617:
With the general acceptance of the axiom of choice among the mathematical community, these issues relating to infinite and
Dedekind-infinite sets have become less central to most mathematicians. However, the study of Dedekind-infinite sets played an important role in the attempt to clarify the
593:
For a long time, many mathematicians did not even entertain the thought that there might be a distinction between the notions of infinite set and
Dedekind-infinite set. In fact, the distinction was not really realised until after
630:
stating that every set can be well-ordered, clearly the general AC implies that every infinite set is
Dedekind-infinite. However, the equivalence of the two definitions is much weaker than the full strength of AC.
769:
whose members are themselves infinite (and possibly uncountable) sets. By using the axiom of countable choice we may choose one member from each of these sets, and this member is itself a finite subset of
92:
if it is not
Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the
649:), then it follows that every infinite set is Dedekind-infinite. However, the equivalence of these two definitions is in fact strictly weaker than even the CC. Explicitly, there exists a model of
182:
160:
119:
574:, who first explicitly introduced the definition. It is notable that this definition was the first definition of "infinite" that did not rely on the definition of the
1146:
1133:, Springer-Verlag, 2006, Lecture Notes in Mathematics 1876, ISSN print edition 0075–8434, ISSN electronic edition: 1617-9692, in particular Section 4.1.
586:
in 1819. Moreover, Bolzano's definition was more accurately a relation that held between two infinite sets, rather than a definition of an infinite set
988:
is any ring that satisfies the latter condition. Beware that a ring may be
Dedekind-finite even if its underlying set is Dedekind-infinite, e.g. the
665:
That every
Dedekind-infinite set is infinite can be easily proven in ZF: every finite set has by definition a bijection with some finite ordinal
1021:
642:
subset. Hence, in this model, there exists an infinite, Dedekind-finite set. By the above, such a set cannot be well-ordered in this model.
578:(unless one follows Poincaré and regards the notion of number as prior to even the notion of set). Although such a definition was known to
1046:
188:
1124:
1106:
1088:
1074:
305:
203:
582:, he was prevented from publishing his work in any but the most obscure journals by the terms of his political exile from the
1116:
583:
493:
proves the following implications: Dedekind-infinite ⇒ dually
Dedekind-infinite ⇒ weakly Dedekind-infinite ⇒ infinite.
316:"). The full strength of AC is not needed to prove the equivalence; in fact, the equivalence of the two definitions is
1151:
677:
321:
950:
234:
in the usual sense. However, there exists a model of
Zermelo–Fraenkel set theory without the axiom of choice (
603:
363:
554:
When sets have additional structures, both kinds of infiniteness can sometimes be proved equivalent over
627:
984:
961:
301:
250:
219:
207:
126:
31:
746:
618:
boundary between the finite and the infinite, and also an important role in the history of the AC.
384:
626:
Since every infinite well-ordered set is
Dedekind-infinite, and since the AC is equivalent to the
165:
143:
102:
380:
73:
1120:
1102:
1084:
1070:
1042:
1017:
598:
formulated the AC explicitly. The existence of infinite, Dedekind-finite sets was studied by
925:
653:
in which every infinite set is
Dedekind-infinite, yet the CC fails (assuming consistency of
599:
571:
211:
192:
50:
242:
are not strong enough to prove that every set that is Dedekind-finite is finite. There are
222:. Using the axioms of Zermelo–Fraenkel set theory with the originally highly controversial
917:
579:
309:
223:
774:. More precisely, according to the axiom of countable choice, a (countable) set exists,
575:
409:
274:
215:
199:
122:
93:
687:
First, define a function over the natural numbers (that is, over the finite ordinals)
1140:
639:
595:
349:
293:
238:) in which there exists an infinite, Dedekind-finite set, showing that the axioms of
134:
289:– an infinite set is one that is literally "not finite", in the sense of bijection.
929:
562:
proves that a well-ordered set is Dedekind-infinite if and only if it is infinite.
262:
196:
65:
680:(denotation: axiom CC) one can prove the converse, namely that every infinite set
17:
1094:
946:
191:
showed the need for a more careful treatment of set theory, most mathematicians
38:
270:
243:
231:
300:
it is Dedekind-infinite. However, this equivalence cannot be proved with the
246:
besides the one given by Dedekind that do not depend on the axiom of choice.
512:
509:
547:
are injective sequences, one could exhibit a countably infinite subset of
543:
had a countably infinite subset, then using the fact that the elements of
1069:. Volume 65. American Mathematical Society. 2nd ed. AMS Bookstore, 2004.
