9632:
27:
7284:
4647:
2536:
4446:
2387:
4642:{\displaystyle \mathbb {N} {\begin{Bmatrix}1&\longleftrightarrow &\{4,5\}\\2&\longleftrightarrow &\{1,2,3\}\\3&\longleftrightarrow &\{4,5,6\}\\4&\longleftrightarrow &\{1,3,5\}\\\vdots &\vdots &\vdots \end{Bmatrix}}{\mathcal {P}}(\mathbb {N} ).}
4127:
314:. For instance, by iteratively taking the power set of an infinite set and applying Cantor's theorem, we obtain an endless hierarchy of infinite cardinals, each strictly larger than the one before it. Consequently, the theorem implies that there is no largest
2531:{\displaystyle {\begin{aligned}\xi \in B&\iff \xi \notin f(\xi )&&{\text{(by definition of }}B{\text{)}};\\\xi \in B&\iff \xi \in f(\xi )&&{\text{(by assumption that }}f(\xi )=B{\text{)}}.\\\end{aligned}}}
6228:
has a theorem (which he calls "Cantor's
Theorem") that is identical to the form above in the paper that became the foundation of modern set theory ("Untersuchungen ĂŒber die Grundlagen der Mengenlehre I"), published in 1908. See
2856:
1202:
519:
6036:
1256:
5806:
4259:
3950:
931:
5896:
2392:
6097:
3836:
1682:
896:
605:
5460:
5421:
5360:
5299:
5260:
5192:
5157:
5059:
4735:
4438:
4381:
4320:
4164:
3942:
3907:
5718:
5586:
3283:
1096:
452:
648:
5639:
3429:
1051:
407:
6624:
6540:
6458:
6502:
6394:
6362:
6318:
6274:
6697:
5486:
5386:
5325:
5225:
5024:
4403:
4346:
4282:
4186:
3865:
2265:
2181:
2001:
1772:
674:
4656:
that contain the very same number. For instance, in our example the number 2 is paired with the subset {1, 2, 3}, which contains 2 as a member. Let us call such numbers
1137:
278:
3651:
3550:
3201:
2604:
2569:
2375:
2340:
2146:
2120:
2036:
1966:
966:
759:
724:
6426:
6723:
6650:
6566:
3576:
3495:
2291:
1931:
1737:
1711:
1544:
2773:
245:
2210:
2065:
1841:
1573:
1518:
847:
159:
132:
6586:
6478:
6338:
6294:
5659:
5606:
5527:
5122:
5102:
5082:
4998:
4975:
4955:
4935:
4915:
4895:
4875:
4855:
4835:
4815:
4795:
4775:
4755:
4693:
3770:
3719:
3699:
3675:
3616:
3596:
3515:
3469:
3449:
3371:
3351:
3327:
3303:
3225:
3166:
3143:
3123:
3103:
3083:
3063:
3043:
3023:
3003:
2983:
2963:
2943:
2919:
2899:
2879:
2793:
2747:
2727:
2704:
2684:
2664:
2644:
2624:
2311:
2230:
2085:
1905:
1881:
1861:
1812:
1792:
1617:
1593:
1489:
1469:
1449:
1429:
1409:
1388:
1367:
1345:
1324:
1296:
1276:
1018:
988:
784:
696:
539:
374:
354:
218:
183:
105:
4664:
that do not contain them. For instance, in our example the number 1 is paired with the subset {4, 5}, which does not contain the number 1. Call these numbers
3207:. For a countable (or finite) set, the argument of the proof given above can be illustrated by constructing a table in which each row is labelled by a unique
6126:). The version of this argument he gave in that paper was phrased in terms of indicator functions on a set rather than subsets of a set. He showed that if
2798:
1144:
461:
9671:
8011:
5529:. In order to distinguish this paradox from the next one discussed below, it is important to note what this contradiction is. By Cantor's theorem
8686:
7748:
5506:
is the name given to a contradiction following from Cantor's theorem together with the assumption that there is a set containing all sets, the
5945:
6106:
emphasized that
Russell's paradox is independent of considerations of cardinality and its underlying notions like one-to-one correspondence.
8769:
7910:
1211:
310:, who first stated and proved it at the end of the 19th century. Cantor's theorem had immediate and important consequences for the
6114:
Cantor gave essentially this proof in a paper published in 1891 "Ăber eine elementare Frage der
Mannigfaltigkeitslehre", where the
5745:
9083:
326:
Cantor's argument is elegant and remarkably simple. The complete proof is presented below, with detailed explanations to follow.
4191:
4122:{\displaystyle {\mathcal {P}}(\mathbb {N} )=\{\varnothing ,\{1,2\},\{1,2,3\},\{4\},\{1,5\},\{3,4,6\},\{2,4,6,\dots \},\dots \}.}
6739:
6123:
287:
Much more significant is Cantor's discovery of an argument that is applicable to any set, and shows that the theorem holds for
9241:
7006:
6966:
6895:
6871:
6835:
6784:
6178:
than objects. "For suppose a correlation of all objects and some propositional functions to have been affected, and let phi-
8029:
7167:
907:
9096:
8419:
7437:
7250:
6776:
5830:
20:
6913:
7099:
6744:
68:
6052:
9101:
9091:
8828:
8681:
8034:
7765:
4440:, so that no element from either infinite set remains unpaired. Such an attempt to pair elements would look like this:
8025:
7162:
3783:
1625:
852:
9237:
7207:
7025:
6998:
3598:
constructed in the previous paragraphs coincides with the row labels for the subset of entries on this main diagonal
8579:
6734:
9334:
9078:
7903:
6241:
551:
5430:
5391:
5330:
5269:
5230:
5162:
5127:
5124:. Then, every number maps to a nonempty set and no number maps to the empty set. But the empty set is a member of
5029:
4705:
4408:
4351:
4290:
4134:
3912:
3877:
8639:
8332:
7743:
7337:
5664:
8073:
7623:
7192:
5532:
3740:
could, albeit with a certain amount of direction in terms of tactics that might perhaps be considered cheating.
9595:
9297:
9060:
9055:
8880:
8301:
7985:
4383:
to show that these infinite sets are equinumerous. In other words, we will attempt to pair off each element of
3230:
6102:
Despite the syntactical similarities between the
Russell set (in either variant) and the Cantor diagonal set,
1056:
412:
9590:
9373:
9290:
9003:
8934:
8811:
8053:
7517:
7396:
7051:
6115:
5923:
3204:
1596:
608:
614:
9515:
9341:
9027:
8661:
8260:
7760:
5905:. As a point of subtlety, the version of Russell's paradox we have presented here is actually a theorem of
300:
9656:
9393:
9388:
8998:
8737:
8666:
7995:
7896:
7753:
7391:
7354:
7217:
7046:
6099:, then the axiom system itself would have entailed the contradiction, with no further hypotheses needed.
9322:
8912:
8306:
8274:
7965:
7212:
7150:
6042:
5611:
3777:
3737:
3376:
1023:
379:
7408:
6591:
6507:
6431:
9676:
9612:
9561:
9458:
8956:
8917:
8394:
8039:
7442:
7327:
7315:
7310:
6249:
6170:
5922:, thus disproving the existence of a set containing all sets. This was possible because we have used
311:
8068:
7041:
6483:
6367:
6343:
6299:
6255:
9661:
9453:
9383:
8922:
8774:
8757:
8480:
7960:
7243:
7135:
6655:
5811:
The proof of Cantor's theorem is straightforwardly adapted to show that assuming a set of all sets
3725:
6863:
5469:
5369:
5308:
5208:
5007:
4386:
4329:
4267:
4169:
3848:
2235:
2151:
1971:
1742:
1471:) can reach every possible subset, i.e., we just need to demonstrate the existence of a subset of
653:
9285:
9262:
9223:
9109:
9050:
8696:
8616:
8460:
8404:
8017:
7862:
7780:
7655:
7607:
7421:
7344:
7155:
7092:
6985:
5497:
5363:
899:
64:
58:
5723:
Another paradox can be derived from the proof of Cantor's theorem by instantiating the function
1101:
250:
9575:
9302:
9280:
9247:
9140:
8986:
8971:
8944:
8895:
8779:
8714:
8539:
8505:
8500:
8374:
8205:
8182:
7814:
7695:
7507:
7320:
7187:
6245:
6175:
5198:
4672:
3733:
3621:
3520:
3171:
2574:
2548:
2345:
2319:
2125:
2090:
2006:
1936:
936:
729:
703:
47:
6399:
3728:
to produce it. The main difficulty lies in an automated discovery of the Cantor diagonal set.
9681:
9505:
9358:
9150:
8868:
8604:
8510:
8369:
8354:
8235:
8210:
7730:
7700:
7644:
7564:
7544:
7522:
281:
6855:
6702:
6629:
6545:
6202:" is a propositional function not contained in this correlation; for it is true or false of
3555:
3474:
2270:
1910:
1716:
1690:
1523:
9478:
9440:
9317:
9121:
8961:
8885:
8863:
8691:
8649:
8548:
8515:
8379:
8167:
8078:
7804:
7794:
7628:
7559:
7512:
7452:
7332:
7182:
7177:
7129:
5902:
2775:. The argument is now complete, and we have established the strict inequality for any set
2752:
2542:
991:
223:
6804:
5909:; we can conclude from the contradiction obtained that we must reject the hypothesis that
4671:
Using this idea, let us build a special set of natural numbers. This set will provide the
2186:
2041:
1817:
1549:
1494:
8:
9666:
9607:
9498:
9483:
9463:
9420:
9307:
9257:
9183:
9128:
9065:
8858:
8853:
8801:
8569:
8558:
8230:
8130:
8058:
8049:
7980:
7975:
7799:
7618:
7613:
7427:
7369:
7300:
7236:
6856:
5503:
3701:
does not agree with any column in at least one entry. Consequently, no column represents
2922:
1884:
791:
3309:
so that such table can be constructed. Each column of the table is labelled by a unique
141:
114:
9636:
9405:
9368:
9353:
9346:
9329:
9133:
9115:
8981:
8907:
8890:
8843:
8656:
8565:
8399:
8384:
8344:
8296:
8281:
8269:
8225:
8200:
7970:
7919:
7722:
7717:
7502:
7457:
7364:
7085:
6571:
6463:
6323:
6279:
6230:
5644:
5591:
5512:
5107:
5087:
5067:
4983:
4960:
4940:
4920:
4900:
4880:
4860:
4857:
is selfish because it is in the corresponding set, which contradicts the definition of
4840:
4820:
4800:
4780:
4760:
4740:
4678:
3773:
3755:
3704:
3684:
3681:(swap "true" and "false") entries of the main diagonal. Thus the indicator function of
3660:
3654:
3601:
3581:
3500:
3454:
3434:
3356:
3336:
3312:
3288:
3210:
3151:
3128:
3108:
3088:
3068:
3048:
3028:
3008:
2988:
2968:
2948:
2928:
2904:
2884:
2864:
2778:
2732:
2712:
2689:
2669:
2649:
2629:
2609:
2296:
2215:
2070:
1890:
1866:
1846:
1797:
1777:
1602:
1578:
1474:
1454:
1434:
1414:
1394:
1373:
1352:
1330:
1309:
1303:
1299:
1281:
1261:
1003:
997:
973:
769:
681:
524:
359:
339:
203:
168:
90:
8589:
1391:. This is the heart of Cantor's theorem: there is no surjective function from any set
9631:
9571:
9378:
9188:
9178:
9070:
8951:
8786:
8762:
8543:
8527:
8432:
8409:
8286:
8255:
8220:
8115:
7950:
7579:
7416:
7379:
7349:
7273:
7060:
7021:
7002:
6994:
6962:
6955:
6891:
6888:
Proceedings of the Tarski
Symposium. Proceedings of Symposia in Pure Mathematics XXV,
6867:
6831:
6780:
5728:
85:
7063:
6943:", p.276. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.
2851:{\displaystyle \operatorname {card} (A)<\operatorname {card} ({\mathcal {P}}(A))}
2316:
Equivalently, and slightly more formally, we have just proved that the existence of
1197:{\displaystyle \operatorname {card} (A)<\operatorname {card} ({\mathcal {P}}(A))}
514:{\displaystyle \operatorname {card} (A)<\operatorname {card} ({\mathcal {P}}(A))}
9585:
9580:
9473:
9430:
9252:
9213:
9208:
9193:
9019:
8976:
8873:
8671:
8621:
8195:
8157:
7867:
7857:
7842:
7837:
7705:
7359:
7145:
6165:
3729:
3678:
42:}, is three, while there are eight elements in its power set (3 < 2 = 8), here
9566:
9556:
9510:
9493:
9448:
9410:
9312:
9232:
9039:
8966:
8939:
8927:
8833:
8747:
8721:
8676:
8644:
8445:
8247:
8190:
8140:
8105:
8063:
7736:
7674:
7492:
7305:
7112:
6990:
315:
26:
7077:
6928:
Georg Cantor, Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
6890:
ed. L. Henkin, Providence RI, Second printing with additions 1979, pp. 297â308.
