
7403:
7399:
7394:
7391:
7385:
7380:
7379:
7374:(June 1969),
7373:
7369:
7368:
7355:
7349:
7345:
7344:
7336:
7330:, p. 187
7329:
7324:
7318:, p. 180
7317:
7312:
7305:
7300:
7294:, p. 182
7293:
7288:
7281:
7276:
7269:
7264:
7256:
7252:
7248:
7242:
7233:
7224:
7217:
7212:
7210:9781439865507
7206:
7202:
7201:
7196:
7190:
7184:, p. 182
7183:
7178:
7170:
7164:
7160:
7159:
7151:
7144:
7139:
7132:
7127:
7119:
7113:
7110:. CRC Press.
7109:
7108:
7100:
7098:
7089:
7083:
7079:
7078:
7070:
7062:
7056:
7052:
7051:
7043:
7041:
7039:
7030:
7024:
7020:
7019:
7011:
7003:
6997:
6993:
6992:
6984:
6977:
6972:
6965:
6960:
6958:
6950:
6945:
6939:, p. 181
6938:
6933:
6926:
6921:
6913:
6907:
6903:
6902:
6894:
6890:
6876:
6872:
6866:
6860:is countable.
6847:
6827:
6816:
6813:
6810:
6807:
6787:
6767:
6747:
6741:
6738:
6735:
6715:
6704:
6701:
6681:
6661:
6641:
6613:
6610:
6607:
6604:
6601:
6581:
6575:
6572:
6569:
6541:
6538:
6535:
6515:
6507:
6502:
6481:
6477:
6471:
6468:
6465:
6461:
6449:
6441:
6438:
6416:
6412:
6367:
6363:
6355:a surjection
6320:
6312:
6291:
6288:
6280:
6275:
6269:is countable.
6254:
6250:
6244:
6241:
6238:
6234:
6208:
6205:
6182:
6174:
6170:
6166:
6160:
6157:
6154:
6148:
6128:
6123:
6119:
6113:
6110:
6107:
6103:
6091:
6088:
6085:
6082:
6060:
6056:
6044:
6039:
6035:
6014:
5991:
5988:
5985:
5982:
5979:
5973:
5970:
5950:
5928:
5924:
5915:
5910:
5903:
5887:
5865:
5861:
5836:
5832:
5824:
5821:
5818:
5814:
5808:
5801:
5797:
5792:
5788:
5781:
5777:
5772:
5768:
5761:
5757:
5752:
5748:
5743:
5740:
5737:
5733:
5729:
5723:
5717:
5684:
5681:
5661:
5641:
5619:
5615:
5609:
5605:
5601:
5598:
5595:
5590:
5586:
5580:
5576:
5572:
5567:
5563:
5557:
5553:
5549:
5544:
5540:
5534:
5530:
5509:
5501:
5496:
5450:
5422:
5419:
5416:
5409:
5405:
5402:
5396:
5393:
5390:
5384:
5351:
5343:
5340:
5298:
5276:
5273:
5269:
5265:
5259:
5253:
5233:
5211:
5207:
5203:
5197:
5191:
5158:
5155:
5126:The integers
5125:
5120:
5113:
5097:
5094:
5091:
5071:
5068:
5065:
5040:
5015:
5012:
5009:
4998:
4990:
4987:
4984:
4981:
4961:
4950:
4947:
4927:
4916:
4913:
4893:
4873:
4851:
4847:
4843:
4838:
4834:
4830:
4824:
4821:
4818:
4812:
4779:
4771:
4768:
4743:
4731:Observe that
4730:
4725:
4718:
4714:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4671:
4666:
4629:
4623:
4621:
4616:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4587:
4581:
4579:
4578:least element
4575:
4571:
4570:least element
4561:
4558:
4557:
4555:
4551:
4546:
4543:
4542:
4540:
4536:
4533:
4532:
4531:
4529:
4519:
4517:
4513:
4508:
4506:
4498:
4494:
4491:
4487:
4486:
4485:
4483:
4479:
4475:
4471:
4461:
4459:
4455:
4450:
4448:
4442:
4440:
4393:
4391:
4372:
4342:
4322:
4314:
4295:
4278:is a set and
4265:
4257:
4256:
4251:
4245:is countable.