317:
989:
446:
if it satisfies any, and then all, of the following equivalent (over
340:
if it satisfies any, and then all, of the following equivalent (over
54:
30:"Dedekind finite" redirects here. For the term from ring theory, see
661:
Proof of equivalence to infinity, assuming axiom of countable choice
1014:
Zermelo's Axiom of Choice: Its Origins, Development & Influence
230:) one can show that a set is Dedekind-finite if and only if it is
355:
there exists an injective map from a countably infinite set to
956:
has the analogous property in the category of (left or right)
297:
1091:, in particular pp. 22-30 and tables 1 and 2 on p. 322-323
129:, there exists a bijection that maps every natural number
202:
it is Dedekind-infinite. In the early twentieth century,
523:
is infinite, the function "drop the last element" from
862:, can be easily defined. We may now define a bijection
1041:. Lecture Notes in Mathematics 1876. Springer-Verlag.
265:" should be compared with the usual definition: a set
168:
146:
105:
722:(i.e. that have a bijection with the finite ordinal
481:, there is no bijection from {0, 1, 2, ..., n−1} to
257:
Comparison with the usual definition of infinite set
738:would be finite (as can be proven by induction on
244:definitions of finiteness and infiniteness of sets
176:
154:
113:
570:The term is named after the German mathematician
292:During the latter half of the 19th century, most
273:when it cannot be put in bijection with a finite
140:. Since the set of squares is a proper subset of
844:, and a bijection from the natural numbers to
638:in which there exists an infinite set with no
527:to itself is surjective but not injective, so
531:is dually Dedekind-infinite. However, since
500:having an infinite Dedekind-finite set. Let
72:. Explicitly, this means that there exists a
8:
606:in 1912; these sets were at first called
170:
169:
167:
148:
147:
145:
107:
106:
104:
1083:, Springer-Verlag, 1982 (out-of-print),
1007:
1005:
1001:
908:is Dedekind-infinite, and we are done.
634:In particular, there exists a model of
206:, today the most commonly used form of
1113:A first course in noncommutative rings
817:) and is therefore a finite subset of
296:simply assumed that a set is infinite
249:A vaguely related notion is that of a
49:(named after the German mathematician
27:Set with an equinumerous proper subset
1147:Basic concepts in infinite set theory
438:that is surjective but not injective;
7:
673:that this is not Dedekind-infinite.
669:, and one can prove by induction on
645:If we assume the axiom CC (i. e., AC
1067:Mathematical surveys and monographs
840:is an infinite countable subset of
702:, so that for every natural number
684:is Dedekind-infinite, as follows:
454:there exists a surjective map from
714:) is the set of finite subsets of
324:(CC). (See the references below.)
189:foundational crisis of mathematics
25:
797:so that for every natural number
881:that takes every member not in
832:as the union of the members of
734:) is never empty, or otherwise
622:Relation to the axiom of choice
535:is Dedekind-finite, then so is
390:there is an injective function
893:) for every natural number to
458:onto a countably infinite set;
1:
1117:Graduate Texts in Mathematics
924:is Dedekind-finite if in the
1101:, Dover Publications, 2008,
328:Dedekind-infinite sets in ZF
177:{\displaystyle \mathbb {N} }
155:{\displaystyle \mathbb {N} }
114:{\displaystyle \mathbb {N} }
1012:Moore, Gregory H. (2013) .
306:Zermelo–Fraenkel set theory
277:, namely a set of the form
204:Zermelo–Fraenkel set theory
1168:
1119:. 2nd ed. Springer, 2001.
749:of f is the countable set
218:free of paradoxes such as
29:
1081:Zermelo's Axiom of Choice
678:axiom of countable choice
322:axiom of countable choice
1037:Herrlich, Horst (2006).
951:von Neumann regular ring
444:weakly Dedekind-infinite
418:dually Dedekind-infinite
285:for some natural number
80:onto some proper subset
504:be such a set, and let
477:for any natural number
408:denotes the set of all
312:(AC) (usually denoted "
184:is Dedekind-infinite.
1016:. Dover Publications.
604:Alfred North Whitehead
496:There exist models of
178:
156:
115:
1065:Faith, Carl Clifton.
