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A consequence of this definition is that every natural number is equal to the set of all preceding natural numbers. The count of elements in each set, at the top level, is the same as the represented natural number, and the nesting depth of the most deeply nested empty set {}, including its nesting
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The negation of the axiom of infinity cannot be derived from the rest of the axioms of ZFC, if they are consistent. (This is tantamount to saying that ZFC is consistent, if the other axioms are consistent.) Thus, ZFC implies neither the axiom of infinity nor its negation and is compatible with
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is an infinite set without saying much about its structure. However, with the help of the other axioms of ZF, we can show that this implies the existence of Ï. First, if we take the powerset of any infinite set
389:{\displaystyle \exists I\ (\exists o\ (o\in I\ \land \ \lnot \exists n\ \ n\in o)\ \land \ \forall x\ (x\in I\Rightarrow \exists y\ (y\in I\ \land \ \forall a\ (a\in y\Leftrightarrow (a\in x\ \lor \ a=x))))).}
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of the same cardinality (or zero, if there is no such ordinal). The result will be an infinite set of ordinals. Then we can apply the axiom of union to that to get an ordinal greater than or equal to Ï.
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is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the
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To extract the natural numbers, we need a definition of which sets are natural numbers. The natural numbers can be defined in a way that does not assume any axioms except the
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also satisfies ZFC â Infinity + ÂŹInfinity, as all of its axioms are universally quantified, and thus trivially satisfied if no set exists.
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For uniqueness, first note that any set that satisfies (*) is itself inductive, since 0 is in all inductive sets, and if an element
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This construction forms the natural numbers. However, the other axioms are insufficient to prove the existence of the set of
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be an inductive set guaranteed by the Axiom of
Infinity. Then we use the axiom schema of specification to define our set
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in the set that represents the number of which it is a part, is also equal to the natural number that the set represents.
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The axiom of infinity cannot be proved from the other axioms of ZFC if they are consistent. (To see why, note that ZFC
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is in all inductive sets, then by the inductive property so is its successor. Thus if there were another set
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603:{\displaystyle \exists I\,(\varnothing \in I\,\land \,\forall x\,(x\in I\Rightarrow \,(x\cup \{x\})\in I)).}
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is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, the
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2442:, and conversely large cardinal axioms are sometimes called stronger axioms of infinity.
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1429:{\displaystyle \Phi (x)=(\emptyset \in x\wedge \forall y(y\in x\to (y\cup \{y\}\in x)))}
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Some old texts use an apparently weaker version of the axiom of infinity, to wit:
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981:{\displaystyle \forall n(n\in \mathbf {N} \iff (\,\,\land \,\,\forall m\in n)).}
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1532:{\displaystyle \forall x(x\in W\leftrightarrow \forall I(\Phi (I)\to x\in I)).}
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2495:. Princeton, NJ: D. Van Nostrand Company. Reprinted 1974 by Springer-Verlag.
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under the operation of taking the successor; that is, for each element of
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4 = {0, 1, 2, 3} = { {}, {{}}, { {}, {{}} }, { {}, {{}}, {{}, {{}}} } }.
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For existence, we will use the Axiom of
Infinity combined with the
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of the
ZermeloâFraenkel axioms, the axiom is expressed as follows:
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In technical language, this formal expression is interpreted as "
2438:. Thus the axiom of infinity is sometimes regarded as the first
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2297:, then that powerset will contain elements that are subsets of
2511:
Set Theory: The Third
Millennium Edition, Revised and Expanded
2364:, we can build a model of ZFC â Infinity + (ÂŹInfinity). It is
2084:
Both these methods produce systems that satisfy the axioms of
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can be applied to remove unwanted elements, leaving the set
1637:{\displaystyle W=\{x\in I:\forall J(\Phi (J)\to x\in J)\}}
409:(the set that is postulated to be infinite) such that the
1128:{\displaystyle \forall n(n\in \mathbf {N} \iff (\;\land }
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Some mathematicians may call a set built this way an
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Extracting the natural numbers from the infinite set
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49:. Unsourced material may be challenged and removed.