9551:
9530:
9488:
9468:
9363:
9218:
8816:
8806:
8796:
8791:
8725:
8599:
8475:
8364:
8359:
8337:
7938:
7872:
7669:
7650:
7554:
7539:
7496:
7432:
7374:
7202:
4323:
3839:
2709:
Finally, to complete the proof, we need to exhibit an injective function from
9650:
9525:
9203:
8710:
8495:
8485:
8455:
8440:
8110:
7877:
7679:
7593:
7588:
6225:
6103:
5507:
2378:
1595:, such a subset is given by the following construction, sometimes called the
7847:
9425:
9272:
9173:
9165:
9045:
8993:
8902:
8838:
8821:
8752:
8611:
8470:
8172:
7955:
7827:
7822:
7640:
7569:
7527:
7386:
7283:
3868:
1411:
to its power set. To establish this, it is enough to show that no function
307:
288:
43:
7020:, Springer Monographs in Mathematics (3rd millennium ed.), Springer,
3497:. Given the order chosen for the row and column labels, the main diagonal
9535:
9415:
8594:
8584:
8531:
8215:
8135:
8120:
8000:
7945:
7852:
7487:
7197:
7140:
7013:
6980:
6119:
5463:
5424:
5302:
5263:
5202:
3724:
Despite the simplicity of the above proof, it is rather difficult for an
3306:
292:
193:
162:
6031:{\displaystyle R_{U}\in R_{U}\iff (R_{U}\in U\wedge R_{U}\notin R_{U}).}
8465:
8320:
8291:
8097:
7832:
7603:
7259:
6940:
455:
189:
77:
9617:
9520:
8573:
8490:
8450:
8414:
8350:
8162:
8152:
8125:
7888:
7635:
7598:
7549:
7447:
7172:
7108:
7068:
6952:
5731:; this turns Cantor's diagonal set into what is sometimes called the
5001:
4700:
4262:
3872:
3330:
197:
135:
1251:{\displaystyle \operatorname {card} (X)<\operatorname {card} (Y)}
9602:
9400:
8848:
8553:
8147:
6396:
that does not have any fixed points; that is, there is no morphism
7001:(Springer-Verlag edition). Reprinted by Martino Fine Books, 2011.
4737:
contains all sets of natural numbers, and so it contains this set
3657:
of the set corresponding to the column. The indicator function of
9198:
7990:
5906:
5000:, we have contradicted our original supposition, that there is a
296:
5423:. Therefore, only one possibility remains, and that is that the
2729:
to its power set. Finding such a function is trivial: just map
7660:
7482:
6244:
provides for a broad generalization of Cantor's theorem to any
6124:
earlier proved the uncountability of the reals by other methods
4661:
4653:
299:, is strictly larger than the cardinality of the integers; see
108:
6049:'s system for instance) by defining the Russell set simply as
4917:, then it is non-selfish and it should instead be a member of
3085:, another contradiction. So the assumption that an element of
8742:
8088:
7933:
7532:
7292:
7228:
7058:
6989:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by
6046:
3005:
because that would contradict the criterion of membership in
5801:{\displaystyle R_{A}=\left\{\,x\in A:x\not \in x\,\right\}.}
4652:
Given such a pairing, some natural numbers are paired with
6957:
Conceptual
Mathematics: A First Introduction to Categories
6812:. University of Cambridge Computer Laboratory. p. 14.
5496:
Cantor's theorem and its proof are closely related to two
4980:
Since there is no natural number which can be paired with
4254:{\displaystyle \{2,4,6,\ldots \}=\{2k:k\in \mathbb {N} \}}
3065:
meaning that it satisfies the criterion for membership in
904:
The previous deduction yields a contradiction of the form
5927:
6914:"Ăber eine elementare Frage der Mannigfaltigskeitslehre"
6174:(1903, section 348), where he shows that there are more
5084:
may be empty. This would mean that every natural number
6853:
6222:." He attributes the idea behind the proof to Cantor.
5366:, by definition, and these singletons form a "copy" of
4460:
926:{\displaystyle \varphi \Leftrightarrow \lnot \varphi }
6705:
6658:
6632:
6594:
6574:
6548:
6510:
6486:
6466:
6434:
6402:
6370:
6346:
6326:
6302:
6282:
6258:
6055:
5948:
5891:{\displaystyle R_{U}\in R_{U}\iff R_{U}\notin R_{U}.}
5833:
5748:
5667:
5647:
5614:
5594:
5535:
5515:
5472:
5433:
5394:
5372:
5333:
5311:
5272:
5233:
5211:
5165:
5130:
5110:
5090:
5070:
5032:
5010:
4986:
4963:
4943:
4923:
4903:
4883:
4863:
4843:
4823:
4803:
4783:
4763:
4743:
4708:
4681:
4449:
4411:
4389:
4354:
4332:
4293:
4270:
4194:
4172:
4137:
3953:
3915:
3880:
3851:
3786:
3758:
3707:
3687:
3663:
3624:
3604:
3584:
3558:
3523:
3503:
3477:
3457:
3437:
3379:
3359:
3339:
3315:
3291:
3233:
3213:
3174:
3154:
3131:
3111:
3091:
3071:
3051:
3031:
3011:
2991:
2971:
2951:
2931:
2907:
2887:
2867:
2801:
2781:
2755:
2735:
2715:
2692:
2672:
2652:
2632:
2612:
2577:
2551:
2390:
2348:
2322:
2299:
2273:
2238:
2218:
2189:
2154:
2128:
2093:
2073:
2044:
2009:
1974:
1939:
1913:
1893:
1869:
1849:
1820:
1800:
1780:
1745:
1719:
1693:
1628:
1605:
1581:
1552:
1526:
1497:
1477:
1457:
1437:
1417:
1397:
1376:
1355:
1349:
It suffices to show that there is no surjection from
1333:
1312:
1284:
1264:
1214:
1147:
1104:
1059:
1026:
1006:
976:
939:
910:
855:
794:
772:
732:
706:
684:
656:
617:
554:
527:
464:
415:
382:
362:
342:
291:
sets also. As a consequence, the cardinality of the
253:
226:
206:
171:
144:
117:
93:
6918:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
3752:
Let us examine the proof for the specific case when
2881:, empty or non-empty, is always in the power set of
2666:
does not map onto every element of the power set of
295:, which is the same as that of the power set of the
192:, Cantor's theorem can be seen to be true by simple
6773:
3431:, ..., in this order. The intersection of each row
2965:. But that leads to a contradiction: no element of
84:is a fundamental result which states that, for any
6954:
6946:
6941:The Empty Set, the Singleton, and the Ordered Pair
6886:Church, A. "Set theory with a universal set." in
6862:. Springer Science & Business Media. pp.
6717:
6691:
6644:
6618:
6580:
6560:
6534:
6496:
6472:
6452:
6420:
6388:
6356:
6332:
6312:
6288:
6268:
6092:{\displaystyle R=\left\{\,x:x\not \in x\,\right\}}
6091:
6030:
5890:
5800:
5712:
5653:
5633:
5600:
5580:
5521:
5480:
5454:
5415:
5380:
5354:
5319:
5293:
5254:
5219:
5186:
5151:
5116:
5104:maps to a subset of natural numbers that contains
5096:
5076:
5053:
5018:
4992:
4969:
4949:
4929:
4909:
4889:
4869:
4849:
4829:
4809:
4789:
4769:
4749:
4729:
4687:
4641:
4432:
4397:
4375:
4340:
4314:
4276:
4253:
4180:
4158:
4121:
3936:
3901:
3859:
3830:
3764:
3713:
3693:
3669:
3645:
3610:
3590:
3570:
3544:
3509:
3489:
3463:
3443:
3423:
3365:
3345:
3321:
3297:
3277:
3219:
3195:
3160:
3137:
3117:
3097:
3077:
3057:
3037:
3017:
2997:
2977:
2957:
2937:
2913:
2893:
2873:
2850:
2787:
2767:
2741:
2721:
2698:
2678:
2658:
2638:
2618:
2598:
2563:
2530:
2369:
2334:
2305:
2285:
2259:
2224:
2204:
2175:
2140:
2114:
2079:
2059:
2030:
1995:
1960:
1925:
1899:
1875:
1855:
1835:
1806:
1786:
1766:
1731:
1705:
1676:
1611:
1587:
1567:
1538:
1512:
1483:
1463:
1443:
1423:
1403:
1382:
1361:
1339:
1318:
1290:
1270:
1250:
1196:
1131:
1090:
1045:
1012:
982:
960:
925:
890:
841:
778:
753:
718:
690:
668:
642:
599:
533:
513:
446:
401:
368:
348:
272:
239:
212:
177:
153:
126:
99:
7107:
4777:must be paired off with some natural number, say
4287:Now that we have an idea of what the elements of
2545:, the assumption must be false. Thus there is no
9648:
6953:F. William Lawvere; Stephen H. Schanuel (2009).
6849:
6847:
6802:
6770:
4699:non-selfish natural numbers. By definition, the
3831:{\displaystyle A=\mathbb {N} =\{1,2,3,\ldots \}}
3653:is false. Each column records the values of the
1677:{\displaystyle B=\{x\in A\mid x\not \in f(x)\}.}
891:{\displaystyle \xi \in B\iff \xi \notin f(\xi )}
6858:Ernst Zermelo: An Approach to His Life and Work
3743:
318:(colloquially, "there's no largest infinity").
6766:
6764:
6762:
6760:
19:For other theorems bearing Cantor's name, see
7904:
7244:
7093:
6844:
6830:. Oxford University Press. pp. 118â119.
6825:
6798:
6796:
3353:; the columns are ordered by the argument to
600:{\displaystyle B=\{x\in A\mid x\notin f(x)\}}
6821:
6819:
5455:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5416:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5355:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5294:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5255:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5187:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5152:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
5054:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4757:as an element. If the mapping is bijective,
4730:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4590:
4572:
4555:
4537:
4520:
4502:
4485:
4473:
4433:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4376:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4315:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4248:
4225:
4219:
4195:
4188:, e.g. the set of all positive even numbers
4159:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
4113:
4104:
4080:
4074:
4056:
4050:
4038:
4032:
4026:
4020:
4002:
3996:
3984:
3975:
3937:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
3902:{\displaystyle {\mathcal {P}}(\mathbb {N} )}
3825:
3801:
3272:
3240:
2762:
2756:
1668:
1635:
1126:
1120:
594:
561:
6757:
5713:{\displaystyle |{\mathcal {P}}(V)|\leq |V|}
9672:Theorems in the foundations of mathematics
8096:
7911:
7897:
7251:
7237:
7100:
7086:
6961:. Cambridge University Press. Session 29.
6793:
5976:
5972:
5861:
5857:
5581:{\displaystyle |{\mathcal {P}}(X)|>|X|}
2861:Another way to think of the proof is that
2471:
2467:
2412:
2408:
869:
865:
817:
813:
6816:
6083:
6067:
5815:exists, then considering its Russell set
5789:
5767:
5474:
5445:
5406:
5374:
5345:
5313:
5284:
5245:
5213:
5177:
5142:
5044:
5012:
4720:
4629:
4451:
4423:
4391:
4366:
4334:
4305:
4244:
4174:
4149:
3965:
3927:
3892:
3853:
3794:
3278:{\displaystyle A=\{x_{1},x_{2},\ldots \}}
4660:. Other natural numbers are paired with
1687:This means, by definition, that for all
1091:{\displaystyle g:A\to {\mathcal {P}}(A)}
447:{\displaystyle f:A\to {\mathcal {P}}(A)}
25:
6134:whose values are 2-valued functions on
5262:cannot be equal. We also know that the
247:subsets, and the theorem holds because
196:of the number of subsets. Counting the
69:question marks, boxes, or other symbols
16:Every set is smaller than its power set
9649:
7918:
6911:
6364:and that there exists an endomorphism
5159:, so the mapping still does not cover
4405:with an element from the infinite set
1208:By definition of cardinality, we have
643:{\displaystyle B\in {\mathcal {P}}(A)}
7892:
7232:
7081:
7059:
6777:Springer Science & Business Media
5608:. On the other hand, all elements of
4797:. However, this causes a problem. If
4668:. Likewise, 3 and 4 are non-selfish.
4322:are, let us attempt to pair off each
7012:
6214:, and therefore it differs from phi-
5491:
3517:of this table thus records whether
3148:Because of the double occurrence of
1907:did not include themselves. For all
6806:Set Theory as a Computational Logic
6740:Cantor's first uncountability proof
6568:. In other words, for every object
5939:above, which in turn entailed that
13:
6489:
6349:
6305:
6261:
6236:
5675:
5617:
5543:
5436:
5397:
5336:
5275:
5236:
5168:
5133:
5035:
4711:
4620:
4414:
4357:
4296:
4140:
3956:
3918:
3883:
2831:
1177:
1074:
1029:
917:
626:
494:
430:
385:
14:
9693:
7034:
5634:{\displaystyle {\mathcal {P}}(V)}
3978:
3471:records a true/false bit whether
3424:{\displaystyle f(x_{1}),f(x_{2})}
1863:was constructed from elements of
1046:{\displaystyle {\mathcal {P}}(A)}
402:{\displaystyle {\mathcal {P}}(A)}
21:Cantor's theorem (disambiguation)
9630:
7282:
6854:Heinz-Dieter Ebbinghaus (2007).