885:to itself, and takes
628:well-ordering theorem
508:be the set of finite
465:is Dedekind-infinite;
210:, was proposed as an
179:
157:
116:
985:Dedekind-finite ring
982:. More generally, a
918:category-theoretical
584:University of Prague
424:there is a function
261:This definition of "
251:Dedekind-finite ring
208:axiomatic set theory
166:
144:
103:
99:A simple example is
32:Dedekind-finite ring
1099:The Axiom of Choice
1079:Moore, Gregory H.,
964:if and only if in
640:countably infinite
612:Dedekind cardinals
350:countably infinite
174:
152:
111:
74:bijective function
1129:Herrlich, Horst,
1023:978-0-486-48841-7
809:) is a member of
608:mediate cardinals
338:Dedekind-infinite
220:Russell's paradox
127:Galileo's paradox
53:) if some proper
47:Dedekind-infinite
18:Dedekind-infinite
16:(Redirected from
1159:
1152:Cardinal numbers
1115:. Volume 131 of
1111:Lam, Tsit-Yuen.
1053:
1052:
1034:
1028:
1027:
1009:
981:
974:
944:
926:category of sets
903:
880:
861:
796:
768:
701:
600:Bertrand Russell
572:Richard Dedekind
558:. For instance,
461:the powerset of
437:
403:
378:
320:weaker than the
284:
212:axiomatic system
183:
181:
180:
175:
173:
161:
159:
158:
153:
151:
120:
118:
117:
112:
110:
51:Richard Dedekind
21:
1167:
1166:
1162:
1161:
1160:
1158:
1157:
1156:
1137:
1136:
1131:Axiom of Choice
1095:Jech, Thomas J.
1062:
1057:
1056:
1049:
1039:Axiom of Choice
1036:
1035:
1031:
1024:
1011:
1010:
1003:
998:
976:
969:
932:
914:
912:Generalizations
894:
863:
849:
828:Now, we define
775:
750:
688:
663:
648:
624:
580:Bernard Bolzano
576:natural numbers
568:
425:
410:natural numbers
391:
366:
330:
310:axiom of choice
279:{0, 1, 2, ...,
278:
259:
224:axiom of choice
214:to formulate a
164:
163:
142:
141:
123:natural numbers
101:
100:
94:natural numbers
90:Dedekind-finite
35:
28:
23:
22:
15:
12:
11:
5:
1165:
1163:
1155:
1154:
1149:
1139:
1138:
1135:
1134:
1127:
1109:
1092:
1077:
1061:
1058:
1055:
1054:
1048:978-3540309895
1047:
1029:
1022:
1000:
999:
997:
994:
913:
910:
696:→ Power(Power(
662:
659:
646:
623:
620:
567:
564:
487:
486:
467:
466:
459:
450:) conditions:
440:
439:
414:
413:
388:
360:
353:
344:) conditions:
329:
326:
298:if and only if
294:mathematicians
258:
255:
216:theory of sets
200:if and only if
195:that a set is
172:
150:
109:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1164:
1153:
1150:
1148:
1145:
1144:
1142:
1132:
1128:
1126:
1125:0-387-95183-0
1122:
1118:
1114:
1110:
1108:
1107:0-486-46624-8
1104:
1100:
1096:
1093:
1090:
1089:0-387-90670-3
1086:
1082:
1078:
1076:
1075:0-8218-3672-2
1072:
1068:
1064:
1063:
1059:
1050:
1044:
1040:
1033:
1030:
1025:
1019:
1015:
1008:
1006:
1002:
995:
993:
991:
987:
986:
979:
972:
967:
963:
959:
955:
952:
948:
943:
939:
935:
931:
927:
923:
920:terms, a set
919:
916:Expressed in
911:
909:
907:
901:
897:
892:
888:
884:
878:
874:
870:
866:
860:
856:
852:
847:
843:
839:
835:
831:
826:
824:
820:
816:
812:
808:
804:
800:
794:
790:
786:
782:
778:
773:
766:
762:
758:
754:
748:
743:
741:
737:
733:
729:
725:
721:
717:
713:
709:
705:
699:
695:
691:
685:
683:
679:
676:By using the
674:
672:
668:
660:
658:
656:
652:
643:
641:
637:
632:
629:
621:
619:
615:
613:
609:
605:
601:
597:
596:Ernst Zermelo
591:
589:
585:
581:
577:
573:
565:
563:
561:
557:
552:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
511:
507:
503:
499:
494:
492:
484:
480:
476:
475:
474:
472:
464:
460:
457:
453:
452:
451:
449:
445:
436:
432:
428:
423:
422:
421:
419:
411:
407:
402:
398:
394:
389:
386:
382:
377:
373:
369:
365:
361:
358:
354:
351:
347:
346:
345:
343:
339:
335:
327:
325:
323:
319:
315:
311:
307:
303:
299:
295:
290:
288:
282:
276:
272:
268:
264:
256:
254:
252:
247:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
198:
194:
190:
185:
139:
136:
132:
128:
124:
121:, the set of
97:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
56:
52:
48:
44:
40:
33:
19:
1130:
1112:
1098:
1080:
1066:
1038:
1032:
1013:
983:
977:
970:
965:
957:
953:
941:
937:
933:
930:monomorphism
921:
915:
905:
899:
895:
890:
886:
882:
876:
872:
868:
864:
858:
854:
850:
845:
841:
837:
833:
829:
827:
822:
818:
814:
810:
806:
802:
798:
792:
788:
784:
780:
776:
771:
764:
760:
756:
752:
744:
739:
735:
731:
727:
723:
719:
715:
711:
707:
703:
697:
693:
689:
686:
681:
675:
670:
666:
664:
654:
650:
644:
635:
633:
625:
616:
611:
607:
592:
587:
569:
559:
555:
553:
548:
544:
540:
536:
532:
528:
524:
520:
516:
505:
501:
497:
495:
490:
488:
482:
478:
470:
468:
462:
455:
447:
443:
441:
434:
430:
426:
417:
415:
405:
400:
396:
392:
375:
371:
367:
356:
341:
337:
333:
331:
313:
308:without the
291:
286:
280:
266:
263:infinite set
260:
248:
239:
235:
227:
186:
137:
130:
98:
89:
85:
81:
77:
69:
66:equinumerous
61:
57:
46:
42:
36:
947:isomorphism
362:there is a
88:. A set is
39:mathematics
1141:Categories
1060:References
469:and it is
385:surjective
226:included (
187:Until the
904:. Hence,
513:sequences
510:injective
381:injective
348:it has a
283:−1}
990:integers
975:implies
936: :
928:, every
867: :
853: :
821:of size
718:of size
692: :
519:. Since
471:infinite
429: :
404:, where
395: :
383:but not
379:that is
370: :
364:function
318:strictly
271:infinite
197:infinite
41:, a set
962:modules
566:History
352:subset;
275:ordinal
193:assumed
133:to its
125:. From
1123:
1105:
1087:
1073:
1045:
1020:
945:is an
588:per se
489:Then,
442:it is
416:it is
332:A set
302:axioms
232:finite
135:square
55:subset
996:Notes
747:image
515:from
76:from
1121:ISBN
1103:ISBN
1085:ISBN
1071:ISBN
1043:ISBN
1018:ISBN
949:. A
902:+ 1)
787:) |
759:) |
745:The
602:and
539:(if
473:if:
420:if:
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779:= {
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657:).
610:or
551:).
336:is
304:of
269:is
253:.
228:ZFC
84:of
68:to
64:is
60:of
45:is
37:In
1143::
1097:,
1004:^
992:.
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971:xy
968:,
940:→
875:\
871:→
857:→
848:,
836:.
825:.
801:,
795:},
791:∈
767:},
763:∈
706:,
700:))
655:ZF
651:ZF
636:ZF
614:.
590:.
560:ZF
556:ZF
498:ZF
491:ZF
448:ZF
433:→
399:→
374:→
342:ZF
314:ZF
240:ZF
236:ZF
162:,
96:.
1051:.
1026:.
966:R
960:-
958:R
954:R
942:A
938:A
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922:A
906:X
900:n
898:(
896:h
891:n
889:(
887:h
883:U
877:h
873:X
869:X
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859:U
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846:U
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830:U
823:n
819:X
815:n
813:(
811:f
807:n
805:(
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799:n
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789:n
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781:g
777:G
772:X
765:N
761:n
757:n
755:(
753:f
751:{
740:n
736:X
732:n
730:(
728:f
724:n
720:n
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694:N
690:f
682:X
671:n
667:n
647:ω
549:A
545:B
541:B
537:B
533:A
529:B
525:B
521:A
517:A
506:B
502:A
485:.
483:A
479:n
463:A
456:A
435:A
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427:f
412:;
406:N
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393:f
387:;
376:A
372:A
368:f
359:;
357:A
334:A
287:n
281:n
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171:N
149:N
138:n
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108:N
86:A
82:B
78:A
70:A
62:A
58:B
43:A
34:.
20:)
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