3293:
2488:
2530:
1892:
2848:
1790:
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:
2946:
1156:
679:
1049:
603:
553:
158:
1973:
503:
2730:
2165:
422:
109:
378:
2404:
2343:
2305:
2071:
1354:
1320:
1251:
2477:
2450:
273:
2029:
1999:
1641:
336:
3106:
2783:
2675:
2565:
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
247:
2756:
2597:
2521:
1428:
454:
2884:
2648:
299:
2247:
1595:
1072:
196:
2976:
2904:
2621:
2271:
2185:
988:
784:
702:
626:
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:
3757:
2790:
1737:
2187:). This graph model enables an implementation of ZF without infinity as data types and thus an interpretation of set theory in expressive
2308:
3049:
3200:
Omodeo, Eugenio G.; Policriti, Alberto; Tomescu, Alexandru I. (2017). "3.3: The
Ackermann encoding of hereditarily finite sets".
1506:
3217:
1162:
finite. For example, the first cannot be hereditarily finite since it contains at least one infinite set as an element, when
3446:
3259:
2529:
3081:
The set of all (well-founded) hereditarily finite sets (which is infinite, and not hereditarily finite itself) is written
1906:
854:
1646:
1458:
3774:
789:
1903:
The
Ackermann coding can be used to construct a model of finitary set theory in the natural numbers. More precisely,
1165:
3121:
2199:
707:
3752:
3346:
2002:
3632:
2209:, the graph whose vertices correspond to hereditarily finite sets and edges correspond to set membership is the
1077:
2913:
1119:
631:
3526:
3405:
2524:
3769:
2949:
1004:
558:
508:
164:
Only sets that can be built by a finite number of applications of these two rules are hereditarily finite.
114:
1949:
461:
3762:
3400:
3363:
2680:
1501:
that maps each hereditarily finite set to a natural number, given by the following recursive definition:
3009:
2114:
1281:
383:
172:
This class of sets is naturally ranked by the number of bracket pairs necessary to represent the sets:
71:
3417:
342:
3451:
3336:
3324:
3319:
3019:
2375:
2314:
2276:
2074:
2042:
1325:
1291:
1222:
2455:
2428:
252:
3252:
1455:
introduced an encoding of hereditarily finite sets as natural numbers. It is defined by a function
2012:
1982:
1608:
306:
3871:
3789:
3664:
3430:
3353:
3084:
2994:
2761:
2653:
2543:
2407:
2365:
1386:
1359:
1256:
2084:
201:
3823:
3704:
3516:
3329:
928:
223:
3739:
3709:
3653:
3573:
3553:
3531:
3126:
2735:
2573:
2493:
2108:
1413:
427:
3813:
3803:
3637:
3568:
3521:
3461:
3341:
3227:
2862:
2626:
2537:
2415:
1431:
278:
2229:
1577:
1054:
178:
8:
3907:
3808:
3719:
3627:
3622:
3436:
3378:
3309:
3245:
3014:
2955:
2421:
Axiomatically characterizing the theory of hereditarily finite sets, the negation of the
2349:
3064:
3731:
3726:
3511:
3466:
3373:
3143:
2889:
2606:
2369:
2357:
2353:
2256:
2170:
2078:
973:
769:
687:
611:
3588:
3425:
3388:
3358:
3282:
3213:
3147:
2422:
2411:
2006:
1976:
1452:
3876:
3866:
3851:
3846:
3714:
3368:
3205:
3180:
3135:
52:
3745:
3683:
3501:
3314:
3223:
3881:
3678:
3659:
3563:
3548:
3505:
3441:
3383:
3004:
2982:
2452:, this establishes that the axiom of infinity is not a consequence these other
2250:
3209:
3185:
3168:
2198:
exist for ZF and also set theories different from
Zermelo set theory, such as
3901:
3886:
3688:
3602:
3597:
1897:
1887:{\displaystyle \displaystyle f^{-1}(i)=\{f^{-1}(j)\mid {\text{BIT}}(i,j)=1\}}
1435:
3856:
3836:
3831:
3649:
3578:
3536:
3395:
3292:
2206:
2195:
2978:
powers of two), and the union of countably many finite sets is countable.
