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ordinals, for example). But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular
879:
693:, Studies in Logic and the Foundations of Mathematics, vol. 105, North-Holland Publishing Co., Amsterdam-New York, p. 238,
939:
137:
admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with
554:
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722:
698:
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365:
420:
814:(1984), ch. 2, "The Constructible Universe, p.95. Perspectives in Mathematical Logic, Springer-Verlag.
963:
338:
119:
202:. There exists a theory of large ordinals in this manner that is highly parallel to that of (small)
958:
775:
Kahle, Reinhard; Setzer, Anton (2010), "An extended predicative definition of the Mahlo universe",
610:
198:-th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called
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203:
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605:
32:
762:
688:
668:
650:, Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, RI, pp. 259–269,
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123:
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211:
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138:
41:
21:
779:, Ontos Math. Log., vol. 2, Ontos Verlag, Heusenstamm, pp. 315–340,
952:
226:
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61:
843:(1978), pp.419--420. Perspectives in Mathematical Logic, ISBN 3-540-07904-1.
130:
643:
45:
905:
17:
866:
854:
Transfinite
Recursion, Constructible Sets, and Analogues of Cardinals
799:
An introduction to the fine structure of the constructible hierarchy
315:, in fact this may be take as the definition of admissibility. The
740:
Friedman, Sy D. (2010), "Constructibility and class forcing",
897:
74:-collection. The term was coined by Richard Platek in 1966.
397:
The
Friedman-Jensen-Sacks theorem states that countable
275:
is an admissible ordinal iff there is a standard model
646:(1985), "Fine structure theory and its applications",
869:" (1973), pp.361--362. Annals of Mathematical Logic 6
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589:{\displaystyle \langle L_{\alpha },\in ,A\rangle }
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267:
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170:
106:
171:{\displaystyle \omega _{\alpha }^{\mathrm {CK} }}
950:
126:uncountable cardinal is an admissible ordinal.
335:th admissible ordinal is sometimes denoted by
933:
483:. Equivalently, for any countable admissible
690:Fundamentals of generalized recursion theory
583:
558:
77:The first two admissible ordinals are ω and
822:
820:
774:
107:{\displaystyle \omega _{1}^{\mathrm {CK} }}
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636:
744:, Springer, Dordrecht, pp. 557–604,
727:(p.151). Association for Symbolic Logic,
679:
677:
517:
817:
739:
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221:is an admissible ordinal if and only if
880:The Theory of Countable Analytical Sets
683:
633:
524:{\displaystyle A\subseteq \mathbb {N} }
951:
674:
463:is the least ordinal not recursive in
742:Handbook of set theory. Vols. 1, 2, 3
648:Recursion theory (Ithaca, N.Y., 1982)
892:
856:" (1967), p.11. Accessed 2023-07-15.
417:is admissible iff there exists some
801:(1974) (p.38). Accessed 2021-05-06.
387:{\displaystyle \tau _{\alpha }^{0}}
13:
867:Gaps in the Constructible Universe
830:(1976). Cambridge University Press
436:{\displaystyle A\subseteq \omega }
162:
159:
98:
95:
14:
975:
896:
872:
859:
846:
841:Recursion-Theoretic Hierarchies
355:{\displaystyle \tau _{\alpha }}
295:of KP whose set of ordinals is
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828:Admissible Sets and Structures
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733:
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626:
912:. You can help Knowledge by
596:is an admissible structure.
206:(one can define recursively
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750:10.1007/978-1-4020-5764-9_9
599:
229:and there does not exist a
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891:
656:10.1090/pspum/042/791062
611:Large countable ordinals
200:recursively inaccessible
141:. One sometimes writes
50:Kripke–Platek set theory
865:W. Marek, M. Srebrny, "
724:Higher Recursion Theory
544:{\displaystyle \alpha }
496:{\displaystyle \alpha }
456:{\displaystyle \alpha }
410:{\displaystyle \alpha }
328:{\displaystyle \alpha }
308:{\displaystyle \alpha }
268:{\displaystyle \alpha }
191:{\displaystyle \alpha }
908:-related article is a
616:Constructible universe
590:
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477:
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237:for which there is a Σ
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120:Church–Kleene ordinal
109:
777:Ways of proof theory
761:. See in particular
667:. See in particular
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56:is admissible when
52:); in other words,
606:α-recursion theory
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551:minimal such that
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118:, also called the
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29:admissible ordinal
921:
920:
476:{\displaystyle A}
288:{\displaystyle M}
971:
964:Set theory stubs
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878:A. S. Kechris, "
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812:Constructibility
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621:Regular cardinal
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644:Friedman, Sy D.
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204:large cardinals
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94:
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73:
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977:
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945:
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937:
930:
922:
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918:
901:
885:
884:
871:
858:
845:
839:P. G. Hinman,
832:
816:
810:K. J. Devlin,
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503:, there is an
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100:
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22:ordinal number
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700:0-444-86171-8
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228:
227:limit ordinal
224:
220:
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213:
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185:
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149:
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62:limit ordinal
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914:expanding it
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669:p. 265
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575:∈
567:α
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