3236:
2431:
2473:
1892:
2791:
1790:
2421:
Axiomatically characterizing the theory of hereditarily finite sets, the negation of the axiom of infinity may be added, thus proving that the axiom of infinity is not a consequence of the other axioms of ZF.
1570:
1944:
923:
1727:
1499:
2111:
leads to an element (another such set) that can act as a root vertex in its own right. No automorphism of this graph exist, corresponding to the fact that equal branches are identified (e.g.
849:
1214:
761:
1114:
2889:
1156:
679:
1049:
603:
553:
158:
1973:
503:
2673:
2165:
422:
109:
378:
2404:
2343:
2305:
2071:
1354:
1320:
1251:
273:
2029:
1999:
1641:
336:
3049:
2726:
2618:
2508:
1408:
1381:
1278:
2105:
216:
961:
35:
whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the
3693:
247:
2699:
2540:
2464:
1428:
454:
2827:
2591:
299:
2247:
1595:
1072:
196:
2919:
2847:
2564:
2271:
2185:
988:
784:
702:
626:
2308:
2992:
2273:. All finite von Neumann ordinals are indeed hereditarily finite and, thus, so is the class of sets representing the natural numbers. In other words,
1796:
3160:
2733:
3382:
3202:
1737:
2187:). This graph model enables an implementation of ZF without infinity as data types and thus an interpretation of set theory in expressive
3710:
3143:
Omodeo, Eugenio G.; Policriti, Alberto; Tomescu, Alexandru I. (2017). "3.3: The
Ackermann encoding of hereditarily finite sets".
2199:
3688:
2002:
1506:
1162:
finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when
2472:
3462:
3341:
3024:
The set of all (well-founded) hereditarily finite sets (which is infinite, and not hereditarily finite itself) is written
3705:
1906:
854:
2892:
1646:
1458:
3698:
3336:
3299:
789:
1903:
The
Ackermann coding can be used to construct a model of finitary set theory in the natural numbers. More precisely,
1165:
3064:
2952:
1281:
707:
3387:
3279:
3267:
3262:
2209:, the graph whose vertices correspond to hereditarily finite sets and edges correspond to set membership is the
3195:
1077:
2856:
1119:
631:
3807:
3725:
3600:
3366:
3289:
2937:
2407:
2365:
1004:
558:
508:
164:
Only sets that can be built by a finite number of applications of these two rules are hereditarily finite.
114:
3759:
3640:
3452:
3272:
1949:
461:
2623:
1501:
that maps each hereditarily finite set to a natural number, given by the following recursive definition:
3675:
3589:
3509:
3489:
3467:
3069:
2467:
2108:
2114:
383:
172:
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:
71:
342:
3749:
3739:
3573:
3504:
3457:
3397:
3284:
2480:
2415:
2375:
2314:
2276:
2042:
1431:
1325:
1291:
1222:
252:
3843:
3744:
3655:
3568:
3563:
3558:
3372:
3314:
3252:
3188:
2962:
2957:
2349:
2074:
1455:
introduced an encoding of hereditarily finite sets as natural numbers. It is defined by a function
2012:
1982:
1608:
306:
3667:
3662:
3447:
3402:
3309:
3086:
3027:
2704:
2596:
2486:
2369:
2357:
2353:
2078:
1386:
1359:
1256:
2084:
201:
928:
3524:
3361:
3353:
3324:
3294:
3225:
3156:
2411:
2006:
1976:
1452:
223:
2678:
2516:
2436:
1413:
427:
3812:
3802:
3787:
3782:
3650:
3304:
3148:
3123:
3078:
52:
3170:
2805:
2569:
278:
3681:
3619:
3437:
3257:
3166:
2229:
1577:
1054:
178:
2898:
3007:
3817:
3614:
3595:
3499:
3484:
3441:
3377:
3319:
2947:
2925:
2832:
2549:
2256:
2250:
2170:
973:
769:
687:
611:
3837:
3822:
3792:
3624:
3538:
3533:
3090:
1897:
1435:
3772:
3767:
3585:
3514:
3472:
3331:
3235:
2206:
2195:
3797:
3432:
2982:
2850:
2188:
20:
3777:
3645:
3548:
3211:
3152:
3128:
3111:
2942:
2210:
2198:
exist for ZF and also set theories different from
Zermelo set theory, such as
32:
24:
1887:{\displaystyle \displaystyle f^{-1}(i)=\{f^{-1}(j)\mid {\text{BIT}}(i,j)=1\}}
3580:
3543:
3494:
3392:
2799:
2543:
1598:
48:
36:
2430:
2921:
powers of two), and the union of countably many finite sets is countable.
2483:. Here, the class of all well-founded hereditarily finite sets is denoted
1051:
is an example for such a hereditarily finite set and so is the empty set
993:
1, 1, 1, 2, 3, 6, 12, 25, 52, 113, 247, 548, 1226, 2770, 6299, 14426, ...