4232:
4212:
4192:
4186:
4183:
4180:
4172:
4170:is countable.
4157:
4137:
4117:
4111:
4108:
4105:
4097:
4096:
4095:
4081:
4061:
4045:
4041:
4039:
4027:
4022:
4020:
4008:
3994:
3991:
3981:
3971:
3956:
3952:
3947:
3928:
3919:
3902:
3899:
3896:
3888:
3879:
3862:
3859:
3856:
3848:
3839:
3822:
3819:
3816:
3808:
3799:
3782:
3779:
3776:
3768:
3759:
3742:
3739:
3736:
3728:
3719:
3702:
3699:
3696:
3688:
3679:
3662:
3659:
3656:
3648:
3639:
3622:
3619:
3616:
3608:
3599:
3582:
3579:
3576:
3568:
3559:
3542:
3539:
3536:
3528:
3519:
3502:
3499:
3496:
3488:
3441:
3438:
3428:
3418:
3398:
3393:
3391:
3379:
3374:
3372:
3360:
3343:
3340:
3337:
3314:
3310:
3306:
3298:
3282:
3262:
3242:
3222:
3213:
3211:
3205:
3152:
3138:
3134:
3130:
3110:
3090:
3087:
3084:
3064:
3044:
3040:
3036:
3028:
3024:
3020:
2973:
2948:
2942:
2940:
2928:
2914:
2872:
2849:
2846:
2843:
2817:
2814:
2811:
2785:
2782:
2779:
2753:
2750:
2744:
2741:
2738:
2725:maps to 5 so
2709:
2706:
2703:
2677:
2674:
2668:
2665:
2662:
2633:
2630:
2627:
2624:
2621:
2598:
2578:
2556:
2552:
2526:
2522:
2518:
2515:
2512:
2507:
2503:
2499:
2494:
2490:
2486:
2481:
2477:
2465:
2449:
2441:
2436:
2422:
2419:
2413:
2410:
2407:
2398:
2395:
2389:
2386:
2383:
2374:
2371:
2365:
2362:
2359:
2350:
2347:
2341:
2338:
2335:
2326:
2323:
2317:
2314:
2311:
2302:
2299:
2293:
2290:
2287:
2278:
2275:
2269:
2266:
2263:
2254:
2246:
2244:
2236:
2231:
2227:
2208:
2195:
2191:
2190:ordered pairs
2185:
2175:
2157:
2152:
2149:
2143:
2138:
2135:
2129:
2124:
2121:
2115:
2110:
2107:
2101:
2096:
2093:
2087:
2080:
2075:
2072:
2066:
2061:
2058:
2052:
2047:
2044:
2038:
2033:
2030:
2024:
2019:
2016:
2010:
1986:
1983:
1977:
1957:
1954:
1951:
1945:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1892:
1889:
1881:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1838:
1835:
1793:
1785:
1769:
1760:
1746:
1738:
1719:
1716:
1713:
1710:
1707:
1696:
1695:precisely one
1677:
1674:
1671:
1668:
1665:
1659:
1656:
1636:
1630:
1624:
1621:
1615:
1609:
1606:
1600:
1577:
1574:
1571:
1568:
1565:
1562:
1559:
1553:
1541:
1525:
1517:
1513:
1497:
1483:
1480:
1476:
1460:
1447:
1442:
1438:
1437:
1432:
1416:
1413:
1407:
1387:
1383:
1375:
1371:
1368:
1360:
1356:
1353:
1345:
1341:
1338:
1334:
1326:
1322:
1318:
1315:
1311:
1303:
1299:
1295:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1236:
1213:
1191:
1187:
1183:
1180:
1160:
1140:
1120:
1112:
1103:
1099:
1085:
1065:
1057:
1053:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
984:
981:
978:
975:
972:
961:
957:
956:
945:
943:
938:
934:
924:
908:
895:
894:
873:
853:
850:
847:
825:
821:
798:
794:
773:
770:
765:
761:
757:
752:
748:
744:
739:
735:
714:
706:
681:
665:
658:
621:
613:
609:
592:
562:
549:
548:
547:
545:
544:
526:
517:
497:
489:
481:
468:
452:
445:
408:
400:
396:
381:
351:
335:
328:
291:
283:
279:
255:
251:
233:
203:
191:
187:
186:
185:
183:
167:
153:
151:
147:
143:
138:
136:
132:
122:
120:
116:
112:
107:
105:
101:
97:
93:
88:
85:
81:
77:
73:
69:
65:
61:
57:
49:
42:
38:
34:
19:
10704:Georg Cantor
10699:Paul Bernays
10630:MorseâKelley
10605:
10538:
10537:Subset
10484:hereditarily
10469:
10446:Venn diagram
10404:ordered pair
10319:Intersection
10263:Axiom schema
10093:
9891:Ultraproduct
9738:Model theory
9703:Independence
9639:Formal proof
9631:Proof theory
9614:
9587:
9544:real numbers
9516:second-order
9427:Substitution
9304:Metalanguage
9245:conservative
9218:Axiom schema
9162:Constructive
9132:MorseâKelley
9098:Set theories
9077:Aleph number
9070:inaccessible
8976:Grothendieck
8905:
8860:intersection
8747:Higher-order
8735:Second-order
8681:Truth tables
8638:Venn diagram
8421:Formal proof
8304:
8294:
8109:Dual numbers
8101:hypercomplex
7891:Real numbers
7581:
7559:
7538:
7517:
7514:Kamke, Erich
7487:
7465:
7447:
7423:
7419:
7397:
7377:
7342:
7335:
7323:
7311:
7299:
7287:
7282:, p. 92
7275:
7270:, p. 91
7263:
7250:
7241:
7232:
7223:
7214:
7199:
7189:
7177:
7157:
7150:
7138:
7126:
7106:
7076:
7069:
7049:
7017:
7010:
6990:
6983:
6976:Apostol 1969
6971:
6944:
6932:
6920:
6900:
6893:
6865:
6505:
6501:
6278:
6274:
5913:
5909:
5499:
5495:
5123:
5119:
4728:
4724:
4590:Aleph number
4577:
4573:
4569:
4567:
4553:
4525:
4522:Total orders
4509:
4502:
4496:
4489:
4481:
4467:
4454:real numbers
4451:
4444:
4395:
4253:
4252:
4249:
4047:
4043:
4029:
4024:
4010:
3954:
3949:We need the
3948:
3403:
3381:
3376:
3362:
3214:
3207:
3154:
2944:
2930:
2437:
2247:
2240:
2187:
2177:
1761:
1736:
1694:
1511:
1489:
1479:Georg Cantor
1477:
1445:
1434:
1430:
1173:is, such as
1110:
1108:
1055:
959:
953:
951:
948:Introduction
937:real numbers
931:In 1874, in
930:
891:
889:
540:
518:
515:
181:
159:
145:
141:
139:
134:
130:
128:
119:real numbers
111:Georg Cantor
108:
103:
95:
89:
79:
63:
53:
10729:Thomas Jech
10572:Alternative
10551:Uncountable
10505:Ultrafilter
10364:Cardinality
10268:replacement
10216:Determinacy
10001:Type theory
9949:undecidable
9881:Truth value
9768:equivalence
9447:non-logical
9060:Enumeration
9050:Isomorphism
8997:cardinality
8981:Von Neumann
8946:Ultrafilter
8911:Uncountable
8845:equivalence
8762:Quantifiers
8752:Fixed-point
8721:First-order
8601:Consistency
