1992:
162:
sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there may be
140:
79:
102:
183:), otherwise the recurrence may not have a unique solution. However, it can be restated non-inductively as follows: a set is hereditary if and only if its
371:
1046:
43:
whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on.
1129:
270:
1443:
1601:
389:
1456:
779:
1461:
1451:
1188:
1041:
394:
107:
385:
1597:
239:
939:
1694:
1438:
263:
999:
692:
155:
433:
1955:
1657:
1420:
1415:
1240:
661:
345:
191:
in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set.
1950:
1733:
1650:
1363:
1294:
1171:
413:
188:
142:
that contains two elements: the empty set and the set that contains only the empty set, is a hereditary set.
1875:
1701:
1387:
1021:
620:
1753:
1748:
1358:
1097:
1026:
355:
256:
1682:
1272:
666:
634:
325:
200:
58:
1972:
1921:
1818:
1316:
1277:
754:
399:
428:
1813:
1743:
1282:
1134:
1117:
840:
320:
17:
184:
87:
1645:
1622:
1583:
1469:
1410:
1056:
976:
820:
764:
377:
205:
1935:
1662:
1640:
1607:
1500:
1346:
1331:
1304:
1255:
1139:
1074:
899:
865:
860:
734:
565:
542:
1865:
1718:
1510:
1228:
964:
870:
729:
714:
595:
570:
176:
1838:
1800:
1677:
1481:
1321:
1245:
1223:
1051:
1009:
908:
875:
739:
527:
438:
151:
8:
2016:
1967:
1858:
1843:
1823:
1780:
1667:
1617:
1543:
1488:
1425:
1218:
1213:
1161:
929:
918:
590:
490:
418:
409:
405:
340:
335:
180:
187:
contains only sets. In this way the concept of hereditary sets can also be extended to
1996:
1765:
1728:
1713:
1706:
1689:
1493:
1475:
1341:
1267:
1250:
1203:
1016:
925:
759:
744:
704:
656:
641:
629:
585:
560:
330:
279:
228:
949:
1991:
1931:
1738:
1548:
1538:
1430:
1311:
1146:
1122:
903:
887:
792:
769:
646:
615:
580:
475:
310:
235:
40:
1945:
1940:
1833:
1790:
1612:
1573:
1568:
1553:
1379:
1336:
1233:
1031:
981:
555:
517:
210:
1926:
1916:
1870:
1853:
1808:
1770:
1672:
1592:
1399:
1326:
1299:
1287:
1193:
1107:
1081:
1036:
1004:
805:
607:
550:
500:
465:
423:
1911:
1890:
1848:
1828:
1723:
1578:
1176:
1166:
1156:
1151:
1085:
959:
835:
724:
719:
298:
175:
The inductive definition of hereditary sets presupposes that set membership is
52:
2010:
1885:
1563:
1070:
855:
845:
815:
800:
470:
223:
1785:
1632:
1533:
1525:
1405:
1353:
1262:
1198:
1181:
1112:
971:
830:
532:
315:
1895:
1775:
954:
944:
891:
575:
495:
480:
360:
305:
150:
In formulations of set theory that are intended to be interpreted in the
825:
680:
651:
457:
28:
1977:
1880:
933:
850:
810:
774:
710:
522:
512:
485:
248:
164:
82:
1962:
1760:
1208:
913:
507:
1558:
350:
1102:
448:
293:
55:
that the empty set is a hereditary set, and thus the set
110:
90:
61:
230:Set Theory: An Introduction to Independence Proofs
227:
134:
96:
73:
135:{\displaystyle \{\varnothing ,\{\varnothing \}\}}
2008:
145:
222:
264:
129:
126:
120:
111:
68:
62:
456:
271:
257:
14:
2009:
278:
104:is a hereditary set. Similarly, a set
252:
24:
25:
2028:
123:
114:
65:
1990:
74:{\displaystyle \{\varnothing \}}
170:
13:
1:
1951:History of mathematical logic
216:
189:non-well-founded set theories
