485:
kreatürlichen Welt vertreten ist; drittens, sofern es als mathematische Größe, Zahl oder
Ordnungstypus vom Denken in abstracto aufgefaßt werden kann. In den beiden letzten Beziehungen, wo es offenbar als beschränktes, noch weiterer Vermehrung fähiges und insofern dem Endlichen verwandtes A.-U. sich darstellt, nenne ich es
390:
Cantor (1) took the absolute to be a manifestation of God When the absolute is first introduced in
Grundlagen, it is linked to God: "the true infinite or absolute, which is in God, admits no kind of determination" (Cantor 1883b, p. 175) This is not an incidental remark, for Cantor is very explicit
85:
The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extra worldly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural
484:
Es wurde das Aktual-Unendliche (A-U.) nach drei
Beziehungen unterschieden: erstens, sofern es in der höchsten Vollkommenheit, im völlig unabhängigen außerweltlichen Sein, in Deo realisiert ist, wo ich es Absolut Unendliches oder kurzweg Absolutes nenne; zweitens, sofern es in der abhängigen,
86:
world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or order type. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it
232:. All of these problems can be traced back to the idea that, for every property that can be logically defined, there exists a set of all objects that have that property. However, as in Cantor's argument (above), this idea leads to difficulties.
527:, ed. Jean van Heijenoort, Cambridge, Massachusetts: Harvard University Press, 1967, pp. 113–117. These references both purport to be a letter from Cantor to Dedekind, dated July 28, 1899. However, as
273:
While this solves the logical problem, one could argue that the philosophical problem remains. It seems natural that a set of individuals ought to exist, so long as the individuals exist. Indeed,
149:
of which one can readily convince oneself that every number γ occurring in it is the type of the sequence of all its preceding elements (including 0). (The sequence
262:, which does not allow the unrestricted formation of sets from arbitrary properties. Rather, we may form the set of all objects that have a given property
683:
270:). This allows for the formation of sets based on properties, in a limited sense, while (hopefully) preserving the consistency of the theory.
130:
Now let us adjoin 0 as an additional element to this sequence, and place it, obviously, in the first position; then we obtain a sequence
673:
955:
633:
965:
878:
589:
575:
567:
425:
734:
523:, Georg Cantor, ed. Ernst Zermelo, Hildesheim: Georg Olms Verlagsbuchhandlung, 1962, pp. 443–447; translated into English in
477:
267:
57:. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or
780:
970:
420:
by Jané; with biography by Adolf
Fraenkel; reprinted Hildesheim: Georg Olms, 1962, and Berlin: Springer-Verlag, 1980,
924:
919:
236:
285:
may have no formal existence (i.e., as a set) within the theory. For example, the class of all sets would be a
888:
698:
298:
626:
785:
693:
688:
528:
330:
109:
724:
678:
325:
252:
219:
70:
893:
719:
228:
to many. This is related to the Burali-Forti's paradox which implies that there can be no greatest
862:
846:
381:
373:
315:
278:
259:
88:
58:
535:, of two letters from Cantor to Dedekind, the first dated July 28 and the second dated August 3.
439:
960:
836:
790:
663:
619:
585:
571:
563:
421:
240:
465:
934:
929:
841:
816:
795:
714:
365:
356:
Ignacio Jané (May 1995). "The role of the absolute infinite in Cantor's conception of set".
274:
97:
775:
760:
647:
409:
310:
294:
883:
857:
831:
806:
801:
478:
https://www.uni-siegen.de/fb6/phima/lehre/phima10/quellentexte/handout-phima-teil4b.pdf
229:
949:
914:
826:
811:
532:
385:
282:
51:
289:. This is philosophically unsatisfying to some and has motivated additional work in
251:. Any such totality would itself have to be a set, thus lying somewhere within the
909:
821:
765:
335:
286:
105:
66:
54:
208:
The system Ω of all numbers is an inconsistent, absolutely infinite multiplicity.
224:
The idea that the collection of all ordinal numbers cannot logically exist seems
17:
770:
559:
225:
73:: every property of the absolute infinite is also held by some smaller object.
729:
320:
290:
182:
would correspond to it which would be greater than all numbers of the system
852:
281:
to describe arbitrary (possibly "large") entities, these predicates of the
277:
might be said to be based on this notion. Although
Zermelo's fix allows a
750:
642:
47:
377:
293:
and other methods of formalizing the foundations of mathematics such as
668:
595:
548:
369:
108:
if it fulfills the condition that every sub-multiplicity has a first
531:
has discovered, this is in fact an amalgamation by Cantor's editor,
525:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879-1931
411:
Gesammelte
Abhandlungen mathematischen und philosophischen Inhalts
39:
615:
611:
550:
The role of the absolute infinite in Cantor's conception of set
501:
128:
in its natural ordering according to magnitude is a "sequence".
391:
and insistent about the relation between the absolute and God.