6745:Controversy over Cantor's theory
6699:must leave out at least one map
6619:{\displaystyle f:T\times T\to Y}
6535:{\displaystyle f:T\times T\to Y}
6453:{\displaystyle \alpha \circ y=y}
5641:are sets, and thus contained in
7168:Gödel's incompleteness theorems
6542:can parameterize all morphisms
1053:exist. For example, a function
7258:
6933:
6905:
6880:
6709:
6683:
6674:
6662:
6636:
6610:
6552:
6526:
6497:{\displaystyle {\mathcal {C}}}
6412:
6389:{\displaystyle \alpha :Y\to Y}
6380:
6357:{\displaystyle {\mathcal {C}}}
6313:{\displaystyle {\mathcal {C}}}
6269:{\displaystyle {\mathcal {C}}}
6118:for the uncountability of the
6022:
5977:
5973:
5858:
5706:
5698:
5690:
5686:
5680:
5669:
5628:
5622:
5574:
5566:
5558:
5554:
5548:
5537:
5449:
5441:
5410:
5402:
5349:
5341:
5288:
5280:
5249:
5241:
5181:
5173:
5146:
5138:
5048:
5040:
4724:
4716:
4633:
4625:
4567:
4532:
4497:
4468:
4427:
4419:
4370:
4362:
4309:
4301:
4153:
4145:
3969:
3961:
3931:
3923:
3909:. Let us see a sample of what
3896:
3888:
3640:
3634:
3539:
3533:
3418:
3405:
3396:
3383:
3190:
3184:
3025:, thus the element mapping to
2845:
2842:
2836:
2826:
2814:
2808:
2587:
2581:
2507:
2501:
2487:
2481:
2468:
2428:
2422:
2409:
2358:
2352:
2254:
2248:
2199:
2193:
2170:
2164:
2109:
2103:
2054:
2048:
2025:
2019:
1990:
1984:
1955:
1949:
1830:
1824:
1761:
1755:
1665:
1659:
1562:
1556:
1507:
1501:
1245:
1239:
1227:
1221:
1191:
1188:
1182:
1172:
1160:
1154:
1114:
1108:
1085:
1079:
1069:
1040:
1034:
949:
943:
914:
885:
879:
866:
836:
833:
827:
814:
801:
742:
736:
637:
631:
591:
585:
508:
505:
499:
489:
477:
471:
441:
435:
425:
396:
390:
1:
9591:History of mathematical logic
6750:
6692:{\displaystyle f(-,x):T\to Y}
6242:Lawvere's fixed-point theorem
6138:, then the 2-valued function
5462:is strictly greater than the
4937:. Therefore, no such element
4166:contains infinite subsets of
3618:where the table records that
3373:, i.e. the column labels are
609:axiom schema of specification
9516:Primitive recursive function
7163:Gödel's completeness theorem
6828:Beyond the Limits of Thought
6276:be such a category, and let
6168:has a very similar proof in
5824:leads to the contradiction:
5488:, proving Cantor's theorem.
5481:{\displaystyle \mathbb {N} }
5381:{\displaystyle \mathbb {N} }
5320:{\displaystyle \mathbb {N} }
5220:{\displaystyle \mathbb {N} }
5019:{\displaystyle \mathbb {N} }
4398:{\displaystyle \mathbb {N} }
4341:{\displaystyle \mathbb {N} }
4277:{\displaystyle \varnothing }
4181:{\displaystyle \mathbb {N} }
3860:{\displaystyle \mathbb {N} }
2260:{\displaystyle x\notin f(x)}
2176:{\displaystyle x\notin f(x)}
1996:{\displaystyle x\notin f(x)}
1767:{\displaystyle x\notin f(x)}
669:{\displaystyle B\subseteq A}
301:Cardinality of the continuum
30:The cardinality of the set {
7:
7047:Encyclopedia of Mathematics
6728:
6626:, an attempt to write maps
6122:also first appears (he had
1298:if and only if there is an
10:
9698:
8580:SchröderâBernstein theorem
8307:Monadic predicate calculus
7966:Foundations of mathematics
7749:von NeumannâBernaysâGödel
7151:Foundations of mathematics
6900:International Logic Review
6735:SchröderâBernstein theorem
6460:. Then there is no object
6252:in the following way: let
6109:
6043:unrestricted comprehension
5901:This argument is known as
3778:Without loss of generality
3045:must not be an element of
1132:{\displaystyle g(x)=\{x\}}
273:{\displaystyle 2^{n}>n}
18:
9626:
9613:Philosophy of mathematics
9562:Automated theorem proving
9544:
9439:
9271:
9164:
9016:
8733:
8709:
8687:Von NeumannâBernaysâGödel
8632:
8526:
8430:
8328:
8319:
8246:
8181:
8087:
8009:
7926:
7813:
7776:
7688:
7578:
7550:One-to-one correspondence
7466:
7407:
7291:
7280:
7266:
7119:
6803:Lawrence Paulson (1992).
6771:Abhijit Dasgupta (2013).
6171:Principles of Mathematics
6158:) is not in the range of
6130:is a function defined on
3736:could not do it, whereas
3646:{\displaystyle x\in f(x)}
3545:{\displaystyle x\in f(x)}
3196:{\displaystyle x\in f(x)}
2599:{\displaystyle f(\xi )=B}
2564:{\displaystyle \xi \in A}
2495:(by assumption that
2370:{\displaystyle f(\xi )=B}
2335:{\displaystyle \xi \in A}
2141:{\displaystyle x\notin B}
2115:{\displaystyle x\in f(x)}
2031:{\displaystyle x\in f(x)}
1961:{\displaystyle x\in f(x)}
961:{\displaystyle f(\xi )=B}
754:{\displaystyle f(\xi )=B}
719:{\displaystyle \xi \in A}
312:philosophy of mathematics
306:The theorem is named for
7193:LöwenheimâSkolem theorem
6421:{\displaystyle y:1\to Y}
6296:be a terminal object in
5924:restricted comprehension
5301:cannot be less than the
5201:we have proven that the
3726:automated theorem prover
2606: ; in other words,
1843:cannot be equal because
321:
220:elements has a total of
200:as a subset, a set with
9263:Self-verifying theories
9084:Tarski's axiomatization
8035:Tarski's undefinability
8030:incompleteness theorems
7218:Useâmention distinction
6176:propositional functions
5930:) in the definition of
5498:paradoxes of set theory
2626:is not in the image of
2436:(by definition of
1431:(that maps elements in
990:is not surjective, via
900:universal instantiation
161:has a strictly greater
9637:Mathematics portal
9248:Proof of impossibility
8896:propositional variable
8206:Propositional calculus
7508:Constructible universe
7328:Constructibility (V=L)
7213:Typeâtoken distinction
6912:Cantor, Georg (1891),
6826:Graham Priest (2002).
6719:
6718:{\displaystyle T\to Y}
6693:
6646:
6645:{\displaystyle T\to Y}
6620:
6582:
6562:
6561:{\displaystyle T\to Y}
6536:
6498:
6474:
6454:
6422:
6390:
6358:
6334:
6314:
6290:
6270:
6093:
6032:
5892:
5802:
5714:
5655:
5635:
5602:
5582:
5523:
5482:
5456:
5417:
5382:
5356:
5321:
5295:
5256:
5221:
5199:proof by contradiction
5188:
5153:
5118:
5098:
5078:
5055:
5020:
4994:
4971:
4951:
4931:
4911:
4891:
4871:
4851:
4831:
4811:
4791:
4771:
4751:
4731:
4689:
4643:
4434:
4399:
4377:
4342:
4316:
4278:
4255:
4182:
4160:
4123:
3938:
3903:
3861:
3832:
3766:
3715:
3695:
3671:
3647:
3612:
3592:
3572:
3571:{\displaystyle x\in A}
3546:
3511:
3491:
3490:{\displaystyle x\in y}
3465:
3445:
3425:
3367:
3347:
3323:
3305:is assumed to admit a
3299:
3279:
3221:
3197:
3162:
3139:
3119:
3099:
3079:
3059:
3039:
3019:
2999:
2979:
2959:
2939:
2915:
2895:
2875:
2852:
2789:
2769:
2743:
2723:
2700:
2680:
2660:
2640:
2620:
2600:
2565:
2532:
2377:implies the following
2371:
2336:
2307:
2287:
2286:{\displaystyle x\in B}
2261:
2226:
2206:
2177:
2142:
2116:
2081:
2061:
2032:
1997:
1962:
1927:
1926:{\displaystyle x\in A}
1901:
1877:
1857:
1837:
1808:
1788:
1768:
1733:
1732:{\displaystyle x\in B}
1707:
1706:{\displaystyle x\in A}
1678:
1613:
1589:
1569:
1546:. Recalling that each
1540:
1539:{\displaystyle x\in A}
1514:
1485:
1465:
1445:
1425:
1405:
1384:
1363:
1341:
1320:
1292:
1272:
1252:
1198:
1133:
1092:
1047:
1014:
984:
962:
927:
892:
843:
780:
755:
720:
692:
670:
644:
601:
535:
515:
448:
403:
370:
350:
274:
241:
214:
179:
155:
128:
101:
57:This article contains
51:
9506:Kolmogorov complexity
9459:Computably enumerable
9359:Model complete theory
9151:Principia Mathematica
8211:Propositional formula
8040:BanachâTarski paradox
7731:Principia Mathematica
7565:Transfinite induction
7424:(i.e. set difference)
6720:
6694:
6647:
6621:
6583:
6563:
6537:
6504:such that a morphism
6499:
6475:
6455:
6423:
6391:
6359:
6335:
6315:
6291:
6271:
6094:
6033:
5893:
5803:
5715:
5656:
5636:
5603:
5583:
5524:
5483:
5457:
5418:
5383:
5357:
5322:
5296:
5257:
5222:
5189:
5154:
5119:
5099:
5079:
5056:
5021:
4995:
4972:
4952:
4932:
4912:
4892:
4872:
4852:
4832:
4812:
4792:
4772:
4752:
4732:
4690:
4644:
4435:
4400:
4378:
4348:with each element of
4343:
4317:
4279:
4256:
4183:
4161:
4124:
3939:
3904:
3862:
3833:
3767:
3748:is countably infinite
3716:
3696:
3672:
3648:
3613:
3593:
3573:
3547:
3512:
3492:
3466:
3446:
3426:
3368:
3348:
3324:
3300:
3280:
3222:
3198:
3163:
3140:
3120:
3100:
3080:
3060:
3040:
3020:
3000:
2980:
2960:
2940:
2916:
2896:
2876:
2853:
2790:
2770:
2768:{\displaystyle \{x\}}
2749:to the singleton set
2744:
2724:
2701:
2681:
2661:
2641:
2621:
2601:
2566:
2533:
2372:
2337:
2308:
2293:by the definition of
2288:
2262:
2227:
2207:
2178:
2143:
2117:
2082:
2062:
2033:
1998:
1963:
1928:
1902:
1878:
1858:
1838:
1809:
1789:
1769:
1734:
1708:
1679:
1614:
1590:
1570:
1541:
1515:
1491:that is not equal to
1486:
1466:
1446:
1426:
1406:
1385:
1364:
1342:
1321:
1293:
1273:
1253:
1199:
1134:
1093:
1048:
1015:
985:
963:
928:
893:
844:
781:
756:
721:
693:
671:
645:
602:
536:
516:
458:. As a consequence,
449:
404:
371:
351:
282:non-negative integers
275:
242:
240:{\displaystyle 2^{n}}
215:
180:
156:
129:
102:
29:
9454:ChurchâTuring thesis
9441:Computability theory
8650:continuum hypothesis
8168:Square of opposition
8026:Gödel's completeness
7805:Burali-Forti paradox
7560:Set-builder notation
7513:Continuum hypothesis
7453:Symmetric difference
7136:ChurchâTuring thesis
7130:Entscheidungsproblem
7009:(Paperback edition).
6898:. Also published in
6779:. pp. 362â363.
