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 [math]\displaystyle{ \omega_1^{\mathrm{CK}} }[/math] (the least nonrecursive 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 [math]\displaystyle{ \omega_\alpha^{\mathrm{CK}} }[/math] for the [math]\displaystyle{ \alpha }[/math]-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] [math]\displaystyle{ \alpha }[/math] is an admissible ordinal iff there is a standard model [math]\displaystyle{ M }[/math] of KP whose set of ordinals is [math]\displaystyle{ \alpha }[/math], in fact this may be take as the definition of admissibility.[7][8] The [math]\displaystyle{ \alpha }[/math]th admissible ordinal is sometimes denoted by [math]\displaystyle{ \tau_\alpha }[/math][9][8]p. 174 or [math]\displaystyle{ \tau^0_\alpha }[/math].[10]
The Friedman-Jensen-Sacks theorem states that countable [math]\displaystyle{ \alpha }[/math] is admissible iff there exists some [math]\displaystyle{ A\subseteq\omega }[/math] such that [math]\displaystyle{ \alpha }[/math] is the least ordinal not recursive in [math]\displaystyle{ A }[/math].[11]
See also
- α-recursion theory
- Large countable ordinals
- Constructible universe
- Regular cardinal
References
- ↑ 1.0 1.1 "Fine structure theory and its applications", Recursion theory (Ithaca, N.Y., 1982), Proc. Sympos. Pure Math., 42, Amer. Math. Soc., Providence, RI, 1985, pp. 259–269, doi:10.1090/pspum/042/791062. See in particular p. 265.
- ↑ 2.0 2.1 Fundamentals of generalized recursion theory, Studies in Logic and the Foundations of Mathematics, 105, North-Holland Publishing Co., Amsterdam-New York, 1981, p. 238, ISBN 0-444-86171-8, https://books.google.com/books?id=GRE7AAAAQBAJ&pg=PA238.
- ↑ 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. 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., 2, Ontos Verlag, Heusenstamm, pp. 315–340.
- ↑ K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06.
- ↑ K. J. Devlin, Constructibility (1984), ch. 2, "The Constructible Universe, p.95. Perspectives in Mathematical Logic, Springer-Verlag.
- ↑ 8.0 8.1 J. Barwise, Admissible Sets and Structures (1976). Cambridge University Press
- ↑ P. G. Hinman, Recursion-Theoretic Hierarchies (1978), pp.419--420. Perspectives in Mathematical Logic, ISBN 3-540-07904-1.
- ↑ S. Kripke, "Transfinite Recursion, Constructible Sets, and Analogues of Cardinals" (1967), p.11. Accessed 2023-07-15.
- ↑ W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), pp.361--362. Annals of Mathematical Logic 6
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