2540:. Here, the class of all well-founded hereditarily finite sets is denoted
3861:
3496:
3039:
2907:
2188:
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, ...
20:
2081:
is the identity): The root vertex corresponds to the top level bracket
3841:
3612:
3268:
3139:
2999:
2843:{\displaystyle \displaystyle V_{\omega }=\bigcup _{k=0}^{\infty }V_{k}}
2210:
32:
24:
3644:
3607:
3558:
3456:
2856:
2600:
1785:{\displaystyle \displaystyle f^{-1}\colon \omega \to H_{\aleph _{0}}}
1598:
48:
36:
2487:
3669:
3491:
2361:
2981:
Equivalently, a set is hereditarily finite if and only if its
1253:, meaning that the cardinality of each member is smaller than
3541:
3301:
3237:
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
3108:
to show its place in the von
Neumann hierarchy of pure sets.
3202:
3043:
1602:
16:
Finite sets whose elements are all hereditarily finite sets
2425:
may be added. As the theory validates the other axioms of
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
3199:
3122:"Die Widerspruchsfreiheit der allgemeinen Mengenlehre"
3038:
2031:
relation models the membership relation between sets.
3087:
2958:
2916:
2892:
2865:
2794:
2793:
2764:
2738:
2683:
2656:
2629:
2609:
2576:
2546:
2496:
2458:
2431:
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:
2855:This formulation shows, again, that there are only
2536: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 \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}}
3100:
2970:
2940:
2898:
2878:
2842:
2777:
2750:
2724:
2669:
2642:
2615:
2591:
2559:
2515:
2471:
2444:
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
3899:
2567:. Note that this is also a set in this context.
844:{\displaystyle \{\{\{\{\{\{\{\{\}\}\}\}\}\}\}\}}
1209:{\displaystyle {\mathbb {N} }=\{0,1,2,\dots \}}
3253:
62:: The empty set is a hereditarily finite set.
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:
3260:
3246:
2221:
3184:
3119:
3050:On-Line Encyclopedia of Integer Sequences
1914:
1171:
1158:are examples of finite sets that are not
1137:
1109:{\displaystyle \{7,{\mathbb {N} },\pi \}}
1092:
1074:, as noted. On the other hand, the sets
3113:
2941:{\displaystyle 2\uparrow \uparrow (n-1)}
2486:
1151:{\displaystyle \{3,\{{\mathbb {N} }\}\}}
674:{\displaystyle \{\{\{\{\{\{\}\}\}\}\}\}}
55:hereditarily finite sets is as follows:
2345:must necessarily contain them as well.
3900:
2464:
2461:
2437:
2434:
2249:also represents the first von Neumann
3241:
3166:
2523: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}\}}
1968:{\displaystyle {\text{BIT}}^{\top }}
1356:is in bijective correspondence with
498:{\displaystyle \{\{\{\{\{\}\}\}\}\}}
111:are hereditarily finite, then so is
42:
2725:{\displaystyle V_{i+1}=\wp (V_{i})}
1446:
13:
3173:Notre Dame Journal of Formal Logic
2824:
2703:
2577:
2385:
2324:
2286:
2216:
2052:
1960:
1928:
1770:
1474:
1364:
1335:
1301:
1261:
1232:
259:
205:
14:
3919:
2852:and all its elements are finite.
2309:standard model of natural numbers
2160:{\displaystyle \{t,t,s\}=\{t,s\}}
417:{\displaystyle \{\{\},\{\{\}\}\}}
167:
104:{\displaystyle a_{1},\dots a_{k}}
3291:
2528:
2410:involving these axioms and e.g.
