937:
160:
325:
However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms: Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.
556:
722:
446:
275:
638:
1394:
72:
615:
591:
295:
249:
561:
Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions.
457:
341:
In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a
650:
351:
805:
1083:
903:
768:
is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a
1411:
875:
857:
829:
1389:
32:
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1042:
641:
1406:
1399:
1037:
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330:
179:
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968:
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570:
896:
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1426:
1301:
1253:
1067:
990:
1460:
1341:
1153:
973:
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1210:
1190:
1168:
306:
The axiom of union allows one to unpack a set of sets and thus create a flatter set. Together with the
1450:
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1274:
1205:
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1098:
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20:
1445:
1356:
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953:
889:
795:
213:
1368:
1363:
1148:
1103:
1010:
311:
254:
620:
333:, the axiom of union implies that one can form the union of a family of sets indexed by a set.
155:{\displaystyle \forall A\,\exists B\,\forall c\,(c\in B\iff \exists D\,(c\in D\land D\in A)\,)}
1225:
1062:
1054:
1025:
995:
926:
871:
853:
852:. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.
825:
801:
172:
1513:
1503:
1488:
1483:
1351:
1005:
307:
1382:
1320:
1138:
958:
63:
322:
The axiom of replacement allows one to form many unions, such as the union of two sets.
1518:
1315:
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1200:
1185:
1142:
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280:
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198:
1538:
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1234:
817:
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36:
1473:
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1215:
1173:
1032:
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1133:
863:
845:
1478:
1346:
1249:
912:
551:{\displaystyle \forall {\mathcal {F}}\,\exists A\forall x\Rightarrow x\in A].}
788:
Ernst
Zermelo, 1908, "Untersuchungen ĂĽber die Grundlagen der Mengenlehre I",
1281:
1244:
1195:
1093:
733:
169:
800:
From Frege to Gödel: A Source Book in
Mathematical Logic, pp. 199–215
342:
717:{\displaystyle \bigcap A=\{c\in E:\forall D(D\in A\Rightarrow c\in D)\}}
441:{\displaystyle \forall {\mathcal {F}}\,\exists A\,\forall Y\,\forall x.}
1306:
1128:
310:, this implies that for any two sets, there is a set (called their
1178:
945:
28:
277:
which consists of just the elements of the elements of that set
885:
345:
of the union of a set. For example, Kunen states the axiom as
868:
Set Theory: The Third
Millennium Edition, Revised and Expanded
881:
50:
whose elements are precisely the elements of the elements of
519:
466:
412:
360:
728:
so no separate axiom of intersection is necessary. (If
314:) that contains exactly the elements of the two sets.
653:
623:
603:
579:
460:
354:
283:
257:
237:
75:
1459:
1422:
1334:
1224:
1112:
1053:
944:
919:
822:Set Theory: An Introduction to Independence Proofs
716:
632:
609:
585:
550:
440:
289:
269:
243:
154:
772:is antithetical to Zermelo–Fraenkel set theory.)
66:of the Zermelo–Fraenkel axioms, the axiom reads:
42:Informally, the axiom states that for each set
897:
8:
711:
663:
904:
890:
882:
736:, then trying to form the intersection of
617:, it is possible to form the intersection
113:
109:
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518:
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365:
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236:
148:
120:
96:
89:
82:
74:
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7:
569:There is no corresponding axiom of
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461:
380:
373:
366:
355:
114:
90:
83:
76:
14:
935:
35:. This axiom was introduced by
16:Concept in axiomatic set theory
708:
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684:
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432:
420:
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389:
386:
149:
145:
121:
110:
97:
1:
642:axiom schema of specification
331:axiom schema of replacement
270:{\displaystyle \bigcup A\ }
186:such that, for any element
33:Zermelo–Fraenkel set theory
1561:
1395:von Neumann–Bernays–Gödel
860:(Springer-Verlag edition).
1196:One-to-one correspondence
933:
633:{\displaystyle \bigcap A}
792:65(2), pp. 261–281.