154:or to express the content of
146:In formulations of set theory
1876:Primitive recursive function
97:{\displaystyle \varnothing }
7:
194:
156:ZermeloâFraenkel set theory
46:
10:
2033:
940:SchröderâBernstein theorem
667:Monadic predicate calculus
326:Foundations of mathematics
201:Hereditarily countable set
1986:
1973:Philosophy of mathematics
1922:Automated theorem proving
1904:
1799:
1631:
1524:
1376:
1093:
1069:
1047:Von NeumannâBernaysâGödel
992:
886:
790:
688:
679:
606:
541:
447:
369:
286:
1623:Self-verifying theories
1444:Tarski's axiomatization
395:Tarski's undefinability
390:incompleteness theorems
206:Hereditarily finite set
1997:Mathematics portal
1608:Proof of impossibility
1256:propositional variable
566:Propositional calculus
136:
98:
75:
1866:Kolmogorov complexity
1819:Computably enumerable
1719:Model complete theory
1511:Principia Mathematica
571:Propositional formula
400:BanachâTarski paradox
137:
99:
76:
1814:ChurchâTuring thesis
1801:Computability theory
1010:continuum hypothesis
528:Square of opposition
386:Gödel's completeness
152:von Neumann universe
108:
88:
81:containing only the
59:
1968:Mathematical object
1859:P versus NP problem
1824:Computable function
1618:Reverse mathematics
1544:Logical consequence
1421:primitive recursive
1416:elementary function
1189:Free/bound variable
1042:TarskiâGrothendieck
561:Logical connectives
491:Logical equivalence
341:Logical consequence
181:axiom of regularity
51:For example, it is
1766:Transfer principle
1729:Semantics of logic
1714:Categorical theory
1690:Non-standard model
1204:Logical connective
331:Information theory
280:Mathematical logic
185:transitive closure
132:
94:
71:
2004:
2003:
1936:Abstract category
1739:Theories of truth
1549:Rule of inference
1539:Natural deduction
1520:
1519:
1065:
1064:
770:Cartesian product
675:
674:
581:Many-valued logic
556:Boolean functions
439:Russell's paradox
414:diagonal argument
311:First-order logic
234:. North-Holland.
16:(Redirected from
2024:
1995:
1994:
1946:History of logic
1941:Category of sets
1834:Decision problem
1613:Ordinal analysis
1554:Sequent calculus
1452:Boolean algebras
1392:
1391:
1366:
1337:logical/constant
1091:
1090:
1077:
1000:ZermeloâFraenkel
751:Set operations:
686:
685:
623:
454:
453:
434:LöwenheimâSkolem
321:Formal semantics
273:
266:
259:
250:
249:
245:
233:
211:Well-founded set
141:
139:
138:
133:
103:
101:
100:
95:
80:
78:
77:
72:
21:
2032:
2031:
2027:
2026:
2025:
2023:
2022:
2021:
2007:
2006:
2005:
2000:
1989:
1982:
1927:Category theory
1917:Algebraic logic
1900:
1871:Lambda calculus
1809:Church encoding
1795:
1771:Truth predicate
1627:
1593:Complete theory
1516:
1385:
1381:
1377:
1372:
1364:
1084: and
1080:
1075:
1061:
1037:New Foundations
1005:axiom of choice
988:
950:Gödel numbering
890: and
882:
786:
671:
621:
602:
551:Boolean algebra
537:
501:Equiconsistency
466:Classical logic
443:
424:Halting problem
412: and
388: and
376: and
375:
370:Theorems (
365:
282:
277:
242:
219:
197:
173:
148:
109:
106:
105:
89:
86:
85:
60:
57:
56:
49:
23:
22:
15:
12:
11:
5:
2030:
2020:
2019:
2002:
2001:
1987:
1984:
1983:
1981:
1980:
1975:
1970:
1965:
1960:
1959:
1958:
1948:
1943:
1938:
1929:
1924:
1919:
1914:
1912:Abstract logic
1908:
1906:
1902:
1901:
1899:
1898:
1893:
1891:Turing machine
1888:
1883:
1878:
1873:
1868:
1863:
1862:
1861:
1856:
1851:
1846:
1841:
1831:
1829:Computable set
1826:
1821:
1816:
1811:
1805:
1803:
1797:
1796:
1794:
1793:
1788:
1783:
1778:
1773:
1768:
1763:
1758:
1757:
1756:
1751:
1746:
1736:
1731:
1726:
1724:Satisfiability
1721:
1716:
1711:
1710:
1709:
1699:
1698:
1697:
1687:
1686:
1685:
1680:
1675:
1670:
1665:
1655:
1654:
1653:
1648:
1641:Interpretation
1637:
1635:
1629:
1628:
1626:
1625:
1620:
1615:
1610:
1605:
1595:
1590:
1589:
1588:
1587:
1586:
1576:
1571:
1561:
1556:
1551:
1546:
1541:
1536:
1530:
1528:
1522:
1521:
1518:
1517:
1515:
1514:
1506:
1505:
1504:
1503:
1498:
1497:
1496:
1491:
1486:
1466:
1465:
1464:
1462:minimal axioms
1459:
1448:
1447:
1446:
1435:
1434:
1433:
1428:
1423:
1418:
1413:
1408:
1395:
1393:
1374:
1373:
1371:
1370:
1369:
1368:
1356:
1351:
1350:
1349:
1344:
1339:
1334:
1324:
1319:
1314:
1309:
1308:
1307:
1302:
1292:
1291:
1290:
1285:
1280:
1275:
1265:
1260:
1259:
1258:
1253:
1248:
1238:
1237:
1236:
1231:
1226:
1221:
1216:
1211:
1201:
1196:
1191:
1186:
1185:
1184:
1179:
1174:
1169:
1159:
1154:
1152:Formation rule
1149:
1144:
1143:
1142:
1137:
1127:
1126:
1125:
1115:
1110:
1105:
1100:
1094:
1088:
1071:Formal systems
1067:
1066:
1063:
1062:
1060:
1059:
1054:
1049:
1044:
1039:
1034:
1029:
1024:
1019:
1014:
1013:
1012:
1007:
996:
994:
990:
989:
987:
986:
985:
984:
974:
969:
968:
967:
960:Large cardinal
957:
952:
947:
942:
937:
923:
922:
921:
916:
911:
896:
894:
884:
883:
881:
880:
879:
878:
873:
868:
858:
853:
848:
843:
838:
833:
828:
823:
818:
813:
808:
803:
797:
795:
788:
787:
785:
784:
783:
782:
777:
772:
767:
762:
757:
749:
748:
747:
742:
732:
727:
725:Extensionality
722:
720:Ordinal number
717:
707:
702:
701:
700:
689:
683:
677:
676:
673:
672:
670:
669:
664:
659:
654:
649:
644:
639:
638:
637:
627:
626:
625:
612:
610:
604:
603:
601:
600:
599:
598:
593:
588:
578:
573:
568:
563:
558:
553:
547:
545:
539:
538:
536:
535:
530:
525:
520:
515:
510:
505:
504:
503:
493:
488:
483:
478:
473:
468:
462:
460:
451:
445:
444:
442:
441:
436:
431:
426:
421:
416:
404:Cantor's
402:
397:
392:
382:
380:
367:
366:
364:
363:
358:
353:
348:
343:
338:
333:
328:
323:
318:
313:
308:
303:
302:
301:
290:
288:
284:
283:
276:
275:
268:
261:
253:
247:
246:
240:
224:Kunen, Kenneth
218:
215:
214:
213:
208:
203:
196:
193:
172:
169:
147:
144:
131:
128:
125:
122:
119:
116:
113:
93:
70:
67:
64:
53:vacuously true
48:
45:
33:hereditary set
9:
6:
4:
3:
2:
2029:
2018:
2015:
2014:
2012:
1999:
1998:
1993:
1985:
1979:
1976:
1974:
1971:
1969:
1966:
1964:
1961:
1957:
1954:
1953:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1909:
1907:
1903:
1897:
1894:
1892:
1889:
1887:
1886:Recursive set
1884:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1837:
1836:
1835:
1832:
1830:
1827:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1806:
1804:
1802:
1798:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1755:
1752:
1750:
1747:
1745:
1742:
1741:
1740:
1737:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1708:
1705:
1704:
1703:
1700:
1696:
1695:of arithmetic
1693:
1692:
1691:
1688:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1660:
1659:
1656:
1652:
1649:
1647:
1644:
1643:
1642:
1639:
1638:
1636:
1634:
1630:
1624:
1621:
1619:
1616:
1614:
1611:
1609:
1606:
1603:
1602:from ZFC
1599:
1596:
1594:
1591:
1585:
1582:
1581:
1580:
1577:
1575:
1572:
1570:
1567:
1566:
1565:
1562:
1560:
1557:
1555:
1552:
1550:
1547:
1545:
1542:
1540:
1537:
1535:
1532:
1531:
1529:
1527:
1523:
1513:
1512:
1508:
1507:
1502:
1501:non-Euclidean
1499:
1495:
1492:
1490:
1487:
1485:
1484:
1480:
1479:
1477:
1474:
1473:
1471:
1467:
1463:
1460:
1458:
1455:
1454:
1453:
1449:
1445:
1442:
1441:
1440:
1436:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1409:
1407:
1404:
1403:
1401:
1397:
1396:
1394:
1389:
1383:
1378:Example
1375:
1367:
1362:
1361:
1360:
1357:
1355:
1352:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1329:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1306:
1303:
1301:
1298:
1297:
1296:
1293:
1289:
1286:
1284:
1281:
1279:
1276:
1274:
1271:
1270:
1269:
1266:
1264:
1261:
1257:
1254:
1252:
1249:
1247:
1244:
1243:
1242:
1239:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1206:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1183:
1180:
1178:
1175:
1173:
1170:
1168:
1165:
1164:
1163:
1160:
1158:
1155:
1153:
1150:
1148:
1145:
1141:
1138:
1136:
1135:by definition
1133:
1132:
1131:
1128:
1124:
1121:
1120:
1119:
1116:
1114:
1111:
1109:
1106:
1104:
1101:
1099:
1096:
1095:
1092:
1089:
1087:
1083:
1078:
1072:
1068:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1022:KripkeâPlatek
1020:
1018:
1015:
1011:
1008:
1006:
1003:
1002:
1001:
998:
997:
995:
991:
983:
980:
979:
978:
975:
973:
970:
966:
963:
962:
961:
958:
956:
953:
951:
948:
946:
943:
941:
938:
935:
931:
927:
924:
920:
917:
915:
912:
910:
907:
906:
905:
901:
898:
897:
895:
893:
889:
885:
877:
874:
872:
869:
867:
866:constructible
864:
863:
862:
859:
857:
854:
852:
849:
847:
844:
842:
839:
837:
834:
832:
829:
827:
824:
822:
819:
817:
814:
812:
809:
807:
804:
802:
799:
798:
796:
794:
789:
781:
778:
776:
773:
771:
768:
766:
763:
761:
758:
756:
753:
752:
750:
746:
743:
741:
738:
737:
736:
733:
731:
728:
726:
723:
721:
718:
716:
712:
708:
706:
703:
699:
696:
695:
694:
691:
690:
687:
684:
682:
678:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
636:
633:
632:
631:
628:
624:
619:
618:
617:
614:
613:
611:
609:
605:
597:
594:
592:
589:
587:
584:
583:
582:
579:
577:
574:
572:
569:
567:
564:
562:
559:
557:
554:
552:
549:
548:
546:
544:
543:Propositional
540:
534:
531:
529:
526:
524:
521:
519:
516:
514:
511:
509:
506:
502:
499:
498:
497:
494:
492:
489:
487:
484:
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1786:Ultraproduct
1633:Model theory
