275:
91:(with the bijection definition of cardinality and the injective function order). (If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of
491:, and so his result is one of Zermelo set theory and looks rather different from the modern exposition above. Instead, he considered the set of isomorphism classes of well-ordered subsets of
185:
399:
217:
619:
519:. However, the main purpose of his contribution was to show that trichotomy for cardinal numbers implies the (then 11 year old)
115:
107:. This mapping is used to construct the aleph numbers, which are all the cardinal numbers of infinite well-orderable sets.
368:
353:
667:
144:
570:
520:
191:. As ordinals are well-ordered, this immediately implies the existence of a Hartogs number for any set
131:
24:
662:
595:
532:
345:
319:
289:
278:
52:
565:
615:
556:
508:
488:
111:
96:
40:
587:
579:
384:
270:{\displaystyle \alpha =\{\beta \in {\textrm {Ord}}\mid \exists i:\beta \hookrightarrow X\}}
591:
119:
68:
83:
to α. However, the cardinal number of α is still a minimal cardinal number (ie. ordinal)
484:
416:
282:
48:
32:
656:
599:
630:
515:. Hartogs showed this to be a well-ordering greater than any well-ordered subset of
537:
64:
607:
20:
360:
326:
195:. Furthermore, the proof is constructive and yields the Hartogs number of
583:
352:
is a definable subclass of the preceding set, so it is a set by the
431:, because if there were, then we would get the contradiction that
79:
cannot be well-ordered then there cannot be an injection from
612:
Set theory, third millennium edition (revised and expanded)
423:
is an ordinal. Furthermore, there is no injection from
130:
Not to be confused with the result in complex analysis,
387:
220:
187:; that is, such that there is no injection from α to
147:
393:
269:
179:
110:The existence of the Hartogs number was proved by
443:is the least such ordinal with no injection into
71:of α is a minimal cardinal greater than that of
16:Certain kind of cardinal number in set theory
8:
264:
227:
137:Hartogs's theorem states that for any set
386:
237:
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219:
172:
164:
156:
148:
146:
35:associated with a set. In particular, if
495:and the relation in which the class of
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180:{\displaystyle |\alpha |\not \leq |X|}
141:, there exists an ordinal α such that
407:can be described by a simple formula.
7:
337:is a set, by the axiom of power set.
523:(and, hence, the axiom of choice).
483:In 1915, Hartogs could use neither
566:"Über das Problem der Wohlordnung"
246:
118:alone (that is, without using the
14:
511:with a proper initial segment of
419:of ordinals is again an ordinal,
447:. This is true because, since
467:so there is an injection from
258:
173:
165:
157:
149:
1:
411:But this last set is exactly
348:well-orderings of subsets of
43:, then the Hartogs number of
318:is a set, as can be seen in
369:axiom schema of replacement
116:Zermelo–Fraenkel set theory
684:
354:axiom schema of separation
129:
103:to α is sometimes called
85:not less than or equal to
451:is an ordinal, for any
51:α such that there is no
640:. University of Bristol
564:Hartogs, Fritz (1915).
551:Goldrei, Derek (1996).
125:
631:"Axiomatic set theory"
395:
394:{\displaystyle \cong }
303:First, we verify that
271:
181:
571:Mathematische Annalen
521:well-ordering theorem
396:
363:of well-orderings in
272:
182:
485:von Neumann-ordinals
385:
218:
145:
25:axiomatic set theory
87:the cardinality of
584:10.1007/BF01458215
557:Chapman & Hall
553:Classic Set Theory
533:Successor cardinal
391:
320:Axiom of power set
290:injective function
267:
177:
105:Hartogs's function
23:, specifically in
499:precedes that of
489:replacement axiom
415:. Now, because a
359:The class of all
240:
132:Hartogs's theorem
126:Hartogs's theorem
112:Friedrich Hartogs
675:
668:Cardinal numbers
648:
646:
645:
635:
629:Charles Morgan.
625:
603:
560:
439:. And finally,
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398:
397:
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367:is a set by the
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529:
481:
479:Historic remark
406:
383:
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283:ordinal numbers
216:
215:
205:
143:
142:
135:
128:
120:axiom of choice
114:in 1915, using
69:cardinal number
17:
12:
11:
5:
681:
679:
671:
670:
665:
655:
654:
650:
649:
626:
620:
604:
578:(4): 438–443.
561:
547:
545:
542:
541:
540:
535:
528:
525:
480:
477:
417:transitive set
409:
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204:
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127:
124:
33:ordinal number
29:Hartogs number
15:
13:
10:
9:
6:
4:
3:
2:
680:
669:
666:
664:
661:
660:
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627:
623:
621:3-540-44085-2
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574:(in German).
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288:for which an
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58:
54:
50:
47:is the least
46:
42:
38:
34:
30:
26:
22:
642:. Retrieved
638:Course Notes
637:
614:. Springer.
611:
608:Jech, Thomas
575:
569:
552:
538:Aleph number
516:
512:
504:
500:
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468:
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349:
341:
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315:
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304:
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297:
293:
292:exists from
285:
213:
209:Goldrei 1996
206:
196:
192:
188:
138:
136:
109:
104:
100:
92:
88:
84:
80:
76:
72:
65:well-ordered
60:
56:
55:from α into
44:
36:
28:
18:
361:order types
21:mathematics
663:Set theory
657:Categories
644:2010-04-10
592:45.0125.01
544:References
509:isomorphic
340:The class
307:is a set.
600:121598654
389:≅
346:reflexive
327:power set
259:↪
256:β
247:∃
244:∣
234:∈
231:β
222:α
154:α
67:then the
53:injection
610:(2002).
527:See also
487:nor the
373:(Domain(
162:≰
344:of all
333:×
314:×
281:of all
277:be the
99:taking
95:.) The
63:can be
49:ordinal
39:is any
618:
598:
590:
75:. If
59:. If
31:is an
634:(PDF)
596:S2CID
471:into
455:<
427:into
371:, as
296:into
279:class
203:Proof
616:ISBN
405:, ≤)
325:The
214:Let
207:See
27:, a
588:JFM
580:doi
507:is
503:if
377:),
329:of
239:Ord
122:).
97:map
41:set
19:In
659::
636:.
594:.
586:.
576:76
568:.
555:.
475:.
463:∈
459:,
435:∈
381:)
300:.
211:.
199:.
647:.
624:.
602:.
582::
559:.
517:X
513:B
505:A
501:B
497:A
493:X
473:X
469:β
465:α
461:β
457:α
453:β
449:α
445:X
441:α
437:α
433:α
429:X
425:α
421:α
413:α
403:β
401:(
379:w
375:w
365:W
356:.
350:X
342:W
335:X
331:X
322:.
316:X
312:X
305:α
298:X
294:β
286:β
265:}
262:X
253::
250:i
228:{
225:=
197:X
193:X
189:X
174:|
170:X
166:|
158:|
150:|
139:X
134:.
101:X
93:X
89:X
81:X
77:X
73:X
61:X
57:X
45:X
37:X
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