373:{\displaystyle \{\{\{\{\}\}\}\}}
2859:many hereditarily finite sets:
2399:{\displaystyle H_{\aleph _{0}}}
2356:, the very small sub-theory of
2338:{\displaystyle H_{\aleph _{0}}}
2311:and so a set theory expressing
2300:{\displaystyle H_{\aleph _{0}}}
2066:{\displaystyle H_{\aleph _{0}}}
2034:
1349:{\displaystyle H_{\aleph _{0}}}
1315:{\displaystyle H_{\aleph _{1}}}
1246:{\displaystyle H_{\aleph _{0}}}
3267:
3193:
3160:
3057:
3032:
2935:
2923:
2920:
2719:
2706:
2586:
2580:
2472:{\displaystyle {\mathsf {ZF}}}
2445:{\displaystyle {\mathsf {ZF}}}
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:
3025:
2307:includes each element in the
1280:. (Analogously, the class of
996:
3204:. Springer. pp. 70–71.
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 \{\{\{\}\}\}}
7:
3120:Ackermann, Wilhelm (1937).
3101:{\displaystyle V_{\omega }}
2988:
2778:{\displaystyle V_{\omega }}
2677:can be obtained by setting
2670:{\displaystyle V_{\omega }}
2560:{\displaystyle V_{\omega }}
2408:constructive axiomatization
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"),
10:
3924:
3758:von Neumann–Bernays–Gödel
3040:Sloane, N. J. A.
3010: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 }
3822:
3785:
3697:
3587:
3559:One-to-one correspondence
3475:
3416:
3300:
3289:
3275:
3210:10.1007/978-3-319-54981-1
3186:10.1215/00294527-2009-009
3065:"hereditarily finite set"
2950:Knuth's up-arrow notation
2886:is finite for any finite
1441:
956:{\displaystyle \{0,1,2\}}
681:. There are six such sets
3167:Kirby, Laurence (2009).
2527:
1732:The inverse is given by
242:{\displaystyle \{\{\}\}}
29:hereditarily finite sets
3044:"Sequence A004111"
2995:Constructive set theory
2751:{\displaystyle i\geq 0}
2592:{\displaystyle \wp (S)}
2516:{\displaystyle ~V_{4}~}
2222:Theories of finite sets
1423:{\displaystyle \omega }
449:{\displaystyle \{0,1\}}
3517:Constructible universe
3337:Constructibility (V=L)
3102:
2972:
2942:
2900:
2880:
2844:
2828:
2779:
2752:
2726:
2671:
2644:
2617:
2593:
2561:
2533:
2517:
2473:
2446:
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:
3740:Principia Mathematica
3574:Transfinite induction
3433:(i.e. set difference)
3169:"Finitary Set Theory"
3127:Mathematische Annalen
3103:
2973:
2943:
2901:
2881:
2879:{\displaystyle V_{n}}
2845:
2808:
2780:
2753:
2727:
2672:
2645:
2643:{\displaystyle V_{0}}
2618:
2594:
2562:
2518:
2490:
2482:
2474:
2447:
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:
3814:Burali-Forti paradox
3569:Set-builder notation
3522:Continuum hypothesis
3462:Symmetric difference
3085:
2956:
2914:
2890:
2863:
2791:
2785:can be expressed as
2762:
2736:
2681:
2654:
2650:the empty set, then
2627:
2607:
2574:
2544:
2538:Von Neumann universe
2494:
2456:
2429:
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:
3775:Tarski–Grothendieck
3015:Hereditary property
2971:{\displaystyle n-1}
2350:Robinson arithmetic
3364:Limitation of size
3140:10.1007/bf01594179
3098:
3053:. OEIS Foundation.