62:
562:, Princeton, New Jersey: Princeton University Press, 1995,
440:"Ueber unendliche, lineare Punktmannichfaltigkeiten (5)"
116:
Now I envisage the system of all numbers and denote it
178:
were consistent, then as a well-ordered set, a number
502:
506:
Jahresbericht der
Deutschen Mathematiker-Vereinigung
112:; such a multiplicity I call for short a "sequence".
902:
871:
743:
707:
656:
100:(text in square brackets not present in original):
584:, A. W. Moore, London, New York: Routledge, 1990,
96:Cantor also mentioned the idea in his letters to
489:und setze es dem Absoluten strengstens entgegen.
258:A standard solution to this problem is found in
482:
102:
83:
171:) cannot be a consistent multiplicity. For if
627:
464:by Michael Heller and W. Hugh Woodin (2011),
403:
401:
399:
8:
92:and strongly contrast it with the absolute.
61:. Cantor linked the absolute infinite with
634:
620:
612:
408:Georg Cantor (1932). Ernst Zermelo (ed.).
243:formation, and thus no such thing as the
194:, because it comprises all numbers. Thus
239:, there can be no end to the process of
570:; orig. pub. Boston: Birkhäuser, 1982,
347:
255:and thus failing to contain every set.
202:, which is a contradiction. Therefore:
190:, however, also belongs to the system
596:Set Theory, Skolem's Paradox and the
511:(1974/75), pp. 104–139, at p. 126 ff.
414:. Berlin: Verlag von Julius Springer.
7:
684:Hilbert's paradox of the Grand Hotel
608:, #1 (January 1985), pp. 13–20.
462:Infinity: New Research and Frontiers
65:, and believed that it had various
46:", is an extension of the idea of
25:
879:Differential geometry of surfaces
674:Controversy over Cantor's theory
735:Synthetic differential geometry
481:Translated quote from German:
1:
153:has this property first for ω
781:Cardinality of the continuum
235:More generally, as noted by
42:), in context often called "
27:Biggest number ever imagined
987:
744:Formalizations of infinity
217:
104:A multiplicity is called
69:properties, including the
956:Philosophy of mathematics
920:Gottfried Wilhelm Leibniz
264:and lie in some given set
966:Superlatives in religion
214:The Burali-Forti paradox
925:August Ferdinand Möbius
708:Branches of mathematics
699:Paradoxes of set theory
521:Gesammelte Abhandlungen
504:, I. Grattan-Guinness,
299:Willard Van Orman Quine
491:
211:
198:would be greater than
94:
889:Möbius transformation
786:Dedekind-infinite set
694:Paradoxes of infinity
689:Infinity (philosophy)
556:Infinity and the Mind
529:Ivor Grattan-Guinness
444:Mathematische Annalen
438:Georg Cantor (1883).
331:Absolute (philosophy)
725:Nonstandard analysis
326:Reflection principle
260:Zermelo's set theory
245:totality of all sets
220:Burali-Forti paradox
167:(and therefore also
76:
71:reflection principle
894:Riemannian manifold
863:Transfinite numbers
720:Internal set theory
268:Axiom of Separation
971:Conceptions of God
847:Sphere at infinity
798:(Complex infinity)
370:10.1007/BF01129011
316:Limitation of size
943:
942:
837:Point at infinity
817:Hyperreal numbers
791:Directed infinity
756:Absolute infinite
679:Galileo's paradox
664:Ananta (infinite)
452:Original article.
139:0, 1, 2, 3, ... ω
32:absolute infinite
18:Absolute Infinite
16:(Redirected from
978:
935:Abraham Robinson
930:Bernhard Riemann
849:(Kleinian group)
842:Regular cardinal
796:Division by zero
776:Cardinal numbers
715:Complex analysis
650:
636:
629:
622:
613:
536:
518:
512:
499:
493:
475:
469:
459:
453:
451:
435:
429:
415:
405:
394:
393:
352:
275:naive set theory
176:
165:
147:+1, ..., γ, ...
135:
98:Richard Dedekind
21:
986:
985:
981:
980:
979:
977:
976:
975:
946:
945:
944:
939:
898:
867:
858:Surreal numbers
832:Ordinal numbers
761:Actual infinity
739:
703:
652:
646:
640:
601:, A. W. Moore,
545:
540:
539:
519:
515:
500:
496:
480:
476:
472:
460:
456:
437:
436:
432:
407:
406:
397:
355:
353:
349:
344:
311:Actual infinity
307:
295:New Foundations
222:
216:
204:
203:
174:
163:
159:
158:
156:
148:
146:
142:
138:
133:
129:
123:
121:
115:
113:
79:
28:
23:
22:
15:
12:
11:
5:
984:
982:
974:
973:
968:
963:
958:
948:
947:
941:
940:
938:
937:
932:
927:
922:
917:
912:
906:
904:
903:Mathematicians
900:
899:
897:
896:
891:
886:
881:
875:
873:
869:
868:
866:
865:
860:
855:
850:
844:
839:
834:
829:
824:
819:
814:
809:
807:Gimel function
804:
802:Epsilon number
799:
793:
788:
783:
778:
773:
768:
763:
758:
753:
747:
745:
741:
740:
738:
737:
732:
727:
722:
717:
711:
709:
705:
704:
702:
701:
696:
691:
686:
681:
676:
671:
666:
660:
658:
654:
653:
641:
639:
638:
631:
624:
616:
610:
609:
593:
579:
553:
544:
541:
538:
537:
513:
494:
470:
454:
430:
395:
364:(3): 375–402.
