Admissible ordinal
In set theory, an ordinal number α is an admissible ordinal if Lα is an admissible set (that is, a transitive model of Kripke–Platek set theory); in other words, α is admissible when α is a limit ordinal and Lα ⊧ Σ0-collection.[1][2] The term was coined by Richard Platek in 1966.[3]
The first two admissible ordinals are ω and (the least non-recursive ordinal, also called the Church–Kleene ordinal).[2] Any regular uncountable cardinal is an admissible ordinal.
By a theorem of Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal, but for Turing machines with oracles.[1] One sometimes writes for the -th ordinal that is either admissible or a limit of admissibles; an ordinal that is both is called recursively inaccessible.[4] There exists a theory of large ordinals in this manner that is highly parallel to that of (small) large cardinals (one can define recursively Mahlo ordinals, for example).[5] But all these ordinals are still countable. Therefore, admissible ordinals seem to be the recursive analogue of regular cardinal numbers.
Notice that α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ < α for which there is a Σ1(Lα) mapping from γ onto α.[6] If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.
See also
References
- ^ a b Friedman, Sy D. (1985), "Fine structure theory and its applications", Recursion theory (Ithaca, N.Y., 1982), Proc. Sympos. Pure Math., vol. 42, Amer. Math. Soc., Providence, RI, pp. 259–269, doi:10.1090/pspum/042/791062, MR 0791062. See in particular p. 265.
- ^ a b Fitting, Melvin (1981), Fundamentals of generalized recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 105, North-Holland Publishing Co., Amsterdam-New York, p. 238, ISBN 0-444-86171-8, MR 0644315.
- ^ G. E. Sacks, Higher Recursion Theory (p.151). Association for Symbolic Logic, Perspectives in Logic
- ^ Friedman, Sy D. (2010), "Constructibility and class forcing", Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, pp. 557–604, doi:10.1007/978-1-4020-5764-9_9, MR 2768687. See in particular p. 560.
- ^ Kahle, Reinhard; Setzer, Anton (2010), "An extended predicative definition of the Mahlo universe", Ways of proof theory, Ontos Math. Log., vol. 2, Ontos Verlag, Heusenstamm, pp. 315–340, MR 2883363.
- ^ K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06.
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