6703:
6656:
6652:as maps of the form
6630:
6592:
6572:
6546:
6508:
6484:
6464:
6432:
6400:
6368:
6344:
6324:
6300:
6280:
6256:
6210:is false or true of
6182:be the correlate of
6053:
5946:
5831:
5746:
5665:
5645:
5612:
5592:
5533:
5513:
5470:
5431:
5392:
5370:
5331:
5309:
5270:
5231:
5209:
5163:
5128:
5108:
5088:
5068:
5030:
5008:
4984:
4961:
4941:
4921:
4901:
4881:
4861:
4841:
4821:
4801:
4781:
4761:
4741:
4706:
4679:
4447:
4409:
4387:
4352:
4330:
4291:
4268:
4192:
4170:
4135:
3951:
3913:
3878:
3849:
3784:
3756:
3705:
3685:
3661:
3622:
3602:
3582:
3556:
3521:
3501:
3475:
3455:
3435:
3377:
3357:
3337:
3313:
3289:
3231:
3211:
3172:
3152:
3129:
3109:
3089:
3069:
3049:
3029:
3009:
2989:
2969:
2949:
2929:
2905:
2885:
2865:
2799:
2779:
2753:
2733:
2713:
2690:
2670:
2650:
2630:
2610:
2575:
2549:
2543:reductio ad absurdum
2388:
2346:
2320:
2297:
2271:
2236:
2216:
2205:{\displaystyle f(x)}
2187:
2152:
2126:
2091:
2071:
2060:{\displaystyle f(x)}
2042:
2007:
1972:
1937:
1911:
1891:
1867:
1847:
1836:{\displaystyle f(x)}
1818:
1798:
1778:
1743:
1717:
1691:
1626:
1603:
1579:
1568:{\displaystyle f(x)}
1550:
1524:
1513:{\displaystyle f(x)}
1495:
1475:
1455:
1435:
1415:
1395:
1374:
1353:
1331:
1310:
1282:
1262:
1212:
1145:
1102:
1057:
1024:
1004:
992:reductio ad absurdum
974:
937:
908:
853:
792:
770:
730:
704:
700:Then there exists a
682:
654:
615:
552:
525:
462:
413:
380:
360:
340:
251:
224:
204:
169:
142:
115:
91:
9608:Mathematical object
9499:P versus NP problem
9464:Computable function
9258:Reverse mathematics
9184:Logical consequence
9061:primitive recursive
9056:elementary function
8829:Free/bound variable
8682:TarskiâGrothendieck
8201:Logical connectives
8131:Logical equivalence
7981:Logical consequence
7766:TarskiâGrothendieck
6930:, E. Zermelo, 1932.
6588:and every morphism
6218:for every value of
3732:noted in 1992 that
3677:coincides with the
3168:in the expression "
3125:must be false; and
2706:is not surjective.
1597:Cantor diagonal set
842:{\displaystyle A\ }
334: —
9657:1891 introductions
9406:Transfer principle
9369:Semantics of logic
9354:Categorical theory
9330:Non-standard model
8844:Logical connective
7971:Information theory
7920:Mathematical logic
7355:Limitation of size
7064:"Cantor's Theorem"
7061:Weisstein, Eric W.
6993:, New York, 1974.
6715:
6689:
6642:
6616:
6578:
6558:
6532:
6494:
6470:
6450:
6418:
6386:
6354:
6330:
6310:
6286:
6266:
6231:Zermelo set theory
6089:
6028:
5888:
5798:
5710:
5651:
5631:
5598:
5578:
5519:
5478:
5452:
5413:
5378:
5352:
5317:
5291:
5252:
5217:
5184:
5149:
5114:
5094:
5074:
5064:Note that the set
5051:
5016:
4990:
4967:
4947:
4927:
4907:
4887:
4867:
4847:
4827:
4807:
4787:
4767:
4747:
4727:
4685:
4639:
4612:
4430:
4395:
4373:
4338:
4312:
4274:
4251:
4178:
4156:
4119:
3934:
3899:
3857:
3828:
3774:countably infinite
3762:
3711:
3691:
3667:
3655:indicator function
3643:
3608:
3588:
3568:
3542:
3507:
3487:
3461:
3441:
3421:
3363:
3343:
3319:
3295:
3275:
3217:
3193:
3158:
3135:
3115:
3095:
3075:
3055:
3035:
3015:
2995:
2975:
2955:
2935:
2925:, some element of
2911:
2891:
2871:
2848:
2785:
2765:
2739:
2719:
2696:
2676:
2656:
2636:
2616:
2596:
2561:
2528:
2526:
2367:
2332:
2303:
2283:
2267:by assumption and
2257:
2222:
2202:
2173:
2148:by definition. If
2138:
2122:by assumption and
2112:
2077:
2057:
2028:
1993:
1958:
1923:
1897:
1873:
1853:
1833:
1804:
1784:
1764:
1729:
1703:
1674:
1609:
1585:
1565:
1536:
1510:
1481:
1461:
1441:
1421:
1401:
1380:
1359:
1337:
1316:
1304:bijective function
1300:injective function
1288:
1268:
1248:
1194:
1129:
1088:
1043:
1010:
980:
958:
923:
888:
849:, we deduce
839:
776:
751:
716:
688:
666:
640:
597:
547:
531:
521:holds for any set
511:
444:
399:
366:
356:be a map from set
346:
332:
270:
237:
210:
175:
154:{\displaystyle A,}
151:
127:{\displaystyle A,}
124:
97:
59:special characters
52:
9644:
9643:
9576:Abstract category
9379:Theories of truth
9189:Rule of inference
9179:Natural deduction
9160:
9159:
8705:
8704:
8410:Cartesian product
8315:
8314:
8221:Many-valued logic
8196:Boolean functions
8079:Russell's paradox
8054:diagonal argument
7951:First-order logic
7886:
7885:
7795:Russell's paradox
7744:ZermeloâFraenkel
7645:Dedekind-infinite
7518:Diagonal argument
7417:Cartesian product
7274:Set (mathematics)
7226:
7225:
7007:978-1-61427-131-4
6968:978-0-521-89485-2
6896:978-0-8218-7360-1
6873:978-3-540-49553-6
6837:978-0-19-925405-7
6786:978-1-4614-8854-5
6581:{\displaystyle T}
6473:{\displaystyle T}
6333:{\displaystyle Y}
6289:{\displaystyle 1}
6206:according as phi-
6198:does not hold of
6116:diagonal argument
5903:Russell's paradox
5729:identity function
5654:{\displaystyle V}
5601:{\displaystyle X}
5522:{\displaystyle V}
5492:Related paradoxes
5117:{\displaystyle x}
5097:{\displaystyle x}
5077:{\displaystyle B}
4993:{\displaystyle B}
4970:{\displaystyle B}
4950:{\displaystyle b}
4930:{\displaystyle B}
4910:{\displaystyle B}
4890:{\displaystyle b}
4870:{\displaystyle B}
4850:{\displaystyle b}
4830:{\displaystyle B}
4810:{\displaystyle b}
4790:{\displaystyle b}
4770:{\displaystyle B}
4750:{\displaystyle B}
4688:{\displaystyle B}
4261:, along with the
3765:{\displaystyle A}
3714:{\displaystyle B}
3694:{\displaystyle B}
3679:logically negated
3670:{\displaystyle B}
3611:{\displaystyle D}
3591:{\displaystyle B}
3510:{\displaystyle D}
3464:{\displaystyle y}
3444:{\displaystyle x}
3366:{\displaystyle f}
3346:{\displaystyle A}
3322:{\displaystyle y}
3298:{\displaystyle A}
3285:, in this order.
3220:{\displaystyle x}
3205:diagonal argument
3161:{\displaystyle x}
3138:{\displaystyle f}
3118:{\displaystyle B}
3098:{\displaystyle A}
3078:{\displaystyle B}
3058:{\displaystyle B}
3038:{\displaystyle B}
3018:{\displaystyle B}
2998:{\displaystyle B}
2978:{\displaystyle B}
2958:{\displaystyle B}
2938:{\displaystyle A}
2914:{\displaystyle f}
2894:{\displaystyle A}
2874:{\displaystyle B}
2788:{\displaystyle A}
2742:{\displaystyle x}
2722:{\displaystyle A}
2699:{\displaystyle f}
2679:{\displaystyle A}
2659:{\displaystyle f}
2639:{\displaystyle f}
2619:{\displaystyle B}
2519:
2496:
2445:
2437:
2306:{\displaystyle B}
2225:{\displaystyle B}
2080:{\displaystyle B}
1900:{\displaystyle f}
1876:{\displaystyle A}
1856:{\displaystyle B}
1807:{\displaystyle B}
1787:{\displaystyle x}
1612:{\displaystyle f}
1588:{\displaystyle A}
1484:{\displaystyle A}
1464:{\displaystyle A}
1444:{\displaystyle A}
1424:{\displaystyle f}
1404:{\displaystyle A}
1383:{\displaystyle Y}
1362:{\displaystyle X}
1340:{\displaystyle Y}
1319:{\displaystyle X}
1291:{\displaystyle Y}
1271:{\displaystyle X}
1258:for any two sets
1013:{\displaystyle A}
983:{\displaystyle f}
800:
779:{\displaystyle x}
691:{\displaystyle f}
545:
534:{\displaystyle A}
376:to its power set
369:{\displaystyle A}
349:{\displaystyle f}
330:
213:{\displaystyle n}
178:{\displaystyle A}
107:, the set of all
100:{\displaystyle A}
65:rendering support
9689:
9677:Cardinal numbers
9635:
9634:
9586:History of logic
9581:Category of sets
9474:Decision problem
9253:Ordinal analysis
9194:Sequent calculus
9092:Boolean algebras
9032:
9031:
9006:
8977:logical/constant
8731:
8730:
8717:
8640:ZermeloâFraenkel
8391:Set operations:
8326:
8325:
8263:
8094:
8093:
8074:LöwenheimâSkolem
7961:Formal semantics
7913:
7906:
7899:
7890:
7889:
7868:Bertrand Russell
7858:John von Neumann
7843:Abraham Fraenkel
7838:Richard Dedekind
7800:Suslin's problem
7711:Cantor's theorem
7428:De Morgan's laws
7286:
7253:
7246:
7239:
7230:
7229:
7146:Effective method
7124:Cantor's theorem
7102:
7095:
7088:
7079:
7078:
7074:
7073:
7055:
7042:"Cantor theorem"
7030:
6986:Naive Set Theory
6973:
6972:
6960:
6950:
6944:
6937:
6931:
6925:
6909:
6903:
6884:
6878:
6877:
6861:
6851:
6842:
6841:
6823:
6814:
6813:
6811:
6800:
6791:
6790:
6768:
6724:
6722:
6721:
6716:
6698:
6696:
6695:
6690:
6651:
6649:
6648:
6643:
6625:
6623:
6622:
6617:
6587:
6585:
6584:
6579:
6567:
6565:
6564:
6559:
6541:
6539:
6538:
6533:
6503:
6501:
6500:
6495:
6493:
6492:
6479:
6477:
6476:
6471:
6459:
6457:
6456:
6451:
6427:
6425:
6424:
6419:
6395:
6393:
6392:
6387:
6363:
6361:
6360:
6355:
6353:
6352:
6340:is an object in
6339:
6337:
6336:
6331:
6319:
6317:
6316:
6311:
6309:
6308:
6295:
6293:
6292:
6287:
6275:
6273:
6272:
6267:
6265:
6264:
6186:. Then "not-phi-
6166:Bertrand Russell
6098:
6096:
6095:
6090:
6088:
6084:
6037:
6035:
6034:
6029:
6021:
6020:
6008:
6007:
5989:
5988:
5971:
5970:
5958:
5957:
5926:(as featured in
5897:
5895:
5894:
5889:
5884:
5883:
5871:
5870:
5856:
5855:
5843:
5842:
5807:
5805:
5804:
5799:
5794:
5790:
5758:
5757:
5719:
5717:
5716:
5711:
5709:
5701:
5693:
5679:
5678:
5672:
5660:
5658:
5657:
5652:
5640:
5638:
5637:
5632:
5621:
5620:
5607:
5605:
5604:
5599:
5587:
5585:
5584:
5579:
5577:
5569:
5561:
5547:
5546:
5540:
5528:
5526:
5525:
5520:
5504:Cantor's paradox
5487:
5485:
5484:
5479:
5477:
5461:
5459:
5458:
5453:
5448:
5440:
5439:
5422:
5420:
5419:
5414:
5409:
5401:
5400:
5387:
5385:
5384:
5379:
5377:
5361:
5359:
5358:
5353:
5348:
5340:
5339:
5326:
5324:
5323:
5318:
5316:
5300:
5298:
5297:
5292:
5287:
5279:
5278:
5261:
5259:
5258:
5253:
5248:
5240:
5239:
5226:
5224:
5223:
5218:
5216:
5193:
5191:
5190:
5185:
5180:
5172:
5171:
5158:
5156:
5155:
5150:
5145:
5137:
5136:
5123:
5121:
5120:
5115:
5103:
5101:
5100:
5095:
5083:
5081:
5080:
5075:
5060:
5058:
5057:
5052:
5047:
5039:
5038:
5025:
5023:
5022:
5017:
5015:
4999:
4997:
4996:
4991:
4976:
4974:
4973:
4968:
4956:
4954:
4953:
4948:
4936:
4934:
4933:
4928:
4916:
4914:
4913:
4908:
4896:
4894:
4893:
4888:
4876:
4874:
4873:
4868:
4856:
4854:
4853:
4848:
4836:
4834:
4833:
4828:
4816:
4814:
4813:
4808:
4796:
4794:
4793:
4788:
4776:
4774:
4773:
4768:
4756:
4754:
4753:
4748:
4736:
4734:
4733:
4728:
4723:
4715:
4714:
4694:
4692:
4691:
4686:
4648:
4646:
4645:
4640:
4632:
4624:
4623:
4617:
4616:
4454:
4439:
4437:
4436:
4431:
4426:
4418:
4417:
4404:
4402:
4401:
4396:
4394:
4382:
4380:
4379:
4374:
4369:
4361:
4360:
4347:
4345:
4344:
4339:
4337:
4321:
4319:
4318:
4313:
4308:
4300:
4299:
4283:
4281:
4280:
4275:
4260:
4258:
4257:
4252:
4247:
4187:
4185:
4184:
4179:
4177:
4165:
4163:
4162:
4157:
4152:
4144:
4143:
4128:
4126:
4125:
4120:
3968:
3960:
3959:
3943:
3941:
3940:
3935:
3930:
3922:
3921:
3908:
3906:
3905:
3900:
3895:
3887:
3886:
3866:
3864:
3863:
3858:
3856:
3837:
3835:
3834:
3829:
3797:
3771:
3769:
3768:
3763:
3730:Lawrence Paulson
3720:
3718:
3717:
3712:
3700:
3698:
3697:
3692:
3676:
3674:
3673:
3668:
3652:
3650:
3649:
3644:
3617:
3615:
3614:
3609:
3597:
3595:
3594:
3589:
3577:
3575:
3574:
3569:
3551:
3549:
3548:
3543:
3516:
3514:
3513:
3508:
3496:
3494:
3493:
3488:
3470:
3468:
3467:
3462:
3450:
3448:
3447:
3442:
3430:
3428:
3427:
3422:
3417:
3416:
3395:
3394:
3372:
3370:
3369:
3364:
3352:
3350:
3349:
3344:
3328:
3326:
3325:
3320:
3304:
3302:
3301:
3296:
3284:
3282:
3281:
3276:
3265:
3264:
3252:
3251:
3226:
3224:
3223:
3218:
3202:
3200:
3199:
3194:
3167:
3165:
3164:
3159:
3145:cannot be onto.