346:
345:
343:
340:
339:
338:
333:
328:
323:
318:
313:
306:
303:
230:ordinal number
218:Main article:
215:
212:
154:
144:
140:
78:
75:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
983:
972:
969:
967:
964:
962:
959:
957:
954:
953:
951:
936:
933:
931:
928:
926:
923:
921:
918:
916:
915:David Hilbert
913:
911:
908:
907:
905:
901:
895:
892:
890:
887:
885:
882:
880:
877:
876:
874:
870:
864:
861:
859:
856:
854:
851:
848:
845:
843:
840:
838:
835:
833:
830:
828:
827:Infinitesimal
825:
823:
820:
818:
815:
813:
812:Hilbert space
810:
808:
805:
803:
800:
797:
794:
792:
789:
787:
784:
782:
779:
777:
774:
772:
769:
767:
764:
762:
759:
757:
754:
752:
749:
748:
746:
742:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
712:
710:
706:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
661:
659:
655:
649:
644:
637:
632:
630:
625:
623:
618:
617:
614:
607:
604:
600:
599:
594:
591:
590:0-415-03307-1
587:
583:
580:
577:
576:3-7643-3034-1
573:
569:
568:0-691-00172-3
565:
561:
557:
554:
552:
551:
547:
546:
542:
534:
533:Ernst Zermelo
530:
526:
522:
517:
514:
510:
507:
503:
498:
495:
490:
488:
479:
474:
471:
467:
463:
458:
455:
450:(4): 545–591.
449:
445:
441:
434:
431:
427:
426:3-540-09849-6
423:
419:
413:
412:
404:
402:
400:
396:
392:
387:
383:
379:
375:
371:
367:
363:
359:
351:
348:
341:
337:
334:
332:
329:
327:
324:
322:
319:
317:
314:
312:
309:
308:
304:
302:
300:
296:
292:
288:
284:
283:meta-language
280:
276:
271:
269:
265:
261:
256:
254:
250:
249:set hierarchy
246:
242:
238:
233:
231:
227:
221:
213:
210:
209:
205:
201:
197:
193:
189:
186:; the number
185:
181:
177:
170:
166:
152:
136:
127:
119:
111:
107:
101:
99:
93:
91:
90:
82:
81:Cantor said:
77:Cantor's view
74:
72:
68:
64:
60:
56:
53:
52:mathematician
49:
45:
41:
37:
33:
19:
910:Georg Cantor
884:Möbius plane
822:Infinite set
766:Aleph number
755:
605:
602:
597:
582:The Infinite
581:
555:
549:
543:Bibliography
524:
520:
516:
508:
505:
497:
487:Transfinitum
486:
483:
473:
461:
457:
447:
443:
433:
418:Cantor 1883b
417:
410:
389:
361:
357:
350:
336:Ineffability
287:proper class
272:
263:
257:
248:
244:
234:
223:
207:
206:
199:
195:
191:
187:
183:
179:
172:
168:
161:
150:
131:
125:
117:
106:well-ordered
103:
95:
89:Transfinitum
87:
84:
80:
67:mathematical
55:Georg Cantor
50:proposed by
43:
35:
31:
29:
771:Beth number
560:Rudy Rucker
266:(Zermelo's
237:A. W. Moore
226:paradoxical
124:The system
59:transfinite
950:Categories
872:Geometries
730:Set theory
358:Erkenntnis
321:Monadology
291:set theory
853:Supertask
598:Tractatus
416:Cited as
386:122487235
253:hierarchy
247:, or the
961:Infinity
751:0.999...
643:Infinity
603:Analysis
378:20012628
305:See also
48:infinity
44:absolute
669:Apeiron
657:History
110:element
588:
574:
566:
424:
384:
376:
354:§3.2,
36:symbol
466:p. 11
382:S2CID
374:JSTOR
342:Notes
279:class
157:+1. )
586:ISBN
572:ISBN
564:ISBN
422:ISBN
160:Now
30:The
366:doi
297:by
241:set
143:, ω
122:...
114:...
63:God
952::
606:45
558:,
509:76
448:21
446:.
442:.
398:^
388:.
380:.
372:.
362:42
360:.
301:.
38::
651:)
648:∞
645:(
635:e
628:t
621:v
592:.
578:.
492:.
468:.
428:.
368::
200:δ
196:δ
192:Ω
188:δ
184:Ω
180:δ
175:′
173:Ω
169:Ω
164:′
162:Ω
155:0
151:Ω
145:0
141:0
137::
134:′
132:Ω
126:Ω
120:.
118:Ω
40:Ω
34:(
20:)
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