2529:Set Theory: An Introduction to Independence Proofs
2477:Untersuchungen ĂŒber die Grundlagen der Mengenlehre
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2124:systems, and since they are isomorphic under the
1335:be the formula that says "x is inductive"; i.e.
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674:2 = 1 âȘ {1} = {0} âȘ {1} = {0, 1} = { {}, {{}} },
2403:The cardinality of the set of natural numbers,
626:von Neumann construction of the natural numbers
145:. It guarantees the existence of at least one
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1306:An alternative method is the following. Let
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109:Learn how and when to remove this message
753:, that includes all the natural numbers.
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2120:. Thus they both completely determine
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2268:This says that there is an element in
1838:that satisfied (*) we would have that
624:This axiom is closely related to the
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2639:Hilbert's paradox of the Grand Hotel
2349:Con(ZFC â Infinity) and use Gödel's
666:1 = 0 âȘ {0} = {} âȘ {0} = {0} = {{}}.
662:The number 1 is the successor of 0:
47:adding citations to reliable sources
16:Axiom of Zermelo-Fraenkel set theory
2434:), has many of the properties of a
670:Likewise, 2 is the successor of 1:
461:consisting of just the elements of
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2048:{\displaystyle \omega \subseteq I}
2022:{\displaystyle I\subseteq \omega }
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2629:Controversy over Cantor's theory
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759:von NeumannâBernaysâGödel axioms
719:{\displaystyle \mathbb {N} _{0}}
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2690:Synthetic differential geometry
2092:allows us to quantify over the
620:Interpretation and consequences
34:needs additional citations for
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2551:(3 ed.). Marcel Dekker.
2385:{\displaystyle V_{\omega }\!}
2351:Second incompleteness theorem
2284:that is a strict superset of
1913:{\displaystyle W\subseteq W'}
1862:{\displaystyle W'\subseteq W}
1545:Axiom schema of specification
775:axiom schema of specification
2736:Cardinality of the continuum
2280:there is another element of
2136:An apparently weaker version
1996:denote this unique element.
628:in set theory, in which the
153:. It was first published by
2427:{\displaystyle \aleph _{0}}
143:ZermeloâFraenkel set theory
3585:
3414:von NeumannâBernaysâGödel
2699:Formalizations of infinity
2549:Introduction to Set Theory
437:, there exists an element
3215:One-to-one correspondence
2952:
2875:Gottfried Wilhelm Leibniz
2074:{\displaystyle I=\omega }
2003:immediately follows: If
2394:hereditarily finite sets
2305:(among other subsets of
2029:is inductive, then also
1770:, so it must also be in
1328:{\displaystyle \Phi (x)}
991:Or, even more formally:
2880:August Ferdinand Möbius
2663:Branches of mathematics
2654:Paradoxes of set theory
2342:{\displaystyle \vdash }
2128:, they must in fact be
2109:{\displaystyle \omega }
2086:second-order arithmetic
1989:{\displaystyle \omega }
790:axiom of extensionality
783:axiom of extensionality
730:that contains 0 and is
508:can be abbreviated as:
3173:Constructible universe
3000:Constructibility (V=L)
2428:
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2272:and for every element
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2001:principle of induction
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1703:{\displaystyle x\in W}
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3396:Principia Mathematica
3230:Transfinite induction
3089:(i.e. set difference)
2844:Möbius transformation
2741:Dedekind-infinite set
2649:Paradoxes of infinity
2644:Infinity (philosophy)
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3564:Axioms of set theory
3470:Burali-Forti paradox
3225:Set-builder notation
3178:Continuum hypothesis
3118:Symmetric difference
2680:Nonstandard analysis
2440:large cardinal axiom
2411:
2368:
2362:von Neumann universe
2333:
2288:. This implies that
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1969:{\displaystyle W=W'}
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1945:is inductive. Thus
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125:and the branches of
123:axiomatic set theory
43:improve this article
3431:TarskiâGrothendieck
2849:Riemannian manifold
2818:Transfinite numbers
2675:Internal set theory
2513:. Springer-Verlag.
