Admissible ordinal explained

In set theory, an ordinal number α is an admissible ordinal if L_α 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 L_α ⊧ Σ₀-collection.^{[1] [2]} The term was coined by Richard Platek in 1966.^[3]

The first two admissible ordinals are ω and

\omega ^{CK} _{1} (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

\omega ^{CK} _{\alpha} for the \alpha -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(L_α) mapping from γ onto α.^[6]

\alpha is an admissible ordinal iff there is a standard model M of KP whose set of ordinals is \alpha , in fact this may be take as the definition of admissibility.^{[7] [8]} The \alpha th admissible ordinal is sometimes denoted by \tau _{\alpha} ^{[9] p. 174} or \tau ^{0} _{\alpha} .^[10]

The Friedman-Jensen-Sacks theorem states that countable

\alpha is admissible iff there exists some A\subseteq\omega such that \alpha is the least ordinal not recursive in A .^[11] Equivalently, for any countable admissible \alpha , there is an A\subseteq N making \alpha minimal such that \langle L _{\alpha,} \in, A\rangle is an admissible structure.^{[12] p. 264}

See also

• α-recursion theory
• Large countable ordinals
• Constructible universe
• Regular cardinal

Notes and References

1. . See in particular p. 265.
2. .
3. G. E. Sacks, Higher Recursion Theory (p.151). Association for Symbolic Logic, Perspectives in Logic
4. . See in particular p. 560.
5. .
6. K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.38). Accessed 2021-05-06.
7. K. J. Devlin, Constructibility (1984), ch. 2, "The Constructible Universe, p.95. Perspectives in Mathematical Logic, Springer-Verlag.
8. J. Barwise, Admissible Sets and Structures (1976). Cambridge University Press
9. P. G. Hinman, Recursion-Theoretic Hierarchies (1978), pp.419--420. Perspectives in Mathematical Logic, ISBN 3-540-07904-1.
10. S. Kripke, "Transfinite Recursion, Constructible Sets, and Analogues of Cardinals" (1967), p.11. Accessed 2023-07-15.
11. W. Marek, M. Srebrny, "Gaps in the Constructible Universe" (1973), pp.361--362. Annals of Mathematical Logic 6
12. A. S. Kechris, "The Theory of Countable Analytical Sets"