8586:Proposition
8563:Traditional
8534:Lindström's
8524:Compactness
8466:Type theory
8411:Cardinality
8271:Other types
8090:Bioctonions
7947:Quaternions
7635:Mathematics
7535:Lang, Serge
7280:Halmos 1960
7268:Halmos 1960
7131:Halmos 1960
6951:, p. 2
6927:, Chapter 2
6873:, and also
6728:. Then if
6562:. Then if
5058:to the set
4535:Well-orders
4488:subsets of
4470:inner model
4452:The set of
4439:uncountable
4397:Proposition
3369:Any finite
3023:denominator
2440:recursively
1806:and all of
1697:element of
893:uncountable
580:is exactly
190:cardinality
146:denumerable
100:cardinality
94:, a set is
56:mathematics
10769:Categories
10724:Kurt Gödel
10709:Paul Cohen
10546:Transitive
10314:Identities
10298:Complement
10285:Operations
10246:Regularity
10184:Adjunction
10143:Set theory
9812:elementary
9505:arithmetic
9373:Quantifier
9351:functional
9223:Expression
8941:Transitive
8885:identities
8870:complement
8803:hereditary
8786:Set theory
8225:Projective
8198:Infinities
7623:Arithmetic
7582:Analysis I
7365:References
7304:Kamke 1950
7143:Kamke 1950
6949:Kamke 1950
6925:Rudin 1976
4537:(see also
4507:for more.
1999:, so that
886:is listed.
541:countably
465:is either
250:aleph-null
156:Definition
142:enumerable
140:The terms
37:Count data
33:Count noun
10657:Paradoxes
10577:Axiomatic
10556:Universal
10532:Singleton
10527:Recursive
10470:Countable
10465:Amorphous
10324:Power set
10241:Power set
10199:dependent
10194:countable
10083:Supertask
9986:Recursion
9944:decidable
9778:saturated
9756:of models
9679:deductive
9674:axiomatic
9594:Hilbert's
9581:Euclidean
9562:canonical
9485:axiomatic
9417:Signature
9346:Predicate
9235:Extension
9157:Ackermann
9082:Operation
8961:Universal
8951:Recursive
8926:Singleton
8921:Inhabited
8906:Countable
8896:Types of
8880:power set
8850:partition
8767:Predicate
8713:Predicate
8628:Syllogism
8618:Soundness
8591:Inference
8581:Tautology
8483:paradoxes
8309:solenoids
8129:Sedenions
7975:Octonions
7440:123695365
6964:Lang 1993
6885:Citations
6825:→
6811:∘
6745:→
6713:→
6617:→
6605:∘
6579:→
6545:→
6469:∈
6462:⋃
6458:→
6450:×
6242:∈
6235:⋃
6209:×
6141:given by
6111:∈
6104:⋃
6100:→
6092:×
6053:→
5986:…
5809:⋯
5789:⋅
5769:⋅
5749:⋅
5741:−
5724:α
5710:given by
5693:→
5642:α
5599:⋯
5510:α
5451:×
5377:given by
5360:→
5352:×
5274:−
5184:given by
5167:→
5112:induction
5095:×
5069:×
5041:×
5013:×
5007:→
4999:×
4985:×
4959:→
4925:→
4844:⋅
4805:given by
4788:→
4780:×
4744:×
4713:ISO 31-11
4696:…
4667:∗
4628:bijection
4458:sequences
4313:power set
4190:→
4115:→
4094:be sets.
