590:
647:
195:
344:
392:
138:, multiplication and exponentiation, and ψ to previously constructed ordinals (except that ψ can only be applied to arguments less than
532:
508:
631:
134:) is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying
688:
326:
385:
433:
239:
Bachmann, Heinz (1950), "Die
Normalfunktionen und das Problem der ausgezeichneten Folgen von Ordnungszahlen",
94:
624:
44:
662:
605:
378:
80:
717:
565:
261:
178:
of ordinals; this extension was carried out by Heinz
Bachmann and is not completely straightforward.
113:
712:
617:
401:
351:
52:
32:
209:
The proof theory of classical and constructive inductive definitions. A 40 year saga, 1968-2008.
681:
722:
707:
489:
551:
462:
452:
336:
290:
252:
64:
8:
208:
537:
307:
294:
278:
674:
442:
322:
221:
48:
298:
314:
270:
135:
36:
412:
332:
286:
248:
165:
658:
601:
421:
56:
40:
318:
701:
370:
313:, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag,
597:
282:
259:
Howard, W. A. (1972), "A system of abstract constructive ordinals.",
274:
196:
An order-theoretic characterization of the Howard-Bachmann-hierarchy
654:
589:
646:
157:
The
Bachmann–Howard ordinal can also be defined as φ
345:"Proof Theory: Part III, Kripke-Platek Set Theory"
306:
699:
79:The Bachmann–Howard ordinal is defined using an
533:the theories of iterated inductive definitions
269:(2), Association for Symbolic Logic: 355–374,
194:J. Van der Meeren, M. Rathjen, A. Weiermann, "
682:
625:
386:
400:
689:
675:
632:
618:
393:
379:
224:" (2006), p.11. Accessed 21 February 2023.
211:" (2008), p.7. Accessed 21 February 2023.
366:(Slides of a talk given at Fischbachau.)
238:
60:
342:
304:
123:is the first epsilon number after Ω = ε
700:
258:
241:Vierteljschr. Naturforsch. Ges. Zürich
68:
374:
142:, to ensure that it is well defined).
641:
584:
198:" (2017). Accessed 21 February 2023.
13:
14:
734:
509:Takeuti–Feferman–Buchholz ordinal
645:
588:
343:Rathjen, Michael (August 2005).
214:
201:
188:
1:
540: < ω
231:
74:
661:. You can help Knowledge by
604:. You can help Knowledge by
531:Proof-theoretic ordinals of
181:
164:(0) for an extension of the
7:
222:The Art of Ordinal Analysis
81:ordinal collapsing function
10:
739:
640:
583:
554: ≥ ω
566:First uncountable ordinal
408:
319:10.1007/978-3-540-46825-7
305:Pohlers, Wolfram (1989),
262:Journal of Symbolic Logic
114:first uncountable ordinal
16:A large countable ordinal
434:Feferman–Schütte ordinal
402:Large countable ordinals
65:William Alvin Howard
51:) and the system CZF of
45:Kripke–Platek set theory
39:of several mathematical
473:Bachmann–Howard ordinal
147:Bachmann–Howard ordinal
55:. It was introduced by
53:constructive set theory
37:proof-theoretic ordinal
33:large countable ordinal
29:Howard-Bachmann ordinal
21:Bachmann–Howard ordinal
600:-related article is a
413:First infinite ordinal
653:This article about a
174:to certain functions
552:Nonrecursive ordinal
463:large Veblen ordinal
453:small Veblen ordinal
19:In mathematics, the
538:Computable ordinals
23:(also known as the
490:Buchholz's ordinal
57:Heinz Bachmann
670:
669:
613:
612:
578:
577:
443:Ackermann ordinal
49:axiom of infinity
730:
718:Set theory stubs
691:
684:
677:
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642:
634:
627:
620:
592:
585:
562:
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372:
371:
365:
363:
362:
356:
350:. Archived from
349:
339:
312:
301:
255:
225:
218:
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205:
199:
192:
166:Veblen functions
136:ordinal addition
738:
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733:
732:
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729:
728:
727:
713:Ordinal numbers
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697:
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695:
639:
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541:
527:
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439:
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422:Epsilon numbers