565:Relation to Intersection
451:which is equivalent to
318:Relation to Replacement
1154:Constructible universe
981:Constructibility (V=L)
718:
634:
611:
587:
552:
442:
337:Relation to Separation
291:
271:
245:
156:
1377:Principia Mathematica
1211:Transfinite induction
1070:(i.e. set difference)
794:English translation:
790:Mathematische Annalen
719:
635:
612:
588:
553:
443:
292:
272:
246:
157:
1545:Axioms of set theory
1451:Burali-Forti paradox
1206:Set-builder notation
1159:Continuum hypothesis
1099:Symmetric difference
651:
621:
601:
577:
458:
352:
281:
255:
235:
73:
21:axiomatic set theory
1412:Tarski–Grothendieck
796:Jean van Heijenoort
302:Relation to Pairing
1001:Limitation of size
714:
630:
607:
583:
548:
438:
329:Together with the
287:
267:
241:
152:
1532:
1531:
1441:Russell's paradox
1390:Zermelo–Fraenkel
1291:Dedekind-infinite
1164:Diagonal argument
1063:Cartesian product
927:Set (mathematics)
806:978-0-674-32449-7
610:{\displaystyle E}
586:{\displaystyle A}
290:{\displaystyle A}
266:
251:, there is a set
244:{\displaystyle A}
227:or, more simply:
1552:
1514:Bertrand Russell
1504:John von Neumann
1489:Abraham Fraenkel
1484:Richard Dedekind
1446:Suslin's problem
1357:Cantor's theorem
1074:De Morgan's laws
939:
906:
899:
892:
883:
850:Naive set theory
833:
815:
809:
786:
723:
721:
720:
715:
639:
637:
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631:
616:
614:
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584:
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549:
523:
522:
470:
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447:
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444:
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416:
415:
364:
363:
308:axiom of pairing
296:
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242:
161:
159:
158:
153:
58:Formal statement
1560:
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1553:
1551:
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1549:
1535:
1534:
1533:
1528:
1455:
1434:
1418:
1383:New Foundations
1330:
1220:
1139:Cardinal number
1122:
1108:
1049:
940:
931:
915:
910:
842:
840:Further reading
837:
836:
816:
812:
793:
787:
783:
778:
649:
648:
619:
618:
599:
598:
597:set containing
575:
574:
567:
456:
455:
350:
349:
339:
320:
304:
279:
278:
253:
252:
233:
232:
219:is a member of
209:is a member of
201:there is a set
194:is a member of
71:
70:
64:formal language
60:
46:there is a set
17:
12:
11:
5:
1558:
1556:
1548:
1547:
1537:
1536:
1530:
1529:
1527:
1526:
1521:
1519:Thoralf Skolem
1516:
1511:
1506:
1501:
1496:
1491:
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1471:
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1457:
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1453:
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1433:
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1429:
1423:
1420:
1419:
1417:
1416:
1415:
1414:
1409:
1404:
1403:
1402:
1387:
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1371:
1360:
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1344:
1338:
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1332:
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1318:
1313:
1304:
1299:
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1279:
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1277:
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1257:
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1242:
1237:
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1229:
1222:
1221:
1219:
1218:
1213:
1208:
1203:
1201:Ordinal number
1198:
1193:
1188:
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1181:
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1161:
1156:
1151:
1146:
1136:
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1125:
1123:
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1120:
1117:
1113:
1110:
1109:
1107:
1106:
1101:
1096:
1091:
1086:
1081:
1079:Disjoint union
1076:
1071:
1065:
1059:
1057:
1051:
1050:
1048:
1047:
1046:
1045:
1040:
1029:
1028:
1026:Martin's axiom
1023:
1018:
1013:
1008:
1003:
998:
993:
991:Extensionality
988:
983:
978:
977:
976:
971:
966:
956:
950:
948:
942:
941:
934:
932:
930:
929:
923:
921:
917:
916:
911:
909:
908:
901:
894:
886:
880:
879:
870:. Springer.