3144:
3142:
3141:
3136:
3124:
3122:
3121:
3116:
3104:
3102:
3101:
3096:
3084:
3082:
3081:
3076:
3064:
3062:
3061:
3056:
3044:
3042:
3041:
3036:
3024:
3022:
3021:
3016:
3004:
3002:
3001:
2996:
2984:
2982:
2981:
2976:
2964:
2962:
2961:
2956:
2944:
2942:
2941:
2936:
2920:
2918:
2917:
2912:
2900:
2898:
2897:
2892:
2880:
2878:
2877:
2872:
2857:
2855:
2854:
2849:
2835:
2834:
2794:
2792:
2791:
2786:
2774:
2772:
2771:
2766:
2748:
2746:
2745:
2740:
2728:
2726:
2725:
2720:
2705:
2703:
2702:
2697:
2685:
2683:
2682:
2677:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2570:
2568:
2567:
2562:
2537:
2535:
2534:
2529:
2527:
2520:
2517:
2497:
2494:
2491:
2446:
2443:
2438:
2435:
2432:
2376:
2374:
2373:
2368:
2341:
2339:
2338:
2333:
2312:
2310:
2309:
2304:
2292:
2290:
2289:
2284:
2266:
2264:
2263:
2258:
2231:
2229:
2228:
2223:
2211:
2209:
2208:
2203:
2182:
2180:
2179:
2174:
2147:
2145:
2144:
2139:
2121:
2119:
2118:
2113:
2086:
2084:
2083:
2078:
2066:
2064:
2063:
2058:
2037:
2035:
2034:
2029:
2002:
2000:
1999:
1994:
1967:
1965:
1964:
1959:
1932:
1930:
1929:
1924:
1906:
1904:
1903:
1898:
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1842:
1840:
1839:
1834:
1813:
1811:
1810:
1805:
1793:
1791:
1790:
1785:
1773:
1771:
1770:
1765:
1738:
1736:
1735:
1730:
1712:
1710:
1709:
1704:
1683:
1681:
1680:
1675:
1618:
1616:
1615:
1610:
1594:
1592:
1591:
1586:
1574:
1572:
1571:
1566:
1545:
1543:
1542:
1537:
1519:
1517:
1516:
1511:
1490:
1488:
1487:
1482:
1470:
1468:
1467:
1462:
1450:
1448:
1447:
1442:
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1390:
1389:
1387:
1386:
1381:
1368:
1366:
1365:
1360:
1348:
1346:
1344:
1343:
1338:
1325:
1323:
1322:
1317:
1297:
1295:
1294:
1289:
1277:
1275:
1274:
1269:
1257:
1255:
1254:
1249:
1203:
1201:
1200:
1195:
1181:
1180:
1138:
1136:
1135:
1130:
1097:
1095:
1094:
1089:
1078:
1077:
1052:
1050:
1049:
1044:
1033:
1032:
1019:
1017:
1016:
1011:
989:
987:
986:
981:
967:
965:
964:
959:
932:
930:
929:
924:
897:
895:
894:
889:
848:
846:
845:
840:
798:
785:
783:
782:
777:
760:
758:
757:
752:
725:
723:
722:
717:
697:
695:
694:
689:
675:
673:
672:
667:
649:
647:
646:
641:
630:
629:
606:
604:
603:
598:
540:
538:
537:
532:
520:
518:
517:
512:
498:
497:
453:
451:
450:
445:
434:
433:
408:
406:
405:
400:
389:
388:
375:
373:
372:
367:
355:
353:
352:
347:
335:
331:Theorem (Cantor)
279:
277:
276:
271:
263:
262:
246:
244:
243:
238:
236:
235:
219:
217:
216:
211:
184:
182:
181:
176:
160:
158:
157:
152:
133:
131:
130:
125:
106:
104:
103:
98:
82:Cantor's theorem
76:In mathematical
9697:
9696:
9692:
9691:
9690:
9688:
9687:
9686:
9662:1891 in science
9647:
9646:
9645:
9640:
9629:
9622:
9567:Category theory
9557:Algebraic logic
9540:
9511:Lambda calculus
9449:Church encoding
9435:
9411:Truth predicate
9267:
9233:Complete theory
9156:
9025:
9021:
9017:
9012:
9004:
8724: and
8720:
8715:
8701:
8677:New Foundations
8645:axiom of choice
8628:
8590:Gödel numbering
8530: and
8522:
8426:
8311:
8261:
8242:
8191:Boolean algebra
8177:
8141:Equiconsistency
8106:Classical logic
8083:
8064:Halting problem
8052: and
8028: and
8016: and
8015:
8010:Theorems (
8005:
7922:
7917:
7887:
7882:
7809:
7788:
7772:
7737:New Foundations
7684:
7574:
7493:Cardinal number
7476:
7462:
7403:
7287:
7278:
7262:
7257:
7227:
7222:
7115:
7113:metamathematics
7106:
7040:
7037:
7028:
6991:Springer-Verlag
6977:
6976:
6969:
6951:
6947:
6938:
6934:
6910:
6906:
6885:
6881:
6874:
6852:
6845:
6838:
6824:
6817:
6809:
6801:
6794:
6787:
6769:
6758:
6753:
6731:
6704:
6701:
6700:
6657:
6654:
6653:
6631:
6628:
6627:
6593:
6590:
6589:
6573:
6570:
6569:
6547:
6544:
6543:
6509:
6506:
6505:
6488:
6487:
6485:
6482:
6481:
6465:
6462:
6461:
6433:
6430:
6429:
6428:that satisfies
6401:
6398:
6397:
6369:
6366:
6365:
6348:
6347:
6345:
6342:
6341:
6325:
6322:
6321:
6320:. Suppose that
6304:
6303:
6301:
6298:
6297:
6281:
6278:
6277:
6260:
6259:
6257:
6254:
6253:
6250:finite products
6239:
6237:Generalizations
6112:
6066:
6062:
6054:
6051:
6050:
6016:
6012:
6003:
5999:
5984:
5980:
5966:
5962:
5953:
5949:
5947:
5944:
5943:
5938:
5917:
5879:
5875:
5866:
5862:
5851:
5847:
5838:
5834:
5832:
5829:
5828:
5823:
5766:
5762:
5753:
5749:
5747:
5744:
5743:
5735:of a given set
5705:
5697:
5689:
5674:
5673:
5668:
5666:
5663:
5662:
5646:
5643:
5642:
5616:
5615:
5613:
5610:
5609:
5593:
5590:
5589:
5573:
5565:
5557:
5542:
5541:
5536:
5534:
5531:
5530:
5514:
5511:
5510:
5494:
5473:
5471:
5468:
5467:
5444:
5435:
5434:
5432:
5429:
5428:
5405:
5396:
5395:
5393:
5390:
5389:
5373:
5371:
5368:
5367:
5344:
5335:
5334:
5332:
5329:
5328:
5312:
5310:
5307:
5306:
5283:
5274:
5273:
5271:
5268:
5267:
5244:
5235:
5234:
5232:
5229:
5228:
5212:
5210:
5207:
5206:
5176:
5167:
5166:
5164:
5161:
5160:
5141:
5132:
5131:
5129:
5126:
5125:
5109:
5106:
5105:
5089:
5086:
5085:
5069:
5066:
5065:
5043:
5034:
5033:
5031:
5028:
5027:
5011:
5009:
5006:
5005:
4985:
4982:
4981:
4962:
4959:
4958:
4942:
4939:
4938:
4922:
4919:
4918:
4902:
4899:
4898:
4882:
4879:
4878:
4862:
4859:
4858:
4842:
4839:
4838:
4822:
4819:
4818:
4802:
4799:
4798:
4782:
4779:
4778:
4762:
4759:
4758:
4742:
4739:
4738:
4719:
4710:
4709:
4707:
4704:
4703:
4680:
4677:
4676:
4628:
4619:
4618:
4611:
4610:
4605:
4600:
4594:
4593:
4570:
4565:
4559:
4558:
4535:
4530:
4524:
4523:
4500:
4495:
4489:
4488:
4471:
4466:
4456:
4455:
4450:
4448:
4445:
4444:
4422:
4413:
4412:
4410:
4407:
4406:
4390:
4388:
4385:
4384:
4365:
4356:
4355:
4353:
4350:
4349:
4333:
4331:
4328:
4327:
4304:
4295:
4294:
4292:
4289:
4288:
4269:
4266:
4265:
4243:
4193:
4190:
4189:
4173:
4171:
4168:
4167:
4148:
4139:
4138:
4136:
4133:
4132:
3964:
3955:
3954:
3952:
3949:
3948:
3926:
3917:
3916:
3914:
3911:
3910:
3891:
3882:
3881:
3879:
3876:
3875:
3852:
3850:
3847:
3846:
3840:natural numbers
3793:
3785:
3782:
3781:
3757:
3754:
3753:
3750:
3706:
3703:
3702:
3686:
3683:
3682:
3662:
3659:
3658:
3623:
3620:
3619:
3603:
3600:
3599:
3583:
3580:
3579:
3557:
3554:
3553:
3522:
3519:
3518:
3502:
3499:
3498:
3476:
3473:
3472:
3456:
3453:
3452:
3436:
3433:
3432:
3412:
3408:
3390:
3386:
3378:
3375:
3374:
3358:
3355:
3354:
3338:
3335:
3334:
3314:
3311:
3310:
3290:
3287:
3286:
3260:
3256:
3247:
3243:
3232:
3229:
3228:
3212:
3209:
3208:
3173:
3170:
3169:
3153:
3150:
3149:
3130:
3127:
3126:
3110:
3107:
3106:
3090:
3087:
3086:
3070:
3067:
3066:
3050:
3047:
3046:
3030:
3027:
3026:
3010:
3007:
3006:
2990:
2987:
2986:
2970:
2967:
2966:
2950:
2947:
2946:
2930:
2927:
2926:
2906:
2903:
2902:
2886:
2883:
2882:
2866:
2863:
2862:
2830:
2829:
2800:
2797:
2796:
2780:
2777:
2776:
2754:
2751:
2750:
2734:
2731:
2730:
2714:
2711:
2710:
2691:
2688:
2687:
2671:
2668:
2667:
2651:
2648:
2647:
2631:
2628:
2627:
2611:
2608:
2607:
2576:
2573:
2572:
2550:
2547:
2546:
2525:
2524:
2516:
2493:
2490:
2463:
2451:
2450:
2442:
2434:
2431:
2404:
2391:
2389:
2386:
2385:
2347:
2344:
2343:
2321:
2318:
2317:
2298:
2295:
2294:
2272:
2269:
2268:
2237:
2234:
2233:
2217:
2214:
2213:
2188:
2185:
2184:
2153:
2150:
2149:
2127:
2124:
2123:
2092:
2089:
2088:
2072:
2069:
2068:
2043:
2040:
2039:
2008:
2005:
2004:
1973:
1970:
1969:
1938:
1935:
1934:
1912:
1909:
1908:
1892:
1889:
1888:
1868:
1865:
1864:
1848:
1845:
1844:
1819:
1816:
1815:
1799:
1796:
1795:
1779:
1776:
1775:
1744:
1741:
1740:
1739:if and only if
1718:
1715:
1714:
1692:
1689:
1688:
1627:
1624:
1623:
1604:
1601:
1600:
1580:
1577:
1576:
1575:is a subset of
1551:
1548:
1547:
1525:
1522:
1521:
1496:
1493:
1492:
1476:
1473:
1472:
1456:
1453:
1452:
1436:
1433:
1432:
1416:
1413:
1412:
1396:
1393:
1392:
1375:
1372:
1371:
1369:
1354:
1351:
1350:
1332:
1329:
1328:
1326:
1311:
1308:
1307:
1283:
1280:
1279:
1263:
1260:
1259:
1213:
1210:
1209:
1206:
1176:
1175:
1146:
1143:
1142:
1140:
1103:
1100:
1099:
1073:
1072:
1058:
1055:
1054:
1028:
1027:
1025:
1022:
1021:
1005:
1002:
1001:
995:
975:
972:
971:
969:
938:
935:
934:
909:
906:
905:
903:
854:
851:
850:
793:
790:
789:
771:
768:
767:
762:
731:
728:
727:
705:
702:
701:
699:
698:is surjective.