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347:
329:
275:10.2307/2272979
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229:
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163:
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152:
126:
122:
111:
97:, the ordinals
95:epsilon numbers
93:enumerates the
92:
77:
17:
12:
11:
5:
736:
726:
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715:
710:
694:
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667:
650:
637:
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629:
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611:
610:
593:
576:
575:
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572:
563:
558:
549:
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529:
521:
519:
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487:
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437:
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383:
375:
368:
367:
340:
327:
302:
256:
235:
233:
230:
227:
226:
213:
207:S. Feferman, "
200:
186:
185:
183:
180:
169:
160:
158:
155:
154:
150:
143:
128:
124:
120:
117:
109:
106:
101:such that ω =
88:
76:
73:
25:Howard ordinal
15:
9:
6:
4:
3:
2:
735:
724:
721:
719:
716:
714:
711:
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706:
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703:
692:
687:
685:
680:
678:
673:
672:
666:
664:
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648:
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643:
635:
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628:
623:
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582:
571:
567:
564:
553:
550:
539:
536:
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524:
518:
514:
510:
507:
503:
495:
491:
488:
482:
478:
474:
471:
468:
464:
461:
458:
454:
451:
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444:
441:
435:
432:
427:
423:
420:
418:
414:
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410:
407:
403:
396:
391:
389:
384:
382:
377:
376:
373:
357:on 2007-06-12
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346:
341:
338:
334:
330:
328:3-540-51842-8
324:
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316:
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276:
272:
268:
264:
263:
257:
254:
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246:
242:
237:
236:
223:
220:M. Rathjen, "
217:
210:
204:
197:
191:
187:
179:
177:
172:
167:
148:
144:
141:
137:
133:
129:
118:
115:
107:
104:
100:
96:
91:
86:
85:
84:
82:
72:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
723:Number stubs
708:Proof theory
663:expanding it
652:
606:expanding it
595:
580:
569:
522:
516:
512:
501:
493:
480:
476:
472:
466:
456:
446:
425:
416:
359:. Retrieved
352:the original
309:Proof theory
308:
266:
260:
244:
240:
216:
203:
190:
175:
170:
156:
146:
139:
131:
102:
98:
89:
78:
35:. It is the
28:
24:
20:
18:
247:: 115–147,
43:, such as
702:Categories
598:set theory
361:2008-04-17
232:References
75:Definition
47:(with the
182:Citations
299:44618354
41:theories
436: Γ
337:1026933
291:0329869
283:2272979
253:0036806
112:is the
67: (
59: (
31:) is a
655:number
568:
511:
492:
475:
465:
455:
445:
424:
415:
335:
325:
297:
289:
281:
251:
149:is ψ(ε
63:) and
657:is a
596:This
355:(PDF)
348:(PDF)
295:S2CID
279:JSTOR
108:Ω = ω
27:, or
659:stub
602:stub
323:ISBN
145:The
69:1972
61:1950
484:Ω+1
469:(Ω)
459:(Ω)
449:(Ω)
315:doi
271:doi
161:Ω+1
151:Ω+1
121:Ω+1
71:).
704::
526:+1
499:(Ω
333:MR
331:,
321:,
293:,
287:MR
285:,
277:,
267:37
265:,
249:MR
245:95
243:,
153:).
130:ψ(
83::
690:e
683:t
676:v
665:.
633:e
626:t
619:v
608:.
570:Ω
559:1
545:1
528:)
523:ω
520:Ω
517:ε
515:(
513:ψ
505:)
502:ω
497:0
494:ψ
486:)
481:ε
479:(
477:ψ
467:θ
457:θ
447:θ
438:0
429:0
426:ε
417:ω
394:e
387:t
380:v
364:.
317::
273::
176:α
171:α
168:φ
159:ε
140:α
132:α
127:.
125:Ω
119:ε
116:.
110:1
105:.
103:ε
99:ε
90:α
87:ε
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