58:"Axiom of infinity"
3020:Limitation of size
2802:Sphere at infinity
2753:(Complex infinity)
2424:
2382:
2360:Indeed, using the
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2255:
2118:second-order logic
2106:
2090:axiom of power set
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2045:
2019:
1986:
1966:
1938:{\displaystyle W'}
1935:
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1889:is inductive, and
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1831:{\displaystyle W'}
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1302:Alternative method
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794:axiom of induction
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3550:
3460:Russell's paradox
3409:ZermeloâFraenkel
3310:Dedekind-infinite
3183:Diagonal argument
3082:Cartesian product
2946:Set (mathematics)
2898:
2897:
2792:Point at infinity
2772:Hyperreal numbers
2746:Directed infinity
2711:Absolute infinite
2634:Galileo's paradox
2619:Ananta (infinite)
1882:{\displaystyle W}
1806:{\displaystyle x}
1783:{\displaystyle W}
1763:{\displaystyle I}
1743:{\displaystyle x}
1723:{\displaystyle x}
1677:{\displaystyle I}
1657:{\displaystyle W}
1560:{\displaystyle I}
1449:{\displaystyle W}
769:The infinite set
697:natural numbers,
494:{\displaystyle x}
474:{\displaystyle x}
450:{\displaystyle y}
426:{\displaystyle x}
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133:that use it, the
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3533:Bertrand Russell
3523:John von Neumann
3508:Abraham Fraenkel
3503:Richard Dedekind
3465:Suslin's problem
3376:Cantor's theorem
3093:De Morgan's laws
2958:
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2918:
2911:
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2890:Abraham Robinson
2885:Bernhard Riemann
2804:(Kleinian group)
2797:Regular cardinal
2751:Division by zero
2731:Cardinal numbers
2670:Complex analysis
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2562:
2543:Hrbacek, Karel;
2493:Naive Set Theory
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500:
498:
497:
492:
480:
478:
477:
472:
456:
454:
453:
448:
432:
430:
429:
424:
395:
393:
392:
387:
356:
350:
320:
311:
305:
290:
266:
257:
251:
236:
233:
221:
215:
200:
188:
165:Formal statement
114:
107:
103:
100:
94:
92:
51:
27:
19:
3584:
3583:
3579:
3578:
3577:
3575:
3574:
3573:
3554:
3553:
3552:
3547:
3474:
3453:
3437:
3402:New Foundations
3349:
3239:
3158:Cardinal number
3141:
3127:
3068:
2959:
2950:
2934:
2929:
2899:
2894:
2853:
2822:
2813:Surreal numbers
2787:Ordinal numbers
2716:Actual infinity
2694:
2658:
2607:
2601:
2595:
2565:
2559:
2542:
2484:
2483:
2474:
2470:
2465:
2448:
2414:
2409:
2408:
2392:, the class of
2371:
2366:
2365:
2331:
2330:
2327:
2145:
2144:
2138:
2098:
2097:
2057:
2056:
2031:
2030:
2005:
2004:
1978:
1977:
1958:
1947:
1946:
1927:
1922:
1921:
1902:
1891:
1890:
1871:
1870:
1845:
1840:
1839:
1820:
1815:
1814:
1795:
1794:
1772:
1771:
1752:
1751:
1732:
1731:
1712:
1711:
1686:
1685:
1666:
1665:
1646:
1645:
1569:
1568:
1549:
1548:
1461:
1460:
1438:
1437:
1337:
1336:
1308:
1307:
1304:
1138:
1137:
996:
995:
801:
800:
767:
704:
699:
698:
652:natural numbers
622:
512:
511:
483:
482:
463:
462:
460:
439:
438:
436:
415:
414:
408:
177:
176:
171:formal language
167:
157:as part of his
151:natural numbers
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3582:
3580:
3572:
3571:
3566:
3556:
3555:
3549:
3548:
3546:
3545:
3540:
3538:Thoralf Skolem
3535:
3530:
3525:
3520:
3515:
3510:
3505:
3500:
3495:
3490:
3484:
3482:
3476:
3475:
3473:
3472:
3467:
3462:
3456:
3454:
3452:
3451:
3448:
3442:
3439:
3438:
3436:
3435:
3434:
3433:
3428:
3423:
3422:
3421:
3406:
3405:
3404:
3392:
3391:
3390:
3379:
3378:
3373:
3368:
3363:
3357:
3355:
3351:
3350:
3348:
3347:
3342:
3337:
3332:
3323:
3318:
3313:
3303:
3298:
3297:
3296:
3291:
3286:
3276:
3266:
3261:
3256:
3250:
3248:
3241:
3240:
3238:
3237:
3232:
3227:
3222:
3220:Ordinal number
3217:
3212:
3207:
3202:
3201:
3200:
3195:
3185:
3180:
3175:
3170:
3165:
3155:
3150:
3144:
3142:
3140:
3139:
3136:
3132:
3129:
3128:
3126:
3125:
3120:
3115:
3110:
3105:
3100:
3098:Disjoint union
3095:
3090:
3084:
3078:
3076:
3070:
3069:
3067:
3066:
3065:
3064:
3059:
3048:
3047:
3045:Martin's axiom
3042:
3037:
3032:
3027:
3022:
3017:
3012:
3010:Extensionality
3007:
3002:
2997:
2996:
2995:
2990:
2985:
2975:
2969:
2967:
2961:
2960:
2953:
2951:
2949:
2948:
2942:
2940:
2936:
2935:
2930:
2928:
2927:
2920:
2913:
2905:
2896:
2895:
2893:
2892:
2887:
2882:
2877:
2872:
2867:
2861:
2859:
2858:Mathematicians
2855:
2854:
2852:
2851:
2846:
2841:
2836:
2830:
2828:
2824:
2823:
2821:
2820:
2815:
2810:
2805:
2799:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2762:Gimel function
2759:
2757:Epsilon number
2754:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2702:
2700:
2696:
2695:
2693:
2692:
2687:
2682:
2677:
2672:
2666:
2664:
2660:
2659:
2657:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2621:
2615:
2613:
2609:
2608:
2596:
2594:
2593:
2586:
2579:
2571:
2564:
2563:
2557:
2540:
2522:
2504:
2485:
2482:
2481:
2467:
2466:
2464:
2461:
2460:
2459:
2454:
2447:
2444:
2436:large cardinal
2421:
2417:
2378:
2374:
2338:
2326:
2323:
2318:ordinal number
2266:
2265:
2254:
2250:
2247:
2244:
2241:
2238:
2235:
2231:
2227:
2224:
2221:
2218:
2215:
2212:
2208:
2204:
2201:
2198:
2195:
2192:
2189:
2185:
2181:
2178:
2175:
2172:
2169:
2165:
2162:
2159:
2155:
2152:
2137:
2134:
2105:
2070:
2067:
2064:
2044:
2041:
2038:
2018:
2015:
2012:
1985:
1964:
1961:
1957:
1954:
1933:
1930:
1908:
1905:
1901:
1898:
1878:
1858:
1855:
1851:
1848:
1826:
1823:
1802:
1779:
1759:
1739:
1719:
1699:
1696:
1693:
1673:
1653:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1556:
1541:
1540:
1528:
1525:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1445:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1324:
1321:
1318:
1315:
1303:
1300:
1299:
1298:
1297:
1296:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1236:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1124:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1086:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1024:
1019:
1015:
1012:
1009:
1006:
1003:
989:
988:
977:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
924:
919:
916:
913:
910:
907:
904:
901:
898:
893:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
850:
845:
842:
839:
836:
833:
829:
824:
820:
817:
814:
811:
808:
766:
763:
755:
754:
713:
708:
687:
686:
683:
676:
675:
668:
667:
660:
659:
638:is defined as
621:
618:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
562:
559:
556:
553:
550:
546:
543:
539:
535:
532:
529:
526:
522:
519:
490:
470:
458:
446:
434:
422:
406:
385:
382:
379:
376:
373:
370:
367:
364:
361:
355:
349:
346:
343:
340:
337:
334:
331:
328:
325:
319:
316:
310:
304:
301:
298:
295:
289:
286:
283:
280:
277:
274:
271:
265:
262:
256:
250:
247:
244:
241:
232:
229:
226:
220:
214:
211:
208:
205:
199:
196:
193:
187:
184:
166:
163:
137:is one of the
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3581:
3570:
3567:
3565:
3562:
3561:
3559:
3544:
3543:Ernst Zermelo
3541:
3539:
3536:
3534:
3531:
3529:
3528:Willard Quine
3526:
3524:
3521:
3519:
3516:
3514:
3511:
3509:
3506:
3504:
3501:
3499:
3496:
3494:
3491:
3489:
3486:
3485:
3483:
3481:
3480:Set theorists
3477:
3471:
3468:
3466:
3463:
3461:
3458:
3457:
3455:
3449:
3447:
3444:
3443:
3440:
3432:
3429:
3427:
3426:KripkeâPlatek
3424:
3420:
3417:
3416:
3415:
3412:
3411:
3410:
3407:
3403:
3400:
3399:
3398:
3397:
3393:
3389:
3386:
3385:
3384:
3381:
3380:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3358:
3356:
3352:
3346:
3343:
3341:
3338:
3336:
3333:
3331:
3329:
3324:
3322:
3319:
3317:
3314:
3311:
3307:
3304:
3302:
3299:
3295:
3292:
3290:
3287:
3285:
3282:
3281:
3280:
3277:
3274:
3270:
3267:
3265:
3262:
3260:
3257:
3255:
3252:
3251:
3249:
3246:
3242:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3199:
3196:
3194:
3191:
3190:
3189:
3186:
3184:
3181:
3179:
3176:
3174:
3171:
3169:
3166:
3163:
3159:
3156:
3154:
3151:
3149:
3146:
3145:
3143:
3137:
3134:
3133:
3130:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3088:
3085:
3083:
3080:
3079:
3077:
3075:
3071:
3063:
3062:specification
3060:
3058:
3055:
3054:
3053:
3050:
3049:
3046:
3043:
3041:
3038:
3036:
3033:
3031:
3028:
3026:
3023:
3021:
3018:
3016:
3013:
3011:
3008:
3006:
3003:
3001:
2998:
2994:
2991:
2989:
2986:
2984:
2981:
2980:
2979:
2976:
2974:
2971:
2970:
2968:
2966:
2962:
2957:
2947:
2944:
2943:
2941:
2937:
2933:
2926:
2921:
2919:
2914:
2912:
2907:
2906:
2903:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2871:
2870:David Hilbert
2868:
2866:
2863:
2862:
2860:
2856:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2831:
2829:
2825:
2819:
2816:
2814:
2811:
2809:
2806:
2803:
2800:
2798:
2795:
2793:
2790:
2788:
2785:
2783:
2782:Infinitesimal
2780:
2778:
2775:
2773:
2770:
2768:
2767:Hilbert space
2765:
2763:
2760:
2758:
2755:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2703:
2701:
2697:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2667:
2665:
2661:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2620:
2617:
2616:
2614:
2610:
2604:
2599:
2592:
2587:
2585:
2580:
2578:
2573:
2572:
2569:
2560:
2558:0-8247-7915-0
2554:
2550:
2546:
2541:
2538:
2537:0-444-86839-9
2534:
2530:
2526:
2525:Kenneth Kunen
2523:
2520:
2519:3-540-44085-2
2516:
2512:
2508:
2505:
2502:
2501:0-387-90092-6
2498:
2494:
2490:
2487:
2486:
2478:
2472:
2469:
2462:
2458:
2455:
2453:
2450:
2449:
2445:
2443:
2441:
2437:
2419:
2406:
2401:
2399:
2395:
2376:
2372:
2363:
2358:
2354:
2352:
2336:
2324:
2322:
2319:
2316:
2312:
2308:
2304:
2300:
2296:
2291:
2287:
2283:
2279:
2275:
2271:
2252:
2239:
2236:
2233:
2229:
2225:
2222:
2219:
2213:
2202:
2199:
2196:
2190:
2183:
2176:
2173:
2170:
2163:
2153:
2143:
2142:
2141:
2135:
2133:
2131:
2127:
2123:
2119:
2103:
2095:
2091:
2087:
2082:
2068:
2065:
2062:
2042:
2039:
2036:
2016:
2013:
2010:
2002:
1997:
1983:
1962:
1959:
1955:
1952:
1931:
1928:
1906:
1903:
1899:
1896:
1876:
1856:
1853:
1849:
1846:
1824:
1821:
1800:
1791:
1777:
1757:
1737:
1717:
1697:
1694:
1691:
1671:
1651:
1625:
1622:
1619:
1610:
1598:
1592:
1589:
1586:
1583:
1577:
1574:
1554:
1546:
1526:
1517:
1514:
1511:
1502:
1490:
1481:
1478:
1475:
1469:
1459:
1458:
1457:
1443:
1414:
1411:
1405:
1399:
1396:
1387:
1384:
1381:
1375:
1369:
1366:
1363:
1354:
1348:
1319:
1301:
1283:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1230:
1227:
1224:
1218:
1212:
1209:
1206:
1203:
1197:
1191:
1185:
1182:
1179:
1170:
1158:
1155:
1152:
1146:
1136:
1135:
1122:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1080:
1077:
1074:
1068:
1062:
1056:
1050:
1047:
1044:
1035:
1013:
1010:
1004:
994:
993:
992:
975:
957:
951:
948:
945:
942:
936:
933:
930:
922:
914:
911:
905:
902:
899:
891:
877:
871:
868:
865:
862:
856:
848:
840:
837:
818:
815:
809:
799:
798:
797:
795:
791:
786:
784:
780:
776:
772:
764:
762:
760:
752:
748:
747:
746:
743:
741:
737:
733:
729:
711:
696:
691:
684:
681:
680:
679:
673:
672:
671:
665:
664:
663:
657:
656:
655:
653:
649:
645:
641:
637:
633:
632:
627:
619:
617:
615:
614:inductive set
610:
597:
588:
585:
576:
570:
567:
557:
554:
551:
544:
537:
533:
530:
520:
509:
507:
502:
488:
468:
444:
420:
412:
405:
401:
396:
383:
365:
362:
359:
353:
347:
344:
341:
332:
329:
326:
317:
308:
302:
299:
296:
287:
278:
275:
272:
263:
254:
245:
242:
239:
230:
218:
212:
209:
206:
197:
185:
174:
172:
164:
162:
160:
156:
155:Ernst Zermelo
152:
148:
144:
140:
136:
132:
128:
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: â
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3493:Georg Cantor
3488:Paul Bernays
3419:MorseâKelley
3394:
3327:
3326:Subset
3273:hereditarily
3235:Venn diagram
3193:ordered pair
3108:Intersection
3052:Axiom schema
3014:
2865:Georg Cantor
2839:Möbius plane
2777:Infinite set
2721:Aleph number
2548:
2545:Jech, Thomas
2531:. Elsevier.