2081:
is the identity): The root vertex corresponds to the top level bracket
3082:
2786:{\displaystyle \displaystyle V_{\omega }=\bigcup _{k=0}^{\infty }V_{k}}
3605:
3427:
2361:
1785:{\displaystyle \displaystyle f^{-1}\colon \omega \to H_{\aleph _{0}}}
3477:
3244:
1602:
3184:
2924:
Equivalently, a set is hereditarily finite if and only if its
1253:, meaning that the cardinality of each member is smaller than
3180:
2226:
In the common axiomatic set theory approaches, the empty set
2167:, trivializing the permutation of the two subgraphs of shape
2077:, namely those without non-trivial symmetries (i.e. the only
3051:
to show its place in the von
Neumann hierarchy of pure sets.
3145:
2986:
16:
Finite sets whose elements are all hereditarily finite sets
2202:
theories. Such models have more intricate edge structure.
2073:
can be seen to be in exact correspondence with a class of
1565:{\displaystyle \displaystyle f(a)=\sum _{b\in a}2^{f(b)}}
1219:
The class of all hereditarily finite sets is denoted by
3065:"Die Widerspruchsfreiheit der allgemeinen Mengenlehre"
2031:
relation models the membership relation between sets.
3030:
2901:
2859:
2835:
2808:
2737:
2736:
2707:
2681:
2626:
2599:
2572:
2552:
2519:
2489:
2439:
2378:
2317:
2279:
2259:
2232:
2173:
2117:
2087:
2045:
2015:
1985:
1952:
1909:
1800:
1799:
1741:
1740:
1649:
1611:
1580:
1510:
1509:
1461:
1416:
1389:
1362:
1328:
1294:
1259:
1225:
1168:
1122:
1080:
1057:
1007:
976:
931:
857:
792:
772:
710:
690:
634:
614:
561:
511:
464:
430:
386:
345:
309:
281:
255:
226:
204:
181:
117:
74:
3758:
3721:
3633:
3523:
3411:
3352:
3243:
3218:
2798:This formulation shows, again, that there are only
2479:The hereditarily finite sets are a subclass of the
1939:{\displaystyle (\mathbb {N} ,{\text{BIT}}^{\top })}
1597:contains no members, and is therefore mapped to an
918:{\displaystyle \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}}
3043:
2913:
2883:
2841:
2821:
2785:
2720:
2693:
2667:
2612:
2585:
2558:
2534:
2502:
2458:
2398:
2337:
2299:
2265:
2241:
2179:
2159:
2099:
2065:
2023:
2009:. Here, each natural number models a set, and the
1993:
1967:
1938:
1886:
1784:
1722:{\displaystyle 2^{f(a)}+2^{f(b)}+2^{f(c)}+\ldots }
1721:
1635:
1589:
1564:
1494:{\displaystyle f\colon H_{\aleph _{0}}\to \omega }
1493:
1422:
1402:
1375:
1348:
1314:
1272:
1245:
1208:
1150:
1108:
1066:
1043:
982:
955:
917:
843:
778:
755:
696:
673:
620:
597:
547:
497:
448:
416:
372:
330:
293:
267:
241:
210:
190:
152:
103:
1605:. On the other hand, a set with distinct members
2510:. Note that this is also a set in this context.
844:{\displaystyle \{\{\{\{\{\{\{\{\}\}\}\}\}\}\}\}}
1209:{\displaystyle {\mathbb {N} }=\{0,1,2,\dots \}}
3196:
62:: The empty set is a hereditarily finite set.
