636:
693:
95:
160:
124:
201:
256:
This ordinal is sometimes said to be the first impredicative ordinal, though this is controversial, partly because there is no generally accepted precise definition of "
57:
There is no standard notation for ordinals beyond the
Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use
438:
211:
The
Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of
578:
554:
677:
310:, Grundlehren der Mathematischen Wissenschaften, Band 225, Springer-Verlag, Berlin, Heidelberg, New York, 1977, xii + 302 pp.
734:
39:
373:
431:
340:
708:
651:
64:
670:
129:
518:
424:
58:
267:
Any recursive path ordering whose function symbols are well-founded with order type less than that of Γ
100:
763:
611:
165:
758:
336:
663:
447:
35:
31:
727:
320:
768:
753:
535:
597:
508:
498:
405:
383:
8:
409:
583:
395:
354:
720:
488:
369:
332:
361:
212:
43:
458:
379:
216:
47:
704:
647:
467:
365:
747:
290:
257:
416:
360:, Lecture Notes in Mathematics, vol. 1407, Berlin: Springer-Verlag,
643:
260:". Sometimes an ordinal is said to be predicative if it is less than Γ
400:
700:
635:
692:
168:
132:
103:
67:
353:
195:
154:
118:
89:
745:
579:the theories of iterated inductive definitions
728:
671:
432:
446:
735:
721:
678:
664:
439:
425:
38:of several mathematical theories, such as
399:
50:, the former of whom suggested the name Γ
90:{\displaystyle \psi (\Omega ^{\Omega })}
351:
746:
389:
420:
155:{\displaystyle \varphi _{\Omega }(0)}
687:
630:
13:
138:
110:
79:
75:
40:arithmetical transfinite recursion
14:
780:
555:Takeuti–Feferman–Buchholz ordinal
271:itself has order type less than Γ
691:
634:
343:(1987). Accessed 3 October 2022.
119:{\displaystyle \theta (\Omega )}
341:Journal of Symbolic Computation
231:). That is, it is the smallest
196:{\displaystyle \varphi (1,0,0)}
326:
313:
300:
284:
190:
172:
149:
143:
113:
107:
84:
71:
1:
586: < ω
278:
251:
206:
707:. You can help Knowledge by
650:. You can help Knowledge by
577:Proof-theoretic ordinals of
392:Predicativity beyond Gamma_0
59:ordinal collapsing functions
7:
10:
785:
686:
629:
600: ≥ ω
612:First uncountable ordinal
454:
366:10.1007/978-3-540-46825-7
352:Pohlers, Wolfram (1989),
480:Feferman–Schütte ordinal
448:Large countable ordinals
337:Termination of Rewriting
21:Feferman–Schütte ordinal
519:Bachmann–Howard ordinal
36:proof-theoretic ordinal
32:large countable ordinal
16:Large countable ordinal
646:-related article is a
459:First infinite ordinal
197:
156:
120:
91:
699:This article about a
198:
157:
121:
92:
598:Nonrecursive ordinal
509:large Veblen ordinal
499:small Veblen ordinal
390:Weaver, Nik (2005),
166:
130:
101:
65:
42:. It is named after
19:In mathematics, the
584:Computable ordinals
410:2005math......9244W
319:Solomon Feferman, "
536:Buchholz's ordinal
193:
152:
116:
87:
716:
715:
659:
658:
624:
623:
489:Ackermann ordinal
333:Nachum Dershowitz
776:
764:Set theory stubs
737:
730:
723:
695:
688:
680:
673:
666:
638:
631:
608:
607:
594:
593:
441:
434:
427:
418:
417:
412:
403:
386:
359:
344:
330:
324:
317:
311:
304:
298:
288:
217:Veblen functions
213:ordinal addition
202:
200:
199:
194:
161:
159:
158:
153:
142:
141:
125:
123:
122:
117:
96:
94:
93:
88:
83:
82:
44:Solomon Feferman
784:
783:
779:
778:
777:
775:
774:
773:
759:Ordinal numbers
744:
743:
742:
741:
685:
684:
627:
625:
620:
606:
603:
602:
601:
592:
589:
588:
587:
573:
571:
550:
544:
531:
485:
476:
468:Epsilon numbers
450:
445:
415:
376:
347:
331:
327:
318:
314:
305:
301:
289:
285:
281:
274:
270:
263:
254:
243:
226:
209:
167:
164:
163:
137:
133:
131:
128:
127:
102:
99:
98:
78:
74:
66:
63:
62:
53:
28:
17:
12:
11:
5:
782:
772:
771:
766:
761:
756:
740:
739:
732:
725:
717:
714:
713:
696:
683:
682:
675:
668:
660:
657:
656:
639:
622:
621:
619:
618:
609:
604:
595:
590:
581:
575:
567:
565:
552:
546:
542:
533:
529:
516:
506:
496:
486:
483:
477:
474:
465:
455:
452:
451:
444:
443:
436:
429:
421:
414:
413:
387:
374:
348:
346:
345:
325:
312:
306:Kurt Schütte,
299:
282:
280:
277:
272:
268:
261:
253:
250:
239:
222:
208:
205:
192:
189:
186:
183:
180:
177:
174:
171:
151:
148:
145:
140:
136:
115:
112:
109:
106:
86:
81:
77:
73:
70:
51:
26:
15:
9:
6:
4:
3:
2:
781:
770:
767:
765:
762:
760:
757:
755:
752:
751:
749:
738:
733:
731:
726:
724:
719:
718:
712:
710:
706:
702:
697:
694:
690:
689:
681:
676:
674:
669:
667:
662:
661:
655:
653:
649:
645:
640:
637:
633:
632:
628:
617:
613:
610:
599:
596:
585:
582:
580:
576:
570:
564:
560:
556:
553:
549:
541:
537:
534:
528:
524:
520:
517:
514:
510:
507:
504:
500:
497:
494:
490:
487:
481:
478:
473:
469:
466:
464:
460:
457:
456:
453:
449:
442:
437:
435:
430:
428:
423:
422:
419:
411:
407:
402:
397:
393:
388:
385:
381:
377:
375:3-540-51842-8
371:
367:
363:
358:
357:
350:
349:
342:
339:(pp.98--99),
338:
334:
329:
322:
321:Predicativity
316:
309:
303:
297:(1975, p.413)
296:
292:
291:Gaisi Takeuti
287:
283:
276:
265:
259:
249:
247:
242:
238:
234:
230:
225:
221:
218:
214:
204:
187:
184:
181:
178:
175:
169:
146:
134:
104:
68:
60:
55:
49:
45:
41:
37:
33:
29:
22:
769:Number stubs
754:Proof theory
709:expanding it
698:
652:expanding it
641:
626:
615:
568:
562:
558:
547:
539:
526:
522:
512:
502:
492:
479:
471:
462:
401:math/0509244
391:
356:Proof theory
355:
328:
315:
308:Proof theory
307:
302:
295:Proof Theory
294:
286:
266:
255:
245:
240:
236:
232:
228:
223:
219:
210:
56:
48:Kurt Schütte
34:. It is the
24:
20:
18:
258:predicative
748:Categories
644:set theory
279:References
252:Properties
235:such that
207:Definition
170:φ
139:Ω
135:φ
111:Ω
105:θ
80:Ω
76:Ω
69:ψ
323:" (2002)
215:and the
482: Γ
406:Bibcode
384:1026933
30:) is a
701:number
614:
557:
538:
521:
511:
501:
491:
470:
461:
382:
372:
244:(0) =
703:is a
642:This
396:arXiv
162:, or
705:stub
648:stub
370:ISBN
46:and
530:Ω+1
515:(Ω)
505:(Ω)
495:(Ω)
362:doi
750::
572:+1
545:(Ω
404:,
394:,
380:MR
378:,
368:,
335:,
293:,
275:.
264:.
248:.
203:.
126:,
97:,
61::
54:.
736:e
729:t
722:v
711:.
679:e
672:t
665:v
654:.
616:Ω
605:1
591:1
574:)
569:ω
566:Ω
563:ε
561:(
559:ψ
551:)
548:ω
543:0
540:ψ
532:)
527:ε
525:(
523:ψ
513:θ
503:θ
493:θ
484:0
475:0
472:ε
463:ω
440:e
433:t
426:v
408::
398::
364::
273:0
269:0
262:0
246:α
241:α
237:φ
233:α
229:β
227:(
224:α
220:φ
191:)
188:0
185:,
182:0
179:,
176:1
173:(
150:)
147:0
144:(
114:)
108:(
85:)
72:(
52:0
27:0
25:Γ
23:(
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