2528:
2510:
2492:
2476:
2471:
2452:Peano axioms
2439:
2402:
2398:empty domain
2359:
2355:
2328:
2325:Independence
2310:
2306:
2298:
2294:
2289:
2285:
2281:
2277:
2273:
2269:
2267:
2139:
2126:identity map
2088:, since the
2083:
1998:
1792:
1542:
1305:
990:
787:
778:
770:
768:
756:
750:
744:
739:
735:
727:
694:
692:
688:
677:
669:
661:
647:
643:
639:
635:
629:
623:
611:
510:
503:
400:there exists
397:
175:
168:
147:infinite set
134:
120:
105:
99:October 2019
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3518:Thomas Jech
3361:Alternative
3340:Uncountable
3294:Ultrafilter
3153:Cardinality
3057:replacement
3005:Determinacy
2726:Beth number
2507:Thomas Jech
2489:Paul Halmos
2303:cardinality
678:and so on:
127:mathematics
3558:Categories
3513:Kurt Gödel
3498:Paul Cohen
3335:Transitive
3103:Identities
3087:Complement
3074:Operations
3035:Regularity
2973:Adjunction
2932:Set theory
2827:Geometries
2685:Set theory
2463:References
2405:aleph null
2122:isomorphic
2055:, so that
1456:such that
159:set theory
131:philosophy
69:newspapers
3446:Paradoxes
3366:Axiomatic
3345:Universal
3321:Singleton
3316:Recursive
3259:Countable
3254:Amorphous
3113:Power set
3030:Power set
2988:dependent
2983:countable
2808:Supertask
2475:Zermelo:
2416:ℵ
2377:ω
2337:⊢
2237:⊊
2230:∧
2223:∈
2211:∃
2207:→
2200:∈
2188:∀
2184:∧
2174:∈
2161:∃
2151:∃
2104:ω
2094:power set
2069:ω
2040:⊆
2037:ω
2017:ω
2014:⊆
1984:ω
1900:⊆
1854:⊆
1695:∈
1623:∈
1617:→
1605:Φ
1596:∀
1587:∈
1515:∈
1509:→
1497:Φ
1488:∀
1485:↔
1479:∈
1467:∀
1412:∈
1400:∪
1391:→
1385:∈
1373:∀
1370:∧
1364:∈
1361:∅
1343:Φ
1314:Φ
1251:∨
1245:∈
1235:⟺
1228:∈
1216:∀
1213:∧
1207:∈
1195:∃
1192:∨
1183:∈
1177:¬
1168:∀
1162:⇒
1156:∈
1144:∀
1123:∧
1101:∨
1095:∈
1085:⟺
1078:∈
1066:∀
1060:∃
1057:∨
1048:∈
1042:¬
1033:∀
1023:⟺
1014:∈
1002:∀
952:∪
934:∈
928:∃
923:∨
918:∅
903:∈
897:∀
892:∧
872:∪
854:∃
849:∨
844:∅
828:⟺
819:∈
807:∀
631:successor
586:∈
571:∪
561:⇒
555:∈
542:∀
538:∧
531:∈
528:∅
518:∃
501:itself."
411:empty set
354:∨
345:∈
336:⇔
330:∈
315:∀
309:∧
300:∈
285:∃
282:⇒
276:∈
261:∀
255:∧
243:∈
228:∃
225:¬
219:∧
210:∈
195:∃
183:∃
161:in 1908.
3569:Infinity
3450:Problems
3354:Theories
3330:Superset
3306:Infinite
3135:Concepts
3015:Infinity
2939:Overview
2706:0.999...
2598:Infinity
2547:(1999).
2457:Finitism
2446:See also
2357:either.
2116:, as in
1963:′
1932:′
1907:′
1850:′
1825:′
792:and the
3388:General
3383:Zermelo
3289:subbase
3271: (
3210:Forcing
3188:Element
3160: (
3138:Methods
3025:Pairing
2624:Apeiron
2612:History
2527:(1980)
2509:(2003)
2491:(1960)
2315:initial
2313:by the
1976:. Let
1710:, then
1644:â i.e.
1547:. Let
658:0 = {}.
506:formula
169:In the
83:scholar
3279:Filter
3269:Finite
3205:Family
3148:Almost
2993:global
2978:Choice
2965:Axioms
2555:
2535:
2517:
2499:
1920:since
1869:since
732:closed
646:}. If
357:
351:
321:
312:
306:
291:
267:
258:
252:
237:
234:
222:
216:
201:
189:
139:axioms
85:
78:
71:
64:
56:
3371:Naive
3301:Fuzzy
3264:Empty
3247:types
3198:tuple
3168:Class
3162:large
3123:Union
3040:Union
2130:equal
504:This
90:JSTOR
76:books
3284:base
2553:ISBN
2533:ISBN
2515:ISBN
2497:ISBN
481:and
129:and
62:news
3245:Set
2353:.)
2276:of
2096:of
1539:(*)
695:all
642:âȘ {
634:of
457:of
433:of
404:set
141:of
121:In
45:by
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2081:.
1790:.
785:.
761:.
742:.
616:.
459:đŒ
435:đŒ
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2220:z
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2171:y
2168:(
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1953:W
1929:W
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860:(
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