683:
680:
679:
677:
655:
652:
651:
625:
624:
616:
613:
612:
607:exists via the
553:
550:
549:
543:
526:
523:
522:
493:
492:
463:
460:
459:
429:
428:
414:
411:
410:
384:
383:
381:
378:
377:
361:
358:
357:
341:
338:
337:
333:
324:
316:cardinal number
258:
254:
252:
249:
248:
231:
227:
225:
222:
221:
205:
202:
201:
170:
167:
166:
143:
140:
139:
116:
113:
112:
92:
89:
88:
74:
73:
72:
63:Without proper
24:
17:
12:
11:
5:
9695:
9685:
9684:
9679:
9674:
9669:
9664:
9659:
9642:
9641:
9627:
9624:
9623:
9621:
9620:
9615:
9610:
9605:
9600:
9599:
9598:
9588:
9583:
9578:
9569:
9564:
9559:
9554:
9552:Abstract logic
9548:
9546:
9542:
9541:
9539:
9538:
9533:
9531:Turing machine
9528:
9523:
9518:
9513:
9508:
9503:
9502:
9501:
9496:
9491:
9486:
9481:
9471:
9469:Computable set
9466:
9461:
9456:
9451:
9445:
9443:
9437:
9436:
9434:
9433:
9428:
9423:
9418:
9413:
9408:
9403:
9398:
9397:
9396:
9391:
9386:
9376:
9371:
9366:
9364:Satisfiability
9361:
9356:
9351:
9350:
9349:
9339:
9338:
9337:
9327:
9326:
9325:
9320:
9315:
9310:
9305:
9295:
9294:
9293:
9288:
9281:Interpretation
9277:
9275:
9269:
9268:
9266:
9265:
9260:
9255:
9250:
9245:
9235:
9230:
9229:
9228:
9227:
9226:
9216:
9211:
9201:
9196:
9191:
9186:
9181:
9176:
9170:
9168:
9162:
9161:
9158:
9157:
9155:
9154:
9146:
9145:
9144:
9143:
9138:
9137:
9136:
9131:
9126:
9106:
9105:
9104:
9102:minimal axioms
9099:
9088:
9087:
9086:
9075:
9074:
9073:
9068:
9063:
9058:
9053:
9048:
9035:
9033:
9014:
9013:
9011:
9010:
9009:
9008:
8996:
8991:
8990:
8989:
8984:
8979:
8974:
8964:
8959:
8954:
8949:
8948:
8947:
8942:
8932:
8931:
8930:
8925:
8920:
8915:
8905:
8900:
8899:
8898:
8893:
8888:
8878:
8877:
8876:
8871:
8866:
8861:
8856:
8851:
8841:
8836:
8831:
8826:
8825:
8824:
8819:
8814:
8809:
8799:
8794:
8792:Formation rule
8789:
8784:
8783:
8782:
8777:
8767:
8766:
8765:
8755:
8750:
8745:
8740:
8734:
8728:
8711:Formal systems
8707:
8706:
8703:
8702:
8700:
8699:
8694:
8689:
8684:
8679:
8674:
8669:
8664:
8659:
8654:
8653:
8652:
8647:
8636:
8634:
8630:
8629:
8627:
8626:
8625:
8624:
8614:
8609:
8608:
8607:
8600:Large cardinal
8597:
8592:
8587:
8582:
8577:
8563:
8562:
8561:
8556:
8551:
8536:
8534:
8524:
8523:
8521:
8520:
8519:
8518:
8513:
8508:
8498:
8493:
8488:
8483:
8478:
8473:
8468:
8463:
8458:
8453:
8448:
8443:
8437:
8435:
8428:
8427:
8425:
8424:
8423:
8422:
8417:
8412:
8407:
8402:
8397:
8389:
8388:
8387:
8382:
8372:
8367:
8365:Extensionality
8362:
8360:Ordinal number
8357:
8347:
8342:
8341:
8340:
8329:
8323:
8317:
8316:
8313:
8312:
8310:
8309:
8304:
8299:
8294:
8289:
8284:
8279:
8278:
8277:
8267:
8266:
8265:
8252:
8250:
8244:
8243:
8241:
8240:
8239:
8238:
8233:
8228:
8218:
8213:
8208:
8203:
8198:
8193:
8187:
8185:
8179:
8178:
8176:
8175:
8170:
8165:
8160:
8155:
8150:
8145:
8144:
8143:
8133:
8128:
8123:
8118:
8113:
8108:
8102:
8100:
8091:
8085:
8084:
8082:
8081:
8076:
8071:
8066:
8061:
8056:
8044:Cantor's
8042:
8037:
8032:
8022:
8020:
8007:
8006:
8004:
8003:
7998:
7993:
7988:
7983:
7978:
7973:
7968:
7963:
7958:
7953:
7948:
7943:
7942:
7941:
7930:
7928:
7924:
7923:
7916:
7915:
7908:
7901:
7893:
7884:
7883:
7881:
7880:
7875:
7873:Thoralf Skolem
7870:
7865:
7860:
7855:
7850:
7845:
7840:
7835:
7830:
7825:
7819:
7817:
7811:
7810:
7808:
7807:
7802:
7797:
7791:
7789:
7787:
7786:
7783:
7777:
7774:
7773:
7771:
7770:
7769:
7768:
7763:
7758:
7757:
7756:
7741:
7740:
7739:
7727:
7726:
7725:
7714:
7713:
7708:
7703:
7698:
7692:
7690:
7686:
7685:
7683:
7682:
7677:
7672:
7667:
7658:
7653:
7648:
7638:
7633:
7632:
7631:
7626:
7621:
7611:
7601:
7596:
7591:
7585:
7583:
7576:
7575:
7573:
7572:
7567:
7562:
7557:
7555:Ordinal number
7552:
7547:
7542:
7537:
7536:
7535:
7530:
7520:
7515:
7510:
7505:
7500:
7490:
7485:
7479:
7477:
7475:
7474:
7471:
7467:
7464:
7463:
7461:
7460:
7455:
7450:
7445:
7440:
7435:
7433:Disjoint union
7430:
7425:
7419:
7413:
7411:
7405:
7404:
7402:
7401:
7400:
7399:
7394:
7383:
7382:
7380:Martin's axiom
7377:
7372:
7367:
7362:
7357:
7352:
7347:
7345:Extensionality
7342:
7341:
7340:
7330:
7325:
7324:
7323:
7318:
7313:
7303:
7297:
7295:
7289:
7288:
7281:
7279:
7277:
7276:
7270:
7268:
7264:
7263:
7256:
7255:
7248:
7241:
7233:
7224:
7223:
7221:
7220:
7215:
7210:
7205:
7203:Satisfiability
7200:
7195:
7190:
7188:Interpretation
7185:
7180:
7175:
7170:
7165:
7160:
7159:
7158:
7148:
7143:
7138:
7133:
7126:
7120:
7117:
7116:
7105:
7104:
7097:
7090:
7082:
7076:
7075:
7056:
7036:
7035:External links
7033:
7032:
7031:
7026:
7010:
6975:
6974:
6967:
6945:
6939:A. Kanamori, "
6932:
6904:
6879:
6872:
6843:
6836:
6815:
6792:
6785:
6755:
6754:
6752:
6749:
6748:
6747:
6742:
6737:
6730:
6727:
6714:
6711:
6708:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6641:
6638:
6635:
6615:
6612:
6609:
6606:
6603:
6600:
6597:
6577:
6557:
6554:
6551:
6531:
6528:
6525:
6522:
6519:
6516:
6513:
6491:
6469:
6449:
6446:
6443:
6440:
6437:
6417:
6414:
6411:
6408:
6405:
6385:
6382:
6379:
6376:
6373:
6351:
6329:
6307:
6285:
6263:
6238:
6235:
6194:)," i.e. "phi-
6146:) = 1 −
6111:
6108:
6087:
6082:
6079:
6076:
6073:
6070:
6065:
6061:
6058:
6039:
6038:
6027:
6024:
6019:
6015:
6011:
6006:
6002:
5998:
5995:
5992:
5987:
5983:
5979:
5975:
5969:
5965:
5961:
5956:
5952:
5934:
5913:
5899:
5898:
5887:
5882:
5878:
5874:
5869:
5865:
5860:
5854:
5850:
5846:
5841:
5837:
5819:
5809:
5808:
5797:
5793:
5788:
5785:
5782:
5779:
5776:
5773:
5770:
5765:
5761:
5756:
5752:
5708:
5704:
5700:
5696:
5692:
5688:
5685:
5682:
5677:
5671:
5650:
5630:
5627:
5624:
5619:
5597:
5576:
5572:
5568:
5564:
5560:
5556:
5553:
5550:
5545:
5539:
5518:
5493:
5490:
5476:
5451:
5447:
5443:
5438:
5412:
5408:
5404:
5399:
5376:
5351:
5347:
5343:
5338:
5315:
5290:
5286:
5282:
5277:
5251:
5247:
5243:
5238:
5215:
5183:
5179:
5175:
5170:
5148:
5144:
5140:
5135:
5113:
5093:
5073:
5050:
5046:
5042:
5037:
5014:
4989:
4966:
4957:which maps to
4946:
4926:
4906:
4886:
4866:
4846:
4826:
4806:
4786:
4766:
4746:
4726:
4722:
4718:
4713:
4695:be the set of
4684:
4650:
4649:
4638:
4635:
4631:
4627:
4622:
4615:
4609:
4606:
4604:
4601:
4599:
4596:
4595:
4592:
4589:
4586:
4583:
4580:
4577:
4574:
4571:
4569:
4566:
4564:
4561:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4534:
4531:
4529:
4526:
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4499:
4496:
4494:
4491:
4490:
4487:
4484:
4481:
4478:
4475:
4472:
4470:
4467:
4465:
4462:
4461:
4459:
4453:
4429:
4425:
4421:
4416:
4393:
4372:
4368:
4364:
4359:
4336:
4311:
4307:
4303:
4298:
4273:
4250:
4246:
4242:
4239:
4236:
4233:
4230:
4227:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4176:
4155:
4151:
4147:
4142:
4130:
4129:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3967:
3963:
3958:
3933:
3929:
3925:
3920:
3898:
3894:
3890:
3885:
3855:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3796:
3792:
3789:
3780:, we may take
3761:
3749:
3742:
3710:
3690:
3666:
3642:
3639:
3636:
3633:
3630:
3627:
3607:
3587:
3567:
3564:
3561:
3541:
3538:
3535:
3532:
3529:
3526:
3506:
3486:
3483:
3480:
3460:
3440:
3420:
3415:
3411:
3407:
3404:
3401:
3398:
3393:
3389:
3385:
3382:
3362:
3342:
3318:
3294:
3274:
3271:
3268:
3263:
3259:
3255:
3250:
3246:
3242:
3239:
3236:
3216:
3192:
3189:
3186:
3183:
3180:
3177:
3157:
3134:
3114:
3094:
3074:
3054:
3034:
3014:
2994:
2974:
2954:
2934:
2910:
2890:
2870:
2847:
2844:
2841:
2838:
2833:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2784:
2764:
2761:
2758:
2738:
2718:
2695:
2675:
2655:
2635:
2615:
2595:
2592:
2589:
2586:
2583:
2580:
2560:
2557:
2554:
2541:Therefore, by
2539:
2538:
2523:
2515:
2512:
2509:
2506:
2503:
2500:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2470:
2466:
2464:
2462:
2459:
2456:
2453:
2452:
2449:
2441:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2411:
2407:
2405:
2403:
2400:
2397:
2394:
2393:
2366:
2363:
2360:
2357:
2354:
2351:
2331:
2328:
2325:
2302:
2282:
2279:
2276:
2256:
2253:
2250:
2247:
2244:
2241:
2221:
2201:
2198:
2195:
2192:
2172:
2169:
2166:
2163:
2160:
2157:
2137:
2134:
2131:
2111:
2108:
2105:
2102:
2099:
2096:
2076:
2056:
2053:
2050:
2047:
2027:
2024:
2021:
2018:
2015:
2012:
1992:
1989:
1986:
1983:
1980:
1977:
1957:
1954:
1951:
1948:
1945:
1942:
1922:
1919:
1916:
1896:
1872:
1852:
1832:
1829:
1826:
1823:
1803:
1783:
1763:
1760:
1757:
1754:
1751:
1748:
1728:
1725:
1722:
1702:
1699:
1696:
1685:
1684:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1608:
1584:
1564:
1561:
1558:
1555:
1535:
1532:
1529:
1509:
1506:
1503:
1500:
1480:
1460:
1451:to subsets of
1440:
1420:
1400:
1379:
1358:
1336:
1315:
1287:
1267:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1193:
1190:
1187:
1184:
1179:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1141:Consequently,
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1107:
1087:
1084:
1081:
1076:
1071:
1068:
1065:
1062:
1042:
1039:
1036:
1031:
1009:
998:injective maps
979:
957:
954:
951:
948:
945:
942:
922:
919:
916:
913:
887:
884:
881:
878:
875:
872:
868:
864:
861:
858:
838:
835:
832:
829:
826:
823:
820:
816:
812:
809:
806:
803:
797:
775:
750:
747:
744:
741:
738:
735:
715:
712:
709:
687:
665:
662:
659:
639:
636:
633:
628:
623:
620:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
544:
530:
510:
507:
504:
501:
496:
491:
488:
485:
482:
479:
476:
473:
470:
467:
443:
440:
437:
432:
427:
424:
421:
418:
398:
395:
392:
387:
365:
345:
328:
323:
320:
269:
266:
261:
257:
234:
230:
209:
174:
150:
147:
123:
120:
96:
67:, you may see
55:
54:
53:
15:
9:
6:
4:
3:
2:
9694:
9683:
9680:
9678:
9675:
9673:
9670:
9668:
9665:
9663:
9660:
9658:
9655:
9654:
9652:
9639:
9638:
9633:
9625:
9619:
9616:
9614:
9611:
9609:
9606:
9604:
9601:
9597:
9594:
9593:
9592:
9589:
9587:
9584:
9582:
9579:
9577:
9573:
9570:
9568:
9565:
9563:
9560:
9558:
9555:
9553:
9550:
9549:
9547:
9543:
9537:
9534:
9532:
9529:
9527:
9526:Recursive set
9524:
9522:
9519:
9517:
9514:
9512:
9509:
9507:
9504:
9500:
9497:
9495:
9492:
9490:
9487:
9485:
9482:
9480:
9477:
9476:
9475:
9472:
9470:
9467:
9465:
9462:
9460:
9457:
9455:
9452:
9450:
9447:
9446:
9444:
9442:
9438:
9432:
9429:
9427:
9424:
9422:
9419:
9417:
9414:
9412:
9409:
9407:
9404:
9402:
9399:
9395:
9392:
9390:
9387:
9385:
9382:
9381:
9380:
9377:
9375:
9372:
9370:
9367:
9365:
9362:
9360:
9357:
9355:
9352:
9348:
9345:
9344:
9343:
9340:
9336:
9335:of arithmetic
9333:
9332:
9331:
9328:
9324:
9321:
9319:
9316:
9314:
9311:
9309:
9306:
9304:
9301:
9300:
9299:
9296:
9292:
9289:
9287:
9284:
9283:
9282:
9279:
9278:
9276:
9274:
9270:
9264:
9261:
9259:
9256:
9254:
9251:
9249:
9246:
9243:
9242:from ZFC
9239:
9236:
9234:
9231:
9225:
9222:
9221:
9220:
9217:
9215:
9212:
9210:
9207:
9206:
9205:
9202:
9200:
9197:
9195:
9192:
9190:
9187:
9185:
9182:
9180:
9177:
9175:
9172:
9171:
9169:
9167:
9163:
9153:
9152:
9148:
9147:
9142:
9141:non-Euclidean
9139:
9135:
9132:
9130:
9127:
9125:
9124:
9120:
9119:
9117:
9114:
9113:
9111:
9107:
9103:
9100:
9098:
9095:
9094:
9093:
9089:
9085:
9082:
9081:
9080:
9076:
9072:
9069:
9067:
9064:
9062:
9059:
9057:
9054:
9052:
9049:
9047:
9044:
9043:
9041:
9037:
9036:
9034:
9029:
9023:
9018:Example
9015:
9007:
9002:
9001:
9000:
8997:
8995:
8992:
8988:
8985:
8983:
8980:
8978:
8975:
8973:
8970:
8969:
8968:
8965:
8963:
8960:
8958:
8955:
8953:
8950:
8946:
8943:
8941:
8938:
8937:
8936:
8933:
8929:
8926:
8924:
8921:
8919:
8916:
8914:
8911:
8910:
8909:
8906:
8904:
8901:
8897:
8894:
8892:
8889:
8887:
8884:
8883:
8882:
8879:
8875:
8872:
8870:
8867:
8865:
8862:
8860:
8857:
8855:
8852:
8850:
8847:
8846:
8845:
8842:
8840:
8837:
8835:
8832:
8830:
8827:
8823:
8820:
8818:
8815:
8813:
8810:
8808:
8805:
8804:
8803:
8800:
8798:
8795:
8793:
8790:
8788:
8785:
8781:
8778:
8776:
8775:by definition
8773:
8772:
8771:
8768:
8764:
8761:
8760:
8759:
8756:
8754:
8751:
8749:
8746:
8744:
8741:
8739:
8736:
8735:
8732:
8729:
8727:
8723:
8718:
8712:
8708:
8698:
8695:
8693:
8690:
8688:
8685:
8683:
8680:
8678:
8675:
8673:
8670:
8668:
8665:
8663:
8662:KripkeâPlatek
8660:
8658:
8655:
8651:
8648:
8646:
8643:
8642:
8641:
8638:
8637:
8635:
8631:
8623:
8620:
8619:
8618:
8615:
8613:
8610:
8606:
8603:
8602:
8601:
8598:
8596:
8593:
8591:
8588:
8586:
8583:
8581:
8578:
8575:
8571:
8567:
8564:
8560:
8557:
8555:
8552:
8550:
8547:
8546:
8545:
8541:
8538:
8537:
8535:
8533:
8529:
8525:
8517:
8514:
8512:
8509:
8507:
8506:constructible
8504:
8503:
8502:
8499:
8497:
8494:
8492:
8489:
8487:
8484:
8482:
8479:
8477:
8474:
8472:
8469:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8447:
8444:
8442:
8439:
8438:
8436:
8434:
8429:
8421:
8418:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8398:
8396:
8393:
8392:
8390:
8386:
8383:
8381:
8378:
8377:
8376:
8373:
8371:
8368:
8366:
8363:
8361:
8358:
8356:
8352:
8348:
8346:
8343:
8339:
8336:
8335:
8334:
8331:
8330:
8327:
8324:
8322:
8318:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8290:
8288:
8285:
8283:
8280:
8276:
8273:
8272:
8271:
8268:
8264:
8259:
8258:
8257:
8254:
8253:
8251:
8249:
8245:
8237:
8234:
8232:
8229:
8227:
8224:
8223:
8222:
8219:
8217:
8214:
8212:
8209:
8207:
8204:
8202:
8199:
8197:
8194:
8192:
8189:
8188:
8186:
8184:
8183:Propositional
8180:
8174:
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8154:
8151:
8149:
8146:
8142:
8139:
8138:
8137:
8134:
8132:
8129:
8127:
8124:
8122:
8119:
8117:
8114:
8112:
8111:Logical truth
8109:
8107:
8104:
8103:
8101:
8099:
8095:
8092:
8090:
8086:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8051:
8047:
8043:
8041:
8038:
8036:
8033:
8031:
8027:
8024:
8023:
8021:
8019:
8013:
8008:
8002:
7999:
7997:
7994:
7992:
7989:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7957:
7954:
7952:
7949:
7947:
7944:
7940:
7937:
7936:
7935:
7932:
7931:
7929:
7925:
7921:
7914:
7909:
7907:
7902:
7900:
7895:
7894:
7891:
7879:
7878:Ernst Zermelo
7876:
7874:
7871:
7869:
7866:
7864:
7863:Willard Quine
7861:
7859:
7856:
7854:
7851:
7849:
7846:
7844:
7841:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7820:
7818:
7816:
7815:Set theorists
7812:
7806:
7803:
7801:
7798:
7796:
7793:
7792:
7790:
7784:
7782:
7779:
7778:
7775:
7767:
7764:
7762:
7761:KripkeâPlatek
7759:
7755:
7752:
7751:
7750:
7747:
7746:
7745:
7742:
7738:
7735:
7734:
7733:
7732:
7728:
7724:
7721:
7720:
7719:
7716:
7715:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7694:
7693:
7691:
7687:
7681:
7678:
7676:
7673:
7671:
7668:
7666:
7664:
7659:
7657:
7654:
7652:
7649:
7646:
7642:
7639:
7637:
7634:
7630:
7627:
7625:
7622:
7620:
7617:
7616:
7615:
7612:
7609:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7586:
7584:
7581:
7577:
7571:
7568:
7566:
7563:
7561:
7558:
7556:
7553:
7551:
7548:
7546:
7543:
7541:
7538:
7534:
7531:
7529:
7526:
7525:
7524:
7521:
7519:
7516:
7514:
7511:
7509:
7506:
7504:
7501:
7498:
7494:
7491:
7489:
7486:
7484:
7481:
7480:
7478:
7472:
7469:
7468:
7465:
7459:
7456:
7454:
7451:
7449:
7446:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7426:
7423:
7420:
7418:
7415:
7414:
7412:
7410:
7406:
7398:
7397:specification
7395:
7393:
7390:
7389:
7388:
7385:
7384:
7381:
7378:
7376:
7373:
7371:
7368:
7366:
7363:
7361:
7358:
7356:
7353:
7351:
7348:
7346:
7343:
7339:
7336:
7335:
7334:
7331:
7329:
7326:
7322:
7319:
7317:
7314:
7312:
7309:
7308:
7307:
7304:
7302:
7299:
7298:
7296:
7294:
7290:
7285:
7275:
7272:
7271:
7269:
7265:
7261:
7254:
7249:
7247:
7242:
7240:
7235:
7234:
7231:
7219:
7216:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7189:
7186:
7184:
7181:
7179:
7176:
7174:
7171:
7169:
7166:
7164:
7161:
7157:
7154:
7153:
7152:
7149:
7147:
7144:
7142:
7139:
7137:
7134:
7132:
7131:
7127:
7125:
7122:
7121:
7118:
7114:
7110:
7103:
7098:
7096:
7091:
7089:
7084:
7083:
7080:
7071:
7070:
7065:
7062:
7057:
7053:
7049:
7048:
7043:
7039:
7038:
7029:
7027:3-540-44085-2
7023:
7019:
7015:
7011:
7008:
7004:
7000:
6999:0-387-90092-6
6996:
6992:
6988:
6987:
6982:
6979:
6978:
6970:
6964:
6959:
6958:
6949:
6942:
6936:
6929:
6923:
6920:(in German),
6919:
6915:
6908:
6902:15 pp. 11â23.