4019:sequences
3995:…
3957:the sets
3953:to index
3929:⋮
3442:…
3088:≠
3019:numerator
2516:…
2423:…
2402:↔
2378:↔
2354:↔
2330:↔
2306:↔
2282:↔
2258:↔
2209:×
2158:…
2147:↔
2133:↔
2119:↔
2105:↔
2091:↔
2081:…
2070:↔
2056:↔
2042:↔
2028:↔
2014:↔
1981:↔
1949:↔
1923:…
1863:…
1784:bijection
1634:↔
1619:↔
1604:↔
1578:…
1516:bijection
1512:countable
1457:ℵ
1436:bijection
1411:→
1388:⋯
1380:→
1365:→
1350:→
1335:−
1331:→
1323:−
1312:−
1308:→
1300:−
1296:…
1273:…
1029:…
905:ℵ
890:A set is
851:≠
774:…
612:bijective
589:ℵ
494:ℵ
399:bijective
230:ℵ
182:countable
131:countable
96:countable
80:countable
64:countable
10785:Infinity
10661:Problems
10565:Theories
10541:Superset
10517:Infinite
10346:Concepts
10226:Infinity
10150:Overview
10068:Logicism
10061:timeline
10037:Concrete
9896:Validity
9866:T-schema
9859:Kripke's
9854:Tarski's
9849:semantic
9839:Strength
9788:submodel
9783:spectrum
9751:function
9599:Tarski's
9588:Elements
9575:geometry
9531:Robinson
9452:variable
9437:function
9410:spectrum
9400:Sentence
9356:variable
9299:Language
9252:Relation
9213:Automata
9203:Alphabet
9187:language
9041:-jection
9019:codomain
9005:Function
8966:Universe
8936:Infinite
8840:Relation
8623:Validity
8613:Argument
8511:theorem,
7718:Integers
7680:Sets of
7558:(1976),
7537:(1993),
7516:(1950),
7485:(1960),
7255:Archived
7197:(2010),
7182:Tao 2016
6937:Tao 2016
6901:Topology
5853:, where
4630:between
4595:Counting
4584:See also
4402:The set
3327:maps to
2947:integers
2769:maps to
1880:integers
1786:between
1518:between
1441:function
1111:infinite
1056:possible
1052:integers
960:elements
942:ordinals
786:, where
543:infinite
10599:General
10594:Zermelo
10500:subbase
10482: (
10421:Forcing
10399:Element
10371: (
10349:Methods
10236:Pairing
10010:Related
9807:Diagram
9705: (
9684:Hilbert
9669:Systems
9664:Theorem
9542:of the
9487:systems
9267:Formula
9262:Grammar
9178: (
9122:General
8835:Forcing
8820:Element
8740:Monadic
8515:paradox
8456:Theorem
8392:General
8299:numbers
8131: (
7977: (
7949: (
7921: (
7893: (
7814: (
7812:Periods
7781: (
7748: (
7720: (
7692: (
7674:systems
7609:Portals
7133:, p. 91
5880:is the
5654:is the
4552:Other (
4476:). The
4311:is its
4049:Theorem
4038:subsets
4031:Theorem
4012:Theorem
3481:Element
3383:Theorem
3364:Theorem
3255:, then
3156:Theorem
2932:Theorem
2801:, then
2693:. Then
2243:mapping
2179:Theorem
927:History
98:if its
10490:Filter
10480:Finite
10416:Family
10359:Almost
10204:global
10189:Choice
10176:Axioms
9773:finite
9536:Skolem
9489:
9464:Theory
9432:Symbol
9422:String
9405:atomic
9282:ground
9277:closed
9272:atomic
9228:ground
9191:syntax
9087:binary
9014:domain
8931:Finite
8696:finite
8554:Logics
8513:
8461:Theory
8099:Other
7672:Number
7596:
7566:
7545:
7524:
7505:
7497:
7472:
7454:
7438:
7404:
7386:
7350:
7207:
7165:
7145:, p. 2
7114:
7084:
7057:
7025:
6998:
6908:
5914:Proof:
5522:, let
5500:Proof:
5124:Proof:
4729:Proof:
3057:where
2544:where
2464:tuples
1882:, and
1628:
1613:
678:has a
519:A set
467:finite
160:A set
68:finite
10582:Naive
10512:Fuzzy
10475:Empty
10458:types
10409:tuple
10379:Class
10373:large
10334:Union
10251:Union
9763:Model
9511:Peano
9368:Proof
9208:Arity
9137:Naive
9024:image
8956:Fuzzy
8916:Empty
8865:union
8810:Class
8451:Model
8441:Lemma
8399:Axiom
8307:-adic