2983:transitive closure
2968:
2938:
2921:↑ ↑
2896:
2876:
2840:
2839:
2775:
2748:
2722:
2667:
2640:
2613:
2589:
2557:
2534:
2513:
2469:
2442:
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:
3895:
3894:
3804:Russell's paradox
3753:Zermelo–Fraenkel
3654:Dedekind-infinite
3527:Diagonal argument
3426:Cartesian product
3283:Set (mathematics)
3219:978-3-319-54980-4
2899:{\displaystyle n}
2732:for each integer
2616:{\displaystyle S}
2512:
2499:
2423:axiom of infinity
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
3915:
3877:Bertrand Russell
3867:John von Neumann
3852:Abraham Fraenkel
3847:Richard Dedekind
3809:Suslin's problem
3720:Cantor's theorem
3437:De Morgan's laws
3295:
3262:
3255:
3248:
3239:
3238:
3232:
3231:
3197:
3191:
3190:
3188:
3164:
3158:
3157:
3155:
3154:
3117:
3111:
3110:
3107:
3105:
3104:
3099:
3097:
3096:
3078:
3076:
3061:
3055:
3054:
3036:
2977:
2975:
2974:
2969:
2947:
2945:
2944:
2939:
2905:
2903:
2902:
2897:
2885:
2883:
2882:
2877:
2875:
2874:
2849:
2847:
2846:
2841:
2838:
2837:
2827:
2822:
2804:
2803:
2784:
2782:
2781:
2776:
2774:
2773:
2757:
2755:
2754:
2749:
2731:
2729:
2728:
2723:
2718:
2717:
2699:
2698:
2676:
2674:
2673:
2668:
2666:
2665:
2649:
2647:
2646:
2641:
2639:
2638:
2622:
2620:
2619:
2614:
2598:
2596:
2595:
2590:
2570:If we denote by
2566:
2564:
2563:
2558:
2556:
2555:
2532:
2522:
2520:
2519:
2514:
2510:
2509:
2508:
2497:
2478:
2476:
2475:
2470:
2468:
2467:
2451:
2449:
2448:
2443:
2441:
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:
3923:
3922:
3918:
3917:
3916:
3914:
3913:
3912:
3898:
3897:
3896:
3891:
3818:
3797:
3781:
3746:New Foundations
3693:
3583:
3502:Cardinal number
3485:
3471:
3412:
3296:
3287:
3271:
3266:
3236:
3235:
3220:
3198:
3194:
3165:
3161:
3152:
3150:
3118:
3114:
3092:
3088:
3086:
3083:
3082:
3074:
3072:
3063:
3062:
3058:
3037:
3033:
3028:
2991:
2957:
2954:
2953:
2915:
2912:
2911:
2891:
2888:
2887:
2870:
2866:
2864:
2861:
2860:
2850:
2833:
2829:
2823:
2812:
2799:
2795:
2792:
2789:
2788:
2769:
2765:
2763:
2760:
2759:
2737:
2734:
2733:
2713:
2709:
2688:
2684:
2682:
2679:
2678:
2661:
2657:
2655:
2652:
2651:
2634:
2630:
2628:
2625:
2624:
2608:
2605:
2604:
2575:
2572:
2571:
2551:
2547:
2545:
2542:
2541:
2504:
2500:
2495:
2492:
2491:
2485:
2460:
2459:
2457:
2454:
2453:
2433:
2432:
2430:
2427:
2426:
2388:
2384:
2383:
2379:
2377:
2374:
2373:
2327:
2323:
2322:
2318:
2316:
2313:
2312:
2289:
2285:
2284:
2280:
2278:
2275:
2274:
2258:
2255:
2254:
2231:
2228:
2227:
2224:
2219:
2217:Axiomatizations
2172:
2169:
2168:
2116:
2113:
2112:
2086:
2083:
2082:
2055:
2051:
2050:
2046:
2044:
2041:
2040:
2037:
2016:
2014:
2011:
2010:
2005:ZF without the
1986:
1984:
1981:
1980:
1959:
1954:
1953:
1951:
1948:
1947:
1927:
1922:
1921:
1913:
1908:
1905:
1904:
1894:
1854:
1833:
1829:
1805:
1801:
1798:
1795:
1794:
1792:
1773:
1769:
1768:
1764:
1746:
1742:
1739:
1736:
1735:
1698:
1694:
1676:
1672:
1654:
1650:
1648:
1645:
1644:
1610:
1607:
1606:
1579:
1576:
1575:
1572:
1546:
1542:
1530:
1508:
1505:
1504:
1477:
1473:
1472:
1468:
1460:
1457:
1456:
1449:
1444:
1415:
1412:
1411:
1394:
1390:
1388:
1385:
1384:
1367:
1363:
1361:
1358:
1357:
1338:
1334:
1333:
1329:
1327:
1324:
1323:
1304:
1300:
1299:
1295:
1293:
1290:
1289:
1264:
1260:
1258:
1255:
1254:
1235:
1231:
1230:
1226:
1224:
1221:
1220:
1170:
1169:
1167:
1164:
1163:
1136:
1135:
1121:
1118:
1117:
1091:
1090:
1079:
1076:
1075:
1056:
1053:
1052:
1006:
1003:
1002:
999:
994:
975:
972:
971:
930:
927:
926:
856:
853:
852:
791:
788:
787:
771:
768:
767:
709:
706:
705:
689:
686:
685:
633:
630:
629:
613:
610:
609:
560:
557:
556:
510:
507:
506:
463:
460:
459:
429:
426:
425:
385:
382:
381:
344:
341:
340:
308:
305:
304:
280:
277:
276:
254:
251:
250:
225:
222:
221:
203:
200:
199:
180:
177:
176:
170:
141:
137:
125:
121:
116:
113:
112:
95:
91:
79:
75:
73:
70:
69:
45:
31:are defined as
17:
12:
11:
5:
3921:
3911:
3910:
3893:
3892:
3890:
3889:
3884:
3882:Thoralf Skolem
3879:
3874:
3869:
3864:
3859:
3854:
3849:
3844:
3839:
3834:
3828:
3826:
3820:
3819:
3817:
3816:
3811:
3806:
3800:
3798:
3796:
3795:
3792:
3786:
3783:
3782:
3780:
3779:
3778:
3777:
3772:
3767:
3766:
3765:
3750:
3749:
3748:
3736:
3735:
3734:
3723:
3722:
3717:
3712:
3707:
3701:
3699:
3695:
3694:
3692:
3691:
3686:
3681:
3676:
3667:
3662:
3657:
3647:
3642:
3641:
3640:
3635:
3630:
3620:
3610:
3605:
3600:
3594:
3592:
3585:
3584:
3582:
3581:
3576:
3571:
3566:
3564:Ordinal number
3561:
3556:
3551:
3546:
3545:
3544:
3539:
3529:
3524:
3519:
3514:
3509:
3499:
3494:
3488:
3486:
3484:
3483:
3480:
3476:
3473:
3472:
3470:
3469:
3464:
3459:
3454:
3449:
3444:
3442:Disjoint union
3439:
3434:
3428:
3422:
3420:
3414:
3413:
3411:
3410:
3409:
3408:
3403:
3392:
3391:
3389:Martin's axiom
3386:
3381:
3376:
3371:
3366:
3361:
3356:
3354:Extensionality
3351:
3350:
3349:
3339:
3334:
3333:
3332:
3327:
3322:
3312:
3306:
3304:
3298:
3297:
3290:
3288:
3286:
3285:
3279:
3277:
3273:
3272:
3265:
3264:
3257:
3250:
3242:
3234:
3233:
3218:
3192:
3179:(3): 227–244.