861:
841:
838:
835:
834:
818:Kunen, Kenneth
810:
780:
779:
777:
774:
766:
765:
726:
725:
713:
710:
707:
704:
701:
698:
695:
692:
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683:
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677:
674:
671:
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665:
662:
659:
656:
629:
626:
606:
582:
566:
563:
559:
558:
547:
544:
541:
538:
535:
532:
529:
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521:
516:
513:
510:
507:
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501:
498:
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492:
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486:
483:
480:
477:
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468:
463:
449:
448:
437:
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428:
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422:
419:
414:
409:
406:
403:
400:
397:
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391:
388:
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382:
378:
375:
371:
368:
362:
357:
338:
335:
319:
316:
303:
300:
299:
298:
286:
263:
260:
240:
225:
224:
199:if and only if
163:
162:
151:
147:
144:
141:
138:
135:
132:
129:
126:
123:
119:
116:
112:
108:
105:
102:
99:
95:
92:
88:
85:
81:
78:
59:
56:
27:is one of the
25:axiom of union
15:
13:
10:
9:
6:
4:
3:
2:
1557:
1546:
1543:
1542:
1540:
1525:
1524:Ernst Zermelo
1522:
1520:
1517:
1515:
1512:
1510:
1509:Willard Quine
1507:
1505:
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1485:
1482:
1480:
1477:
1475:
1472:
1470:
1467:
1466:
1464:
1462:
1461:Set theorists
1458:
1452:
1449:
1447:
1444:
1442:
1439:
1438:
1436:
1430:
1428:
1425:
1424:
1421:
1413:
1410:
1408:
1407:Kripke–Platek
1405:
1401:
1398:
1397:
1396:
1393:
1392:
1391:
1388:
1384:
1381:
1380:
1379:
1378:
1374:
1370:
1367:
1366:
1365:
1362:
1361:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1339:
1337:
1333:
1327:
1324:
1322:
1319:
1317:
1314:
1312:
1310:
1305:
1303:
1300:
1298:
1295:
1292:
1288:
1285:
1283:
1280:
1276:
1273:
1271:
1268:
1266:
1263:
1262:
1261:
1258:
1255:
1251:
1248:
1246:
1243:
1241:
1238:
1236:
1233:
1232:
1230:
1227:
1223:
1217:
1214:
1212:
1209:
1207:
1204:
1202:
1199:
1197:
1194:
1192:
1189:
1187:
1184:
1180:
1177:
1175:
1172:
1171:
1170:
1167:
1165:
1162:
1160:
1157:
1155:
1152:
1150:
1147:
1144:
1140:
1137:
1135:
1132:
1130:
1127:
1126:
1124:
1118:
1115:
1114:
1111:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1069:
1066:
1064:
1061:
1060:
1058:
1056:
1052:
1044:
1043:specification
1041:
1039:
1036:
1035:
1034:
1031:
1030:
1027:
1024:
1022:
1019:
1017:
1014:
1012:
1009:
1007:
1004:
1002:
999:
997:
994:
992:
989:
987:
984:
982:
979:
975:
972:
970:
967:
965:
962:
961:
960:
957:
955:
952:
951:
949:
947:
943:
938:
928:
925:
924:
922:
918:
914:
907:
902:
900:
895:
893:
888:
887:
884:
877:
876:3-540-44085-2
873:
869:
865:
862:
859:
858:0-387-90092-6
855:
851:
847:
844:
843:
839:
831:
830:0-444-86839-9
827:
824:. Elsevier.