8:
2236:
2233:
2154:
2142:
2136:
2118:
2094:
2088:
1880:
1826:
1584:
1581:
1203:
1179:
1145:
1142:
1132:
1123:
1103:
1081:
1061:
1058:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1014:
1011:
1008:
950:
932:
912:
909:
906:
903:
900:
897:
891:
888:
885:
879:
876:
873:
870:
864:
861:
858:
838:
835:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
802:
799:
796:
793:
756:{\displaystyle \{\{\{\{\{\{\{\}\}\}\}\}\}\}}
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
668:
665:
662:
659:
656:
653:
650:
647:
644:
641:
638:
635:
592:
589:
586:
583:
580:
577:
574:
568:
565:
562:
542:
539:
536:
533:
530:
527:
521:
518:
515:
512:
492:
489:
486:
483:
480:
477:
474:
471:
468:
465:
443:
431:
411:
408:
405:
402:
399:
393:
390:
387:
367:
364:
361:
358:
355:
352:
349:
346:
325:
322:
319:
316:
313:
310:
288:
282:
262:
256:
236:
233:
230:
227:
185:
182:
147:
118:
3203:
3189:
3181:
3127:
3035:
3029:
2993:On-Line Encyclopedia of Integer Sequences
2900:
2858:
2834:
2813:
2807:
2776:
2766:
2755:
2742:
2735:
2712:
2706:
2680:
2656:
2631:
2625:
2604:
2598:
2577:
2571:
2551:
2518:
2494:
2488:
2447:
2438:
2388:
2383:
2377:
2327:
2322:
2316:
2289:
2284:
2278:
2258:
2231:
2172:
2116:
2086:
2055:
2050:
2044:
2016:
2014:
1986:
1984:
1959:
1954:
1951:
1927:
1922:
1914:
1913:
1908:
1854:
1833:
1805:
1798:
1773:
1768:
1746:
1739:
1698:
1676:
1654:
1648:
1610:
1579:
1546:
1530:
1508:
1477:
1472:
1460:
1415:
1394:
1388:
1367:
1361:
1338:
1333:
1327:
1304:
1299:
1293:
1264:
1258:
1235:
1230:
1224:
1171:
1170:
1169:
1167:
1158:are examples of finite sets that are not
1137:
1136:
1135:
1121:
1109:{\displaystyle \{7,{\mathbb {N} },\pi \}}
1092:
1091:
1090:
1079:
1074:, as noted. On the other hand, the sets
1056:
1006:
975:
930:
856:
791:
771:
709:
689:
633:
613:
560:
510:
463:
429:
385:
344:
308:
280:
254:
225:
203:
180:
141:
125:
116:
95:
79:
73:
2884:{\displaystyle 2\uparrow \uparrow (n-1)}
2429:
1151:{\displaystyle \{3,\{{\mathbb {N} }\}\}}
674:{\displaystyle \{\{\{\{\{\{\}\}\}\}\}\}}
55:hereditarily finite sets is as follows:
2974:
2249:also represents the first von Neumann
2466:represented with circles in place of
2001:, swapping its two arguments) models
1044:{\displaystyle \{\{\},\{\{\{\}\}\}\}}
970:In this way, the number of sets with
598:{\displaystyle \{\{\},\{\{\{\}\}\}\}}
548:{\displaystyle \{\{\{\},\{\{\}\}\}\}}
153:{\displaystyle \{a_{1},\dots a_{k}\}}
7:
1968:{\displaystyle {\text{BIT}}^{\top }}
1356:is in bijective correspondence with
498:{\displaystyle \{\{\{\{\{\}\}\}\}\}}
111:are hereditarily finite, then so is
2668:{\displaystyle V_{i+1}=\wp (V_{i})}
3116:Notre Dame Journal of Formal Logic
2767:
2646:
2520:
2385:
2324:
2286:
2052:
1960:
1928:
1770:
1474:
1364:
1335:
1301:
1261:
1232:
259:
205:
14:
2795:and all its elements are finite.
2309:standard model of natural numbers
2160:{\displaystyle \{t,t,s\}=\{t,s\}}
417:{\displaystyle \{\{\},\{\{\}\}\}}
104:{\displaystyle a_{1},\dots a_{k}}
3234:
2471:
2410:involving these axioms and e.g.
373:{\displaystyle \{\{\{\{\}\}\}\}}
2802:many hereditarily finite sets:
2399:{\displaystyle H_{\aleph _{0}}}
2356:, the very small sub-theory of
2338:{\displaystyle H_{\aleph _{0}}}
2300:{\displaystyle H_{\aleph _{0}}}
2066:{\displaystyle H_{\aleph _{0}}}
1349:{\displaystyle H_{\aleph _{0}}}
1315:{\displaystyle H_{\aleph _{1}}}
1246:{\displaystyle H_{\aleph _{0}}}
2878:
2866:
2863:
2662:
2649:
2529:
2523:
2352:can already be interpreted in
1933:
1910:
1871:
1859:
1848:
1842:
1820:
1814:
1761:
1708:
1702:
1686:
1680:
1664:
1658:
1556:
1550:
1520:
1514:
1485:
268:{\displaystyle \{\emptyset \}}
1:
2307:includes each element in the
1280:. (Analogously, the class of
3147:. Springer. pp. 70–71.
2311:and a set theory expressing
2024:{\displaystyle {\text{BIT}}}
1994:{\displaystyle {\text{BIT}}}
1636:{\displaystyle a,b,c,\dots }
1383:. It can also be denoted by
763:. There are twelve such sets
331:{\displaystyle \{\{\{\}\}\}}
3063:Ackermann, Wilhelm (1937).
3044:{\displaystyle V_{\omega }}
2721:{\displaystyle V_{\omega }}
2620:can be obtained by setting
2613:{\displaystyle V_{\omega }}
2503:{\displaystyle V_{\omega }}
2408:constructive axiomatization
2345:must contain all of those.