3159:
3112:
3095:
3091:
3071:. January 2023
3056:
3030:
3029:
3027:
3024:
3023:
3022:
3017:
3012:
3007:
3005:Hereditary set
3002:
2997:
2990:
2987:
2967:
2964:
2961:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2895:
2873:
2869:
2836:
2832:
2826:
2821:
2818:
2815:
2811:
2807:
2802:
2798:
2787:
2772:
2768:
2747:
2744:
2741:
2721:
2716:
2712:
2708:
2705:
2702:
2697:
2694:
2691:
2687:
2664:
2660:
2637:
2633:
2612:
2588:
2585:
2582:
2579:
2554:
2550:
2525:curly brackets
2507:
2503:
2484:
2481:
2466:
2463:
2439:
2436:
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:
9:
6:
4:
3:
2:
3920:
3909:
3906:
3905:
3903:
3888:
3887:Ernst Zermelo
3885:
3883:
3880:
3878:
3875:
3873:
3872:Willard Quine
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3853:
3850:
3848:
3845:
3843:
3840:
3838:
3835:
3833:
3830:
3829:
3827:
3825:
3824:Set theorists
3821:
3815:
3812:
3810:
3807:
3805:
3802:
3801:
3799:
3793:
3791:
3788:
3787:
3784:
3776:
3773:
3771:
3770:Kripke–Platek
3768:
3764:
3761:
3760:
3759:
3756:
3755:
3754:
3751:
3747:
3744:
3743:
3742:
3741:
3737:
3733:
3730:
3729:
3728:
3725:
3724:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3702:
3700:
3696:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3673:
3668:
3666:
3663:
3661:
3658:
3655:
3651:
3648:
3646:
3643:
3639:
3636:
3634:
3631:
3629:
3626:
3625:
3624:
3621:
3618:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3595:
3593:
3590:
3586:
3580:
3577:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3550:
3547:
3543:
3540:
3538:
3535:
3534:
3533:
3530:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3507:
3503:
3500:
3498:
3495:
3493:
3490:
3489:
3487:
3481:
3478:
3477:
3474:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3432:
3429:
3427:
3424:
3423:
3421:
3419:
3415:
3407:
3406:specification
3404:
3402:
3399:
3398:
3397:
3394:
3393:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3348:
3345:
3344:
3343:
3340:
3338:
3335:
3331:
3328:
3326:
3323:
3321:
3318:
3317:
3316:
3313:
3311:
3308:
3307:
3305:
3303:
3299:
3294:
3284:
3281:
3280:
3278:
3274:
3270:
3263:
3258:
3256:
3251:
3249:
3244:
3243:
3240:
3229:
3225:
3221:
3215:
3211:
3207:
3203:
3196:
3187:
3182:
3178:
3174:
3170:
3163:
3149:
3145:
3141:
3137:
3133:
3129:
3128:
3123:
3116:
3109:
3093:
3089:
3070:
3066:
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2189:type theories
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2133:
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2127:
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2110:
2091:
2080:
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2047:
2032:
2008:
2004:
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1918:
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1899:
1898:BIT predicate
1877:
1874:
1868:
1865:
1862:
1851:
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1830:
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1817:
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3832:Paul Bernays
3763:Morse–Kelley
3738:
3671:
3670:Subset
3617:hereditarily
3616:
3579:Venn diagram
3537:ordered pair
3452:Intersection
3396:Axiom schema
3201:
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3176:
3172:
3162:
3151:. Retrieved
3131:
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3080:
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3020:Rooted trees
2980:
2952:(a tower of
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2079:automorphism
2075:rooted trees
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2035:Graph models
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1160:hereditarily
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1000:
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3862:Thomas Jech
3705:Alternative
3684:Uncountable
3638:Ultrafilter
3497:Cardinality
3401:replacement
3342:Determinacy
3134:: 305–315.
3075:January 28,
2985:is finite.
2908:cardinality
2416:Replacement
2360:Z with its
555:as well as
33:finite sets
21:mathematics
3908:Set theory
3857:Kurt Gödel
3842:Paul Cohen
3679:Transitive
3447:Identities
3431:Complement
3418:Operations
3379:Regularity
3347:projective
3310:Adjunction
3269:Set theory
3153:2012-01-09
3026:References
3000:Finite set
2370:Adjunction
2253:, denoted
2211:Rado graph
2039:The class
997:Discussion
25:set theory
3790:Paradoxes
3710:Axiomatic
3689:Universal
3665:Singleton
3660:Recursive
3603:Countable
3598:Amorphous
3457:Power set
3374:Power set
3325:dependent
3320:countable
3148:120576556
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3698:Theories
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3650:Infinite
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3276:Overview
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2758:. Thus,
2479:axioms.
1001:The set
966:... etc.
3732:General
3727:Zermelo
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3554:Forcing
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3482:Methods
3369:Pairing
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