823:
819:
814:
811:
808:
807:
803:
797:
791:
785:
782:
775:
773:
771:
770:universal set
763:
759:
755:
751:
747:
743:
742:
741:
739:
735:
731:
705:
702:
699:
693:
690:
687:
681:
675:
672:
669:
666:
660:
657:
654:
647:
646:
645:
643:
627:
624:
604:
596:
580:
572:
564:
562:
545:
539:
536:
533:
514:
511:
508:
505:
502:
499:
493:
481:
475:
454:
453:
452:
435:
429:
426:
423:
407:
404:
401:
398:
395:
392:
383:
376:
369:
348:
347:
346:
344:
336:
334:
332:
327:
323:
317:
315:
313:
309:
301:
284:
261:
258:
238:
230:
229:
228:
222:
218:
215:
212:
208:
204:
200:
197:
193:
189:
185:
181:
177:
174:
171:
168:
167:
166:
165:or in words:
142:
139:
136:
133:
130:
127:
124:
117:
106:
103:
100:
93:
86:
79:
69:
68:
67:
65:
57:
55:
53:
49:
45:
40:
38:
37:Ernst Zermelo
34:
30:
26:
22:
1474:Georg Cantor
1469:Paul Bernays
1400:Morse–Kelley
1375:
1308:
1307:Subset
1254:hereditarily
1216:Venn diagram
1174:ordered pair
1089:Intersection
1033:Axiom schema
1020:
867:
864:Jech, Thomas
849:
821:
813:
799:
789:
784:
767:
761:
757:
753:
749:
745:
737:
729:
727:
594:
571:intersection
568:
560:
450:
340:
328:
324:
321:
305:
231:For any set
226:
220:
216:
210:
206:
202:
195:
191:
187:
183:
175:
164:
61:
51:
47:
43:
41:
24:
18:
1499:Thomas Jech
1342:Alternative
1321:Uncountable
1275:Ultrafilter
1134:Cardinality
1038:replacement
986:Determinacy
846:Paul Halmos
1494:Kurt Gödel
1479:Paul Cohen
1316:Transitive
1084:Identities
1068:Complement
1055:Operations
1016:Regularity
954:Adjunction
913:Set theory
776:References
748:: for all
640:using the
205:such that
1427:Paradoxes
1347:Axiomatic
1326:Universal
1302:Singleton
1297:Recursive
1240:Countable
1235:Amorphous
1094:Power set
1011:Power set
969:dependent
964:countable
734:empty set
703:∈
697:⇒
691:∈
679:∀
670:∈
655:⋂
625:⋂
537:∈
531:⇒
515:∈
509:∧
503:∈
491:∃
479:∀
473:∃
462:∀
427:∈
421:⇒
408:∈
402:∧
396:∈
381:∀
374:∀
367:∃
356:∀
259:⋃
170:Given any
140:∈
134:∧
128:∈
115:∃
111:⟺
104:∈
91:∀
84:∃
77:∀
1539:Category
1431:Problems
1335:Theories
1311:Superset
1287:Infinite
1116:Concepts
996:Infinity
920:Overview
866:, 2003.
820:, 1980.
798:, 1967,
595:nonempty
343:superset
180:there is
1369:General
1364:Zermelo
1270:subbase
1252: (
1191:Forcing
1169:Element
1141: (
1119:Methods
1006:Pairing
732:is the
62:In the
1260:Filter
1250:Finite
1186:Family
1129:Almost
974:global
959:Choice
946:Axioms
874:
856:
828:
804:
760:is in
265:
182:a set
29:axioms
23:, the
1352:Naive
1282:Fuzzy
1245:Empty
1228:types
1179:tuple
1149:Class
1143:large
1104:Union
1021:Union
593:is a
573:. If
312:union
1265:base
872:ISBN
854:ISBN
826:ISBN
802:ISBN
1226:Set
752:in
740:as
644:as
214:and
173:set
31:of
19:In
1541::
848:,
756:,
190:,
178:,
54:.
39:.
1309:·
1293:)
1289:(
1256:)
1145:)
905:e
898:t
891:v
878:.
832:.
764:}
762:D
758:c
754:A
750:D
746:c
744:{
738:A
730:A
724:,
712:}
709:)
706:D
700:c
694:A
688:D
685:(
682:D
676::
673:E
667:c
664:{
661:=
658:A
628:A
605:E
581:A
546:.
543:]
540:A
534:x
528:]
525:)
520:F
512:Y
506:Y
500:x
497:(
494:Y
488:[
485:[
482:x
476:A
467:F
436:.
433:]
430:A
424:x
418:)
413:F
405:Y
399:Y
393:x
390:(
387:[
384:x
377:Y
370:A
361:F
297:.
285:A
262:A
239:A
223:.
221:A
217:D
211:D
207:c
203:D
196:B
192:c
188:c
184:B
176:A
150:)
146:)
143:A
137:D
131:D
125:c
122:(
118:D
107:B
101:c
98:(
94:c
87:B
80:A
52:x
48:y
44:x
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