6901:
6897:
6893:
6889:
6883:
6875:
6869:
6865:
6860:
6859:
6850:
6848:
6839:
6833:
6829:
6822:
6820:
6808:
6807:
6799:
6797:
6788:
6782:
6778:
6774:
6767:
6765:
6763:
6761:
6756:
6746:
6743:
6741:
6738:
6736:
6733:
6732:
6726:
6712:
6706:
6686:
6680:
6677:
6671:
6668:
6665:
6659:
6639:
6633:
6613:
6607:
6604:
6601:
6598:
6595:
6575:
6555:
6549:
6529:
6523:
6520:
6517:
6514:
6511:
6467:
6447:
6444:
6441:
6438:
6435:
6415:
6409:
6406:
6403:
6383:
6377:
6374:
6371:
6327:
6283:
6251:
6247:
6243:
6234:
6232:
6227:
6226:Ernst Zermelo
6223:
6221:
6217:
6213:
6209:
6205:
6201:
6197:
6193:
6189:
6185:
6181:
6177:
6173:
6172:
6167:
6163:
6161:
6157:
6153:
6149:
6145:
6141:
6137:
6133:
6129:
6125:
6121:
6117:
6107:
6105:
6104:Alonzo Church
6100:
6085:
6080:
6077:
6074:
6071:
6068:
6063:
6059:
6056:
6048:
6044:
6025:
6017:
6013:
6009:
6004:
6000:
5996:
5993:
5990:
5985:
5981:
5967:
5963:
5959:
5954:
5950:
5942:
5941:
5940:
5937:
5933:
5929:
5925:
5921:
5916:
5912:
5908:
5904:
5885:
5880:
5876:
5872:
5867:
5863:
5852:
5848:
5844:
5839:
5835:
5827:
5826:
5825:
5822:
5818:
5814:
5795:
5791:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5763:
5759:
5754:
5750:
5742:
5741:
5740:
5738:
5734:
5730:
5726:
5721:
5702:
5694:
5683:
5648:
5625:
5595:
5570:
5562:
5551:
5516:
5509:
5508:universal set
5505:
5501:
5499:
5489:
5465:
5426:
5365:
5362:contains all
5304:
5265:
5204:
5200:
5197:Through this
5195:
5111:
5091:
5071:
5062:
5003:
4987:
4978:
4964:
4944:
4924:
4904:
4884:
4864:
4844:
4824:
4804:
4784:
4764:
4744:
4702:
4698:
4682:
4675:we seek. Let
4674:
4673:contradiction
4669:
4667:
4663:
4659:
4655:
4636:
4613:
4607:
4602:
4597:
4587:
4584:
4581:
4578:
4575:
4562:
4552:
4549:
4546:
4543:
4540:
4527:
4517:
4514:
4511:
4508:
4505:
4492:
4482:
4479:
4476:
4463:
4457:
4443:
4442:
4441:
4325:
4285:
4271:
4264:
4240:
4237:
4234:
4231:
4228:
4222:
4216:
4213:
4210:
4207:
4204:
4201:
4198:
4116:
4110:
4107:
4101:
4098:
4095:
4092:
4089:
4086:
4083:
4077:
4071:
4068:
4065:
4062:
4059:
4053:
4047:
4044:
4041:
4035:
4029:
4023:
4017:
4014:
4011:
4008:
4005:
3999:
3993:
3990:
3987:
3981:
3972:
3947:
3946:
3945:
3874:
3870:
3845:Suppose that
3843:
3841:
3838:, the set of
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3798:
3790:
3787:
3779:
3775:
3759:
3747:
3741:
3739:
3735:
3731:
3727:
3722:
3708:
3688:
3680:
3664:
3656:
3637:
3631:
3628:
3625:
3605:
3585:
3565:
3562:
3559:
3536:
3530:
3527:
3524:
3504:
3484:
3481:
3478:
3458:
3438:
3413:
3409:
3402:
3399:
3391:
3387:
3380:
3360:
3340:
3332:
3316:
3308:
3292:
3269:
3266:
3261:
3257:
3253:
3248:
3244:
3237:
3234:
3214:
3206:
3203:", this is a
3187:
3181:
3178:
3175:
3155:
3146:
3132:
3112:
3092:
3072:
3052:
3032:
3012:
2992:
2972:
2952:
2932:
2924:
2908:
2888:
2868:
2859:
2839:
2823:
2820:
2817:
2811:
2805:
2802:
2782:
2759:
2736:
2716:
2707:
2693:
2673:
2653:
2633:
2613:
2593:
2590:
2584:
2578:
2558:
2555:
2552:
2544:
2521:
2513:
2510:
2504:
2498:
2484:
2478:
2475:
2472:
2465:
2460:
2457:
2454:
2447:
2439:
2425:
2419:
2416:
2413:
2406:
2401:
2398:
2395:
2384:
2383:
2382:
2380:
2379:contradiction
2364:
2361:
2355:
2349:
2329:
2326:
2323:
2314:
2300:
2280:
2277:
2274:
2251:
2245:
2242:
2239:
2219:
2212:cannot equal
2196:
2190:
2167:
2161:
2158:
2155:
2135:
2132:
2129:
2106:
2100:
2097:
2094:
2074:
2067:cannot equal
2051:
2045:
2022:
2016:
2013:
2010:
1987:
1981:
1978:
1975:
1952:
1946:
1943:
1940:
1920:
1917:
1914:
1894:
1886:
1870:
1850:
1827:
1821:
1801:
1781:
1758:
1752:
1749:
1746:
1726:
1723:
1720:
1700:
1697:
1694:
1671:
1662:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1632:
1629:
1622:
1621:
1620:
1606:
1598:
1582:
1559:
1553:
1533:
1530:
1527:
1504:
1498:
1478:
1458:
1438:
1418:
1398:
1377:
1356:
1334:
1313:
1305:
1301:
1285:
1265:
1242:
1236:
1233:
1230:
1224:
1218:
1215:
1205:
1185:
1169:
1166:
1163:
1157:
1151:
1148:
1123:
1117:
1111:
1105:
1082:
1066:
1063:
1060:
1037:
1007:
999:
993:
977:
955:
952:
946:
940:
920:
911:
901:
882:
876:
873:
870:
862:
859:
856:
830:
824:
821:
818:
810:
807:
804:
795:
788:
773:
766:
748:
745:
739:
733:
713:
710:
707:
685:
663:
660:
657:
634:
621:
618:
610:
588:
582:
579:
576:
573:
570:
567:
564:
558:
555:
542:
528:
502:
486:
483:
480:
474:
468:
465:
457:
438:
422:
419:
416:
393:
363:
343:
327:
319:
317:
313:
309:
304:
303:for details.
302:
298:
294:
290:
285:
283:
267:
264:
259:
255:
232:
228:
207:
199:
195:
191:
186:
172:
164:
148:
145:
137:
134:known as the
121:
118:
110:
94:
87:
83:
79:
70:
66:
62:
60:
49:
45:
41:
37:
33:
28:
22:
9682:Georg Cantor
9628:
9426:Ultraproduct
9273:Model theory
9238:Independence
9174:Formal proof
9166:Proof theory
9149:
9122:
9079:real numbers
9051:second-order
8962:Substitution
8839:Metalanguage
8780:conservative
8753:Axiom schema
8697:Constructive
8667:MorseâKelley
8633:Set theories
8612:Aleph number
8605:inaccessible
8511:Grothendieck
8395:intersection
8282:Higher-order
8270:Second-order
8216:Truth tables
8173:Venn diagram
8045:
7956:Formal proof
7828:Georg Cantor
7823:Paul Bernays
7754:MorseâKelley
7729:
7710:
7662:
7661:Subset
7608:hereditarily
7570:Venn diagram
7528:ordered pair
7443:Intersection
7387:Axiom schema
7208:Independence
7183:Decidability
7178:Completeness
7128:
7123:
7067:
7045:
7017:
7014:Jech, Thomas
6984:
6981:Halmos, Paul
6956:
6948:
6935:
6927:
6921:
6917:
6907:
6899:
6887:
6882:
6857:
6827:
6805:
6772:
6240:
6224:
6219:
6215:
6211:
6207:
6203:
6199:
6195:
6191:
6187:
6183:
6179:
6169:
6164:
6159:
6155:
6151:
6147:
6143:
6139:
6135:
6131:
6127:
6113:
6101:
6041:Had we used
6040:
5935:
5931:
5919:
5914:
5910:
5900:
5820:
5816:
5812:
5810:
5736:
5732:
5724:
5722:
5661:, therefore
5588:for any set
5502:
5495:
5196:
5063:
4979:
4696:
4670:
4665:
4657:
4651:
4286:
4131:
3944:looks like:
3869:equinumerous
3844:
3751:
3745:
3723:
3307:linear order
3147:
2945:must map to
2860:
2708:
2540:
2315:
1686:
1207:
786:
764:
548:
329:
325:
308:Georg Cantor
305:
293:real numbers
286:
187:
81:
75:
56:
39:
35:
31:
9536:Type theory
9484:undecidable
9416:Truth value
9303:equivalence
8982:non-logical
8595:Enumeration
8585:Isomorphism
8532:cardinality
8516:Von Neumann
8481:Ultrafilter
8446:Uncountable
8380:equivalence
8297:Quantifiers
8287:Fixed-point
8256:First-order
8136:Consistency
8121:Proposition
8098:Traditional
8069:Lindström's
8059:Compactness
8001:Type theory
7946:Cardinality
7853:Thomas Jech
7696:Alternative
7675:Uncountable
7629:Ultrafilter
7488:Cardinality
7392:replacement
7333:Determinacy
7198:Metatheorem
7156:of geometry
7141:Consistency
5733:Russell set
5464:cardinality
5425:cardinality
5303:cardinality
5264:cardinality
5203:cardinality
4977:can exist.
4666:non-selfish
3451:and column
2985:can map to
970:Therefore,
763:From
194:enumeration
190:finite sets
163:cardinality
9667:Set theory
9651:Categories
9347:elementary
9040:arithmetic
8908:Quantifier
8886:functional
8758:Expression
8476:Transitive
8420:identities
8405:complement
8338:hereditary
8321:Set theory
7848:Kurt Gödel
7833:Paul Cohen
7670:Transitive
7438:Identities
7422:Complement
7409:Operations
7370:Regularity
7338:projective
7301:Adjunction
7260:Set theory
7018:Set Theory
6926:, also in
6751:References
5388:inside of
5364:singletons
4897:is not in
3578:. The set
2571:such that
2342:such that
1774:. For all
1098:such that
898: via
726:such that
456:surjective
78:set theory
9618:Supertask
9521:Recursion
9479:decidable
9313:saturated
9291:of models
9214:deductive
9209:axiomatic
9129:Hilbert's
9116:Euclidean
9097:canonical
9020:axiomatic
8952:Signature
8881:Predicate
8770:Extension
8692:Ackermann
8617:Operation
8496:Universal
8486:Recursive
8461:Singleton
8456:Inhabited
8441:Countable
8431:Types of
8415:power set
8385:partition
8302:Predicate
8248:Predicate
8163:Syllogism
8153:Soundness
8126:Inference
8116:Tautology
8018:paradoxes
7781:Paradoxes
7701:Axiomatic
7680:Universal
7656:Singleton
7651:Recursive
7594:Countable
7589:Amorphous
7448:Power set
7365:Power set
7316:dependent
7311:countable
7173:Soundness
7109:Metalogic
7069:MathWorld
7052:EMS Press
6710:→
6684:→
6666:−
6637:→
6611:→
6605:×
6553:→
6527:→
6521:×
6439:∘
6436:α
6413:→
6381:→
6372:α
6010:∉
5997:∧
5991:∈
5974:⟺
5960:∈
5873:∉
5859:⟺
5845:∈
5772:∈
5727:with the
5695:≤
5002:bijection
4701:power set
4608:⋮
4603:⋮
4598:⋮
4568:⟷
4533:⟷
4498:⟷
4469:⟷
4272:∅
4263:empty set
4241:∈
4217:…
4111:…
4102:…
3979:∅
3873:power set
3871:with its
3823:…
3629:∈
3563:∈
3552:for each
3528:∈
3482:∈
3331:power set
3329:from the
3270:…
3179:∈
2824:
2806:
2585:ξ
2556:∈
2553:ξ
2505:ξ
2485:ξ
2476:∈
2473:ξ
2469:⟺
2458:∈
2455:ξ
2426:ξ
2417:∉
2414:ξ
2410:⟺
2399:∈
2396:ξ
2356:ξ
2327:∈
2324:ξ
2278:∈
2243:∉
2159:∉
2133:∉
2098:∈
2014:∈
1979:∉
1944:∈
1918:∈
1794:the sets
1750:∉
1724:∈
1698:∈
1648:∣
1642:∈
1531:∈
1237:
1219:
1170:
1152:
1070:→
947:ξ
921:φ
918:¬
915:⇔
912:φ
883:ξ
874:∉
871:ξ
867:⟺
860:∈
857:ξ
822:∉
815:⟺
808:∈
740:ξ
711:∈
708:ξ
661:⊆
622:∈
580:∉
574:∣
568:∈
487:
469:
426:→
198:empty set
185:itself.
136:power set
48:inclusion
9603:Logicism
9596:timeline
9572:Concrete
9431:Validity
9401:T-schema
9394:Kripke's
9389:Tarski's
9384:semantic
9374:Strength
9323:submodel
9318:spectrum
9286:function
9134:Tarski's
9123:Elements
9110:geometry
9066:Robinson
8987:variable
8972:function
8945:spectrum
8935:Sentence
8891:variable
8834:Language
8787:Relation
8748:Automata
8738:Alphabet
8722:language
8576:-jection
8554:codomain
8540:Function
8501:Universe
8471:Infinite
8375:Relation
8158:Validity
8148:Argument
8046:theorem,
7785:Problems
7689:Theories
7665:Superset
7641:Infinite
7470:Concepts
7350:Infinity
7267:Overview
7016:(2002),
6729:See also
6246:category
6078:∉
5784:∉
5327:because
5004:between
3738:Isabelle
3105:maps to
2686:, i.e.,
2232:because
2087:because
1654:∉
1520:for any
996:We know
933:, since
650:because
409:. Then
297:integers
289:infinite
280:for all
9545:Related
9342:Diagram
9240: (
9219:Hilbert
9204:Systems
9199:Theorem
9077:of the
9022:systems
8802:Formula
8797:Grammar
8713: (
8657:General
8370:Forcing
8355:Element
8275:Monadic
8050:paradox
7991:Theorem
7927:General
7723:General
7718:Zermelo
7624:subbase
7606: (
7545:Forcing
7523:Element
7495: (
7473:Methods
7360:Pairing
7054:, 2001
6924:: 75â78
6110:History
6045:(as in
5907:Zermelo
4837:, then
4662:subsets
4658:selfish
4654:subsets
4324:element
1933:either
1302:but no
765:for all
678:Assume
454:is not
109:subsets
44:ordered
9308:finite
9071:Skolem
9024:
8999:Theory
8967:Symbol
8957:String
8940:atomic
8817:ground
8812:closed
8807:atomic
8763:ground
8726:syntax
8622:binary
8549:domain
8466:Finite
8231:finite
8089:Logics
8048:
7996:Theory
7614:Filter
7604:Finite
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