1598:Independence
1534:Formal proof
1526:Proof theory
1509:
1482:
1439:real numbers
1411:second-order
1322:Substitution
1199:Metalanguage
1140:conservative
1113:Axiom schema
1057:Constructive
1027:MorseâKelley
993:Set theories
972:Aleph number
965:inaccessible
871:Grothendieck
755:intersection
697:
642:Higher-order
630:Second-order
576:Truth tables
533:Venn diagram
316:Formal proof
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177:well-founded
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1896:Type theory
1844:undecidable
1776:Truth value
1663:equivalence
1342:non-logical
955:Enumeration
945:Isomorphism
892:cardinality
876:Von Neumann
841:Ultrafilter
806:Uncountable
740:equivalence
657:Quantifiers
647:Fixed-point
616:First-order
496:Consistency
481:Proposition
458:Traditional
429:Lindström's
419:Compactness
361:Type theory
306:Cardinality
179:(i.e., the
171:Assumptions
2017:Set theory
1707:elementary
1400:arithmetic
1268:Quantifier
1246:functional
1118:Expression
836:Transitive
780:identities
765:complement
698:hereditary
681:Set theory
217:References
165:urelements
29:set theory
1978:Supertask
1881:Recursion
1839:decidable
1673:saturated
1651:of models
1574:deductive
1569:axiomatic
1489:Hilbert's
1476:Euclidean
1457:canonical
1380:axiomatic
1312:Signature
1241:Predicate
1130:Extension
1052:Ackermann
977:Operation
856:Universal
846:Recursive
821:Singleton
816:Inhabited
801:Countable
791:Types of
775:power set
745:partition
662:Predicate
608:Predicate
523:Syllogism
513:Soundness
486:Inference
476:Tautology
378:paradoxes
124:∅
115:∅
92:∅
83:empty set
66:∅
2011:Category
1963:Logicism
1956:timeline
1932:Concrete
1791:Validity
1761:T-schema
1754:Kripke's
1749:Tarski's
1744:semantic
1734:Strength
1683:submodel
1678:spectrum
1646:function
1494:Tarski's
1483:Elements
1470:geometry
1426:Robinson
1347:variable
1332:function
1305:spectrum
1295:Sentence
1251:variable
1194:Language
1147:Relation
1108:Automata
1098:Alphabet
1082:language
936:-jection
914:codomain
900:Function
861:Universe
831:Infinite
735:Relation
518:Validity
508:Argument
406:theorem,
226:(1980).
195:See also
47:Examples
37:pure set
18:Pure set
1905:Related
1702:Diagram
1600: (
1579:Hilbert
1564:Systems
1559:Theorem
1437:of the
1382:systems
1162:Formula
1157:Grammar
1073: (
1017:General
730:Forcing
715:Element
635:Monadic
410:paradox
351:Theorem
287:General
39:) is a
1668:finite
1431:Skolem
1384:
1359:Theory
1327:Symbol
1317:String
1300:atomic
1177:ground
1172:closed
1167:atomic
1123:ground
1086:syntax
982:binary
909:domain
826:Finite
591:finite
449:Logics
408:
356:Theory
238:
1658:Model
1406:Peano
1263:Proof
1103:Arity
1032:Naive
919:image
851:Fuzzy
811:Empty
760:union
705:Class
346:Model
336:Lemma
294:Axiom
1781:Type
1584:list
1388:list
1365:list
1354:Term
1288:rank
1182:open
1076:list
888:Maps
793:sets
652:Free
622:list
372:list
299:list
236:ISBN
35:(or
31:, a
1468:of
1450:of
1398:of
930:Sur
904:Map
711:Ur-
693:Set
160:all
41:set
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