2003:Zermelo–Fraenkel set theory
1574:For example, the empty set
1403:{\displaystyle V_{\omega }}
1376:{\displaystyle \aleph _{0}}
1273:{\displaystyle \aleph _{0}}
456:, the Neumann ordinal "2"),
3860:
3694:von Neumann–Bernays–Gödel
2983:Sloane, N. J. A.
2953:Hereditarily countable set
2100:{\displaystyle \{\dots \}}
963:, the Neumann ordinal "3")
766:... sets represented with
684:... sets represented with
608:... sets represented with
301:, the Neumann ordinal "1")
218:, the Neumann ordinal "0")
211:{\displaystyle \emptyset }
3495:One-to-one correspondence
3232:
3153:10.1007/978-3-319-54981-1
3129:10.1215/00294527-2009-009
3008:"hereditarily finite set"
2893:Knuth's up-arrow notation
2829:is finite for any finite
956:{\displaystyle \{0,1,2\}}
681:. There are six such sets
3110:Kirby, Laurence (2009).
2470:
1732:The inverse is given by
242:{\displaystyle \{\{\}\}}
29:hereditarily finite sets
2987:"Sequence A004111"
2938:Constructive set theory
2694:{\displaystyle i\geq 0}
2535:{\displaystyle \wp (S)}
2459:{\displaystyle ~V_{4}~}
2222:Theories of finite sets
1423:{\displaystyle \omega }
449:{\displaystyle \{0,1\}}
3453:Constructible universe
3280:Constructibility (V=L)
3045:
2915:
2885:
2843:
2823:
2787:
2771:
2722:
2695:
2669:
2614:
2587:
2560:
2536:
2504:
2476:
2460:
2400:
2339:
2301:
2267:
2243:
2181:
2161:
2101:
2067:
2025:
1995:
1969:
1940:
1896:where BIT denotes the
1888:
1786:
1723:
1637:
1601:, that is, the number
1591:
1566:
1495:
1424:
1404:
1377:
1350:
1316:
1274:
1247:
1210:
1152:
1110:
1068:
1045:
984:
957:
919:
845:
780:
757:
698:
675:
622:
599:
549:
499:
450:
418:
374:
332:
295:
269:
243:
212:
192:
154:
105:
3676:Principia Mathematica
3510:Transfinite induction
3369:(i.e. set difference)
3112:"Finitary Set Theory"
3070:Mathematische Annalen
3046:
2916:
2886:
2844:
2824:
2822:{\displaystyle V_{n}}
2788:
2751:
2723:
2696:
2670:
2615:
2588:
2586:{\displaystyle V_{0}}
2561:
2537:
2505:
2461:
2433:
2401:
2340:
2302:
2268:
2244:
2182:
2162:
2102:
2068:
2026:
1996:
1970:
1941:
1889:
1787:
1724:
1638:
1592:
1567:
1496:
1425:
1405:
1378:
1351:
1317:
1275:
1248:
1211:
1153:
1111:
1069:
1046:
985:
958:
920:
846:
781:
758:
699:
676:
623:
600:
550:
500:
451:
419:
375:
333:
296:
294:{\displaystyle \{0\}}
270:
244:
213:
193:
155:
106:
3750:Burali-Forti paradox
3505:Set-builder notation
3458:Continuum hypothesis
3398:Symmetric difference
3028:
2899:
2857:
2833:
2806:
2734:
2728:can be expressed as
2705:
2679:
2624:
2597:
2593:the empty set, then
2570:
2550:
2517:
2487:
2481:Von Neumann universe
2437:
2376:
2315:
2277:
2257:
2242:{\displaystyle \{\}}
2230:
2171:
2115:
2085:
2043:
2013:
1983:
1950:
1907:
1797:
1738:
1647:
1609:
1590:{\displaystyle \{\}}
1578:
1507:
1459:
1432:von Neumann universe
1414:
1410:, which denotes the
1387:
1360:
1326:
1292:
1257:
1223:
1166:
1120:
1078:
1067:{\displaystyle \{\}}
1055:
1005:
974:
929:
855:
790:
786:bracket pairs, e.g.
770:
708:
704:bracket pairs, e.g.
688:
632:
628:bracket pairs, e.g.
612:
559:
509:
462:
428:
384:
343:
307:
279:
253:
224:
202:
191:{\displaystyle \{\}}
179:
115:
72:
3711:Tarski–Grothendieck
2958:Hereditary property
2914:{\displaystyle n-1}
2350:Robinson arithmetic
3300:Limitation of size
3083:10.1007/bf01594179
3041:
2996:. OEIS Foundation.