8297:-adic
8054:Over
8015:Over
8009:types
8007:Split
7436:S2CID
7251:expii
6506:Proof
6279:Proof
5902:prime
4974:. So
4611:Notes
3474:Tuple
3467:Index
3371:union
3025:of a
682:with
352:from
284:from
10495:base
9886:Type
9689:list
9493:list
9470:list
9459:Term
9393:rank
9287:open
9181:list
8993:Maps
8898:sets
8757:Free
8727:list
8477:list
8404:list
8343:List
8200:and
7594:ISBN
7564:ISBN
7543:ISBN
7522:ISBN
7503:ISBN
7495:ISBN
7470:ISBN
7452:ISBN
7424:1878
7402:ISBN
7384:ISBN
7348:ISBN
7205:ISBN
7163:ISBN
7112:ISBN
7082:ISBN
7055:ISBN
7023:ISBN
6996:ISBN
6906:ISBN
6869:See
6780:and
5900:-th
4940:and
4886:and
4652:and
4574:some
4074:and
4054:Let
3077:and
3021:and
2937:The
2571:and
2233:The
1970:and
1433:(or
1192:1000
840:for
634:and
546:if:
490:<
188:Its
184:if:
144:and
58:, a
10456:Set
9573:of
9555:of
9503:of
9035:Sur
9009:Map
8816:Ur-
8798:Set
7586:doi
7428:doi
6404:to
6333:in
5963:in
5916:If
5291:if
5226:if
4554:not
4541:):
4355:to
3955:all
1737:and
1510:is
1098:".
1035:100
955:set
539:is
374:to
304:to
180:is
62:is
60:set
54:In
10771::
9959:NP
9583::
9577::
9507::
9184:),
9039:Bi
9031:In
7889::
7592:.
7580:.
7434:,
7422:,
7418:,
7249:.
7213:,
7096:^
7037:^
6956:^
4619:^
4449:.
4441:.
3889:10
1735:,
1475:.
1188:10
952:A
923:.
152:.
121:.
106:.
10539:·
10523:)
10519:(
10486:)
10375:)
10135:e
10128:t
10121:v
10039:/
9954:P
9709:)
9495:)
9491:(
9388:â
9383:!
9378:â
9339:=
9334:â
9329:â
9324:â§
9319:âš
9314:ÂŹ
9037:/
9033:/
9007:/
8818:)
8814:(
8701:â
8691:3
8479:)
8377:e
8370:t
8363:v
8311:)
8305:p
8301:(
8295:p
8169:/
8153:)
8140:S
8076::
8063:C
8037::
8024:R
7999:)
7986:O
7971:)
7958:H
7943:)
7930:C
7915:)
7902:R
7838:)
7824:P
7803:)
7790:A
7770:)
7757:Q
7742:)
7729:Z
7714:)
7701:N
7664:e
7657:t
7650:v
7611::
7602:.
7588::
7430::
7356:.
7171:.
7120:.
7090:.
7063:.
7031:.
7004:.
6914:.
6848:T
6828:T
6821:N
6817::
6814:h
6808:g
6788:T
6768:S
6748:T
6742:S
6739::
6736:g
6716:S
6709:N
6705::
6702:h
6682:S
6662:S
6642:S
6621:N
6614:S
6611::
6608:f
6602:h
6582:T
6576:S
6573::
6570:f
6549:N
6542:T
6539::
6536:h
6516:T
6482:i
6478:A
6472:I
6466:i
6454:N
6446:N
6442::
6439:G
6417:i
6413:A
6391:N
6368:i
6364:g
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4988:g
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2279:1
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2017:1
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1920:,
1917:6
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1911:4
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1860:,
1857:3
1854:,
1851:2
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1575:,
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607:.
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567:|
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527:S
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486:|
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478:|
469:(
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443:.
430:N
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394:.
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361:N
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326:.
313:N
292:S
277:.
264:N
248:(
234:0
208:|
204:S
200:|
168:S
50:.
43:.
20:)
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