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.. See in particular p. 265.. The term was coined by Richard Platek in 1966.G. E. Sacks, Higher Recursion Theory (p.151). Association for Symbolic Logic, Perspectives in Logic
The first two admissible ordinals are ω and (the least nonrecursive ordinal, also called the Church–Kleene ordinal). 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. 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.. See in particular p. 560. 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).. 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.K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06. is an admissible ordinal if and only if there is a standard model of KP whose set of ordinals is , in fact this may be take as the definition of admissibility.K. J. Devlin, Constructibility (1984), ch. 2, "The Constructible Universe, p.95. Perspectives in Mathematical Logic, Springer-Verlag.J. Barwise, Admissible Sets and Structures (1976). Cambridge University Press The th admissible ordinal is sometimes denoted by P. G. Hinman, Recursion-Theoretic Hierarchies (1978), pp.419--420. Perspectives in Mathematical Logic, ISBN 3-540-07904-1.p. 174 or .
The Friedman–Jensen–Sacks theorem states that a countable is admissible if and only if there exists some such that is the least ordinal not recursive in .W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), pp.361--362. Annals of Mathematical Logic 6 Equivalently, for any countable admissible , there is an making minimal such that is an admissible structure.A. S. Kechris, "The Theory of Countable Analytical Sets"p. 264
See also α-recursion theory Large countable ordinals Constructible universe Regular cardinal
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