2926:transitive closure
2911:
2881:
2864:↑ ↑
2839:
2819:
2783:
2782:
2718:
2691:
2665:
2610:
2583:
2556:
2532:
2500:
2477:
2456:
2396:
2358:Zermelo set theory
2335:
2297:
2263:
2239:
2177:
2157:
2097:
2063:
2021:
1991:
1965:
1936:
1884:
1883:
1782:
1781:
1719:
1633:
1587:
1562:
1561:
1541:
1491:
1434:. So here it is a
1420:
1400:
1373:
1346:
1312:
1270:
1243:
1206:
1148:
1106:
1064:
1041:
980:
953:
915:
841:
776:
753:
694:
671:
618:
595:
545:
495:
446:
414:
370:
328:
291:
265:
239:
208:
188:
150:
101:
3831:
3830:
3740:Russell's paradox
3689:Zermelo–Fraenkel
3590:Dedekind-infinite
3463:Diagonal argument
3362:Cartesian product
3226:Set (mathematics)
3162:978-3-319-54980-4
2842:{\displaystyle n}
2675:for each integer
2559:{\displaystyle S}
2455:
2442:
2266:{\displaystyle 0}
2213:or random graph.
2180:{\displaystyle t}
2019:
2007:axiom of infinity
1989:
1977:converse relation
1957:
1925:
1857:
1526:
1453:Wilhelm Ackermann
990:bracket pairs is
983:{\displaystyle n}
779:{\displaystyle 8}
697:{\displaystyle 7}
621:{\displaystyle 6}
43:Formal definition
3851:
3813:Bertrand Russell
3803:John von Neumann
3788:Abraham Fraenkel
3783:Richard Dedekind
3745:Suslin's problem
3656:Cantor's theorem
3373:De Morgan's laws
3238:
3205:
3198:
3191:
3182:
3175:
3174:
3140:
3134:
3133:
3131:
3107:
3101:
3100:
3098:
3097:
3060:
3054:
3053:
3050:
3048:
3047:
3042:
3040:
3039:
3021:
3019:
3004:
2998:
2997:
2979:
2920:
2918:
2917:
2912:
2890:
2888:
2887:
2882:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2818:
2817:
2792:
2790:
2789:
2784:
2781:
2780:
2770:
2765:
2747:
2746:
2727:
2725:
2724:
2719:
2717:
2716:
2700:
2698:
2697:
2692:
2674:
2672:
2671:
2666:
2661:
2660:
2642:
2641:
2619:
2617:
2616:
2611:
2609:
2608:
2592:
2590:
2589:
2584:
2582:
2581:
2565:
2563:
2562:
2557:
2541:
2539:
2538:
2533:
2513:If we denote by
2509:
2507:
2506:
2501:
2499:
2498:
2475:
2465:
2463:
2462:
2457:
2453:
2452:
2451:
2440:
2405:
2403:
2402:
2397:
2395:
2394:
2393:
2392:
2368:, Empty Set and
2344:
2342:
2341:
2336:
2334:
2333:
2332:
2331:
2306:
2304:
2303:
2298:
2296:
2295:
2294:
2293:
2272:
2270:
2269:
2264:
2248:
2246:
2245:
2240:
2200:non-well founded
2186:
2184:
2183:
2178:
2166:
2164:
2163:
2158:
2106:
2104:
2103:
2098:
2072:
2070:
2069:
2064:
2062:
2061:
2060:
2059:
2030:
2028:
2027:
2022:
2020:
2017:
2000:
1998:
1997:
1992:
1990:
1987:
1974:
1972:
1971:
1966:
1964:
1963:
1958:
1955:
1945:
1943:
1942:
1937:
1932:
1931:
1926:
1923:
1917:
1893:
1891:
1890:
1885:
1858:
1855:
1841:
1840:
1813:
1812:
1791:
1789:
1788:
1783:
1780:
1779:
1778:
1777:
1754:
1753:
1728:
1726:
1725:
1720:
1712:
1711:
1690:
1689:
1668:
1667:
1642:
1640:
1639:
1634:
1596:
1594:
1593:
1588:
1571:
1569:
1568:
1563:
1560:
1559:
1540:
1500:
1498:
1497:
1492:
1484:
1483:
1482:
1481:
1447:Ackermann coding
1430:th stage of the
1429:
1427:
1426:
1421:
1409:
1407:
1406:
1401:
1399:
1398:
1382:
1380:
1379:
1374:
1372:
1371:
1355:
1353:
1352:
1347:
1345:
1344:
1343:
1342:
1321:
1319:
1318:
1313:
1311:
1310:
1309:
1308:
1279:
1277:
1276:
1271:
1269:
1268:
1252:
1250:
1249:
1244:
1242:
1241:
1240:
1239:
1215:
1213:
1212:
1207:
1175:
1174:
1157:
1155:
1154:
1149:
1141:
1140:
1115:
1113:
1112:
1107:
1096:
1095:
1073:
1071:
1070:
1065:
1050:
1048:
1047:
1042:
989:
987:
986:
981:
962:
960:
959:
954:
924:
922:
921:
916:
850:
848:
847:
842:
785:
783:
782:
777:
762:
760:
759:
754:
703:
701:
700:
695:
680:
678:
677:
672:
627:
625:
624:
619:
604:
602:
601:
596:
554:
552:
551:
546:
504:
502:
501:
496:
455:
453:
452:
447:
423:
421:
420:
415:
379:
377:
376:
371:
337:
335:
334:
329:
300:
298:
297:
292:
274:
272:
271:
266:
248:
246:
245:
240:
217:
215:
214:
209:
197:
195:
194:
189:
159:
157:
156:
151:
146:
145:
130:
129:
110:
108:
107:
102:
100:
99:
84:
83:
3859:
3858:
3854:
3853:
3852:
3850:
3849:
3848:
3834:
3833:
3832:
3827:
3754:
3733:
3717:
3682:New Foundations
3629:
3519:
3438:Cardinal number
3421:
3407:
3348:
3239:
3230:
3214:
3209:
3179:
3178:
3163:
3142:
3141:
3137:
3109:
3108:
3104:
3095:
3093:
3062:
3061:
3057:
3031:
3026:
3025:
3017:
3015:
3006:
3005:
3001:
2981:
2980:
2976:
2971:
2934:
2897:
2896:
2855:
2854:
2831:
2830:
2809:
2804:
2803:
2793:
2772:
2738:
2732:
2731:
2708:
2703:
2702:
2677:
2676:
2652:
2627:
2622:
2621:
2600:
2595:
2594:
2573:
2568:
2567:
2548:
2547:
2515:
2514:
2490:
2485:
2484:
2443:
2435:
2434:
2428:
2384:
2379:
2374:
2373:
2323:
2318:
2313:
2312:
2285:
2280:
2275:
2274:
2255:
2254:
2228:
2227:
2224:
2219:
2217:Axiomatizations
2169:
2168:
2113:
2112:
2083:
2082:
2051:
2046:
2041:
2040:
2037:
2011:
2010:
2005:ZF without the
1981:
1980:
1953:
1948:
1947:
1921:
1905:
1904:
1894:
1829:
1801:
1795:
1794:
1792:
1769:
1764:
1742:
1736:
1735:
1694:
1672:
1650:
1645:
1644:
1607:
1606:
1576:
1575:
1572:
1542:
1505:
1504:
1473:
1468:
1457:
1456:
1449:
1444:
1412:
1411:
1390:
1385:
1384:
1363:
1358:
1357:
1334:
1329:
1324:
1323:
1300:
1295:
1290:
1289:
1260:
1255:
1254:
1231:
1226:
1221:
1220:
1164:
1163:
1118:
1117:
1076:
1075:
1053:
1052:
1003:
1002:
999:
994:
972:
971:
927:
926:
853:
852:
788:
787:
768:
767:
706:
705:
686:
685:
630:
629:
610:
609:
557:
556:
507:
506:
460:
459:
426:
425:
382:
381:
341:
340:
305:
304:
277:
276:
251:
250:
222:
221:
200:
199:
177:
176:
170:
137:
121:
113:
112:
91:
75:
70:
69:
45:
31:are defined as
17:
12:
11:
5:
3857:
3855:
3847:
3846:
3836:
3835:
3829:
3828:
3826:
3825:
3820:
3818:Thoralf Skolem
3815:
3810:
3805:
3800:
3795:
3790:
3785:
3780:
3775:
3770:
3764:
3762:
3756:
3755:
3753:
3752:
3747:
3742:
3736:
3734:
3732:
3731:
3728:
3722:
3719:
3718:
3716:
3715:
3714:
3713:
3708:
3703:
3702:
3701:
3686:
3685:
3684:
3672:
3671:
3670:
3659:
3658:
3653:
3648:
3643:
3637:
3635:
3631:
3630:
3628:
3627:
3622:
3617:
3612:
3603:
3598:
3593:
3583:
3578:
3577:
3576:
3571:
3566:
3556:
3546:
3541:
3536:
3530:
3528:
3521:
3520:
3518:
3517:
3512:
3507:
3502:
3500:Ordinal number
3497:
3492:
3487:
3482:
3481:
3480:
3475:
3465:
3460:
3455:
3450:
3445:
3435:
3430:
3424:
3422:
3420:
3419:
3416:
3412:
3409:
3408:
3406:
3405:
3400:
3395:
3390:
3385:
3380:
3378:Disjoint union
3375:
3370:
3364:
3358:
3356:
3350:
3349:
3347:
3346:
3345:
3344:
3339:
3328:
3327:
3325:Martin's axiom
3322:
3317:
3312:
3307:
3302:
3297:
3292:
3290:Extensionality
3287:
3282:
3277:
3276:
3275:
3270:
3265:
3255:
3249:
3247:
3241:
3240:
3233:
3231:
3229:
3228:
3222:
3220:
3216:
3215:
3210:
3208:
3207:
3200:
3193:
3185:
3177:
3176:
3161:
3135:
3122:(3): 227–244.
3102:
3055:
3038:
3034:
3014:. January 2023
2999:
2973:
2972:
2970:
2967:
2966:
2965:
2960:
2955:
2950:
2948:Hereditary set
2945:
2940:
2933:
2930:
2910:
2907:
2904:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2838:
2816:
2812:
2779:
2775:
2769:
2764:
2761:
2758:
2754:
2750:
2745:
2741:
2730:
2715:
2711:
2690:
2687:
2684:
2664:
2659:
2655:
2651:
2648:
2645:
2640:
2637:
2634:
2630:
2607:
2603:
2580:
2576:
2555:
2531:
2528:
2525:
2522:
2497:
2493:
2468:curly brackets
2450:
2446:
2427:
2424:
2391:
2387:
2382:
2366:Extensionality
2348:Now note that
2330:
2326:
2321:
2292:
2288:
2283:
2262:
2251:ordinal number
2238:
2235:
2223:
2220:
2218:
2215:
2176:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2096:
2093:
2090:
2058:
2054:
2049:
2036:
2033:
1962:
1935:
1930:
1920:
1916:
1912:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1853:
1850:
1847:
1844:
1839:
1836:
1832:
1828:
1825:
1822:
1819:
1816:
1811:
1808:
1804:
1793:
1776:
1772:
1767:
1763:
1760:
1757:
1752:
1749:
1745:
1734:
1718:
1715:
1710:
1707:
1704:
1701:
1697:
1693:
1688:
1685:
1682:
1679:
1675:
1671:
1666:
1663:
1660:
1657:
1653:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1586:
1583:
1558:
1555:
1552:
1549:
1545:
1539:
1536:
1533:
1529:
1525:
1522:
1519:
1516:
1513:
1503:
1490:
1487:
1480:
1476:
1471:
1467:
1464:
1448:
1445:
1443:
1440:
1419:
1397:
1393:
1370:
1366:
1341:
1337:
1332:
1307:
1303:
1298:
1288:is denoted by
1267:
1263:
1238:
1234:
1229:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1173:
1147:
1144:
1139:
1134:
1131:
1128:
1125:
1105:
1102:
1099:
1094:
1089:
1086:
1083:
1063:
1060:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
998:
995:
992:
979:
968:
967:
964:
952:
949:
946:
943:
940:
937:
934:
914:
911:
908:
905:
902:
899:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
866:
863:
860:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
775:
764:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
713:
693:
682:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
617:
606:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
544:
541:
538:
535:
532:
529:
526:
523:
520:
517:
514:
494:
491:
488:
485:
482:
479:
476:
473:
470:
467:
457:
445:
442:
439:
436:
433:
413:
410:
407:
404:
401:
398:
395:
392:
389:
380:and then also
369:
366:
363:
360:
357:
354:
351:
348:
338:
327:
324:
321:
318:
315:
312:
302:
290:
287:
284:
264:
261:
258:
238:
235:
232:
229:
219:
207:
187:
184:
169:
168:Representation
166:
162:
161:
149:
144:
140:
136:
133:
128:
124:
120:
98:
94:
90:
87:
82:
78:
66:Recursion rule
63:
51:definition of
44:
41:
15:
13:
10:
9:
6:
4:
3:
2:
3856:
3845:
3842:
3841:
3839:
3824:
3823:Ernst Zermelo
3821:
3819:
3816:
3814:
3811:
3809:
3808:Willard Quine
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3784:
3781:
3779:
3776:
3774:
3771:
3769:
3766:
3765:
3763:
3761:
3760:Set theorists
3757:
3751:
3748:
3746:
3743:
3741:
3738:
3737:
3735:
3729:
3727:
3724:
3723:
3720:
3712:
3709:
3707:
3706:Kripke–Platek
3704:
3700:
3697:
3696:
3695:
3692:
3691:
3690:
3687:
3683:
3680:
3679:
3678:
3677:
3673:
3669:
3666:
3665:
3664:
3661:
3660:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3638:
3636:
3632:
3626:
3623:
3621:
3618:
3616:
3613:
3611:
3609:
3604:
3602:
3599:
3597:
3594:
3591:
3587:
3584:
3582:
3579:
3575:
3572:
3570:
3567:
3565:
3562:
3561:
3560:
3557:
3554:
3550:
3547:
3545:
3542:
3540:
3537:
3535:
3532:
3531:
3529:
3526:
3522:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3479:
3476:
3474:
3471:
3470:
3469:
3466:
3464:
3461:
3459:
3456:
3454:
3451:
3449:
3446:
3443:
3439:
3436:
3434:
3431:
3429:
3426:
3425:
3423:
3417:
3414:
3413:
3410:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3379:
3376:
3374:
3371:
3368:
3365:
3363:
3360:
3359:
3357:
3355:
3351:
3343:
3342:specification
3340:
3338:
3335:
3334:
3333:
3330:
3329:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3274:
3271:
3269:
3266:
3264:
3261:
3260:
3259:
3256:
3254:
3251:
3250:
3248:
3246:
3242:
3237:
3227:
3224:
3223:
3221:
3217:
3213:
3206:
3201:
3199:
3194:
3192:
3187:
3186:
3183:
3172:
3168:
3164:
3158:
3154:
3150:
3146:
3139:
3136:
3130:
3125:
3121:
3117:
3113:
3106:
3103:
3092:
3088:
3084:
3080:
3076:
3072:
3071:
3066:
3059:
3056:
3052:
3036:
3032:
3013:
3009:
3003:
3000:
2995:
2994:
2988:
2984:
2978:
2975:
2968:
2964:
2961:
2959:
2956:
2954:
2951:
2949:
2946:
2944:
2941:
2939:
2936:
2935:
2931:
2929:
2927:
2922:
2908:
2905:
2902:
2894:
2875:
2872:
2869:
2860:
2852:
2836:
2814:
2810:
2801:
2796:
2777:
2773:
2762:
2759:
2756:
2752:
2748:
2743:
2739:
2729:
2713:
2709:
2688:
2685:
2682:
2657:
2653:
2643:
2638:
2635:
2632:
2628:
2605:
2601:
2578:
2574:
2553:
2545:
2526:
2511:
2495:
2491:
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2197:
2192:
2190:
2189:type theories
2174:
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2127:
2124:
2121:
2110:
2091:
2080:
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2047:
2034:
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2004:
1978:
1918:
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1899:
1898:BIT predicate
1877:
1874:
1868:
1865:
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1837:
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1661:
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1651:
1643:is mapped to
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1621:
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1433:
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1330:
1305:
1296:
1287:
1285:
1282:hereditarily
1265:
1236:
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1200:
1197:
1194:
1191:
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3773:Georg Cantor
3768:Paul Bernays
3699:Morse–Kelley
3674:
3607:
3606:Subset
3553:hereditarily
3552:
3515:Venn diagram
3473:ordered pair
3388:Intersection
3332:Axiom schema
3144:
3138:
3119:
3115:
3105:
3094:. Retrieved
3074:
3068:
3058:
3023:
3016:. Retrieved
3011:
3002:
2990:
2977:
2963:Rooted trees
2923:
2895:(a tower of
2797:
2794:
2512:
2478:
2420:
2347:
2225:
2207:graph theory
2204:
2193:
2079:automorphism
2075:rooted trees
2038:
2035:Graph models
1902:
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1731:
1573:
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1283:
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1160:hereditarily
1159:
1000:
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3798:Thomas Jech
3641:Alternative
3620:Uncountable
3574:Ultrafilter
3433:Cardinality
3337:replacement
3285:Determinacy
3077:: 305–315.
3018:January 28,
2928:is finite.
2851:cardinality
2416:Replacement
2360:Z with its
555:as well as
33:finite sets
21:mathematics
3844:Set theory
3793:Kurt Gödel
3778:Paul Cohen
3615:Transitive
3383:Identities
3367:Complement
3354:Operations
3315:Regularity
3253:Adjunction
3212:Set theory
3096:2012-01-09
2969:References
2943:Finite set
2370:Adjunction
2253:, denoted
2211:Rado graph
2039:The class
997:Discussion
25:set theory
3726:Paradoxes
3646:Axiomatic
3625:Universal
3601:Singleton
3596:Recursive
3539:Countable
3534:Amorphous
3393:Power set
3310:Power set
3268:dependent
3263:countable
3091:120576556
3037:ω
2906:−
2873:−
2800:countably
2768:∞
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2647:℘
2606:ω
2566:, and by
2544:power set
2521:℘
2496:ω
2386:ℵ
2364:given by
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2287:ℵ
2107:and each
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1436:countable
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1365:ℵ
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206:∅
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3838:Category
3730:Problems
3634:Theories
3610:Superset
3586:Infinite
3415:Concepts
3295:Infinity
3219:Overview
2932:See also
2701:. Thus,
1001:The set
966:... etc.
3668:General
3663:Zermelo
3569:subbase
3551: (
3490:Forcing
3468:Element
3440: (
3418:Methods
3305:Pairing
3171:3558535
2985:(ed.).
1975:is the
1946:(where
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3549:Finite
3485:Family
3428:Almost
3273:global
3258:Choice
3245:Axioms
3169:
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2362:axioms
2196:models
2194:Graph
1442:Models
925:(i.e.
424:(i.e.
249:(i.e.
198:(i.e.
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3544:Empty
3527:types
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1438:set.
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3157:ISBN
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