# Nonrecursive ordinal

(Redirected from Church–Kleene ordinal)

In mathematics, particularly set theory, non-recursive ordinals are large countable ordinals greater than all the recursive ordinals, and therefore can not be expressed using ordinal collapsing functions.

## The Church–Kleene ordinal and variants

The smallest non-recursive ordinal is the Church Kleene ordinal, ${\displaystyle \omega _{1}^{\mathsf {CK}}}$ , named after Alonzo Church and S. C. Kleene; its order type is the set of all recursive ordinals. Since the successor of a recursive ordinal is recursive, the Church–Kleene ordinal is a limit ordinal. It is also the smallest ordinal that is not hyperarithmetical, and the smallest admissible ordinal after ω. The notation ${\displaystyle \omega _{1}^{\mathsf {CK}}}$  is in reference to ω
1
, the first uncountable ordinal, which is set of all countable ordinals. The ${\displaystyle \omega _{1}^{\mathsf {CK}}}$ -recursive subsets of ω are exactly the ${\displaystyle \Delta _{1}^{1}}$  subsets of ω.

An ordinal α is called admissible if ${\displaystyle L_{\alpha }\models {\mathsf {KP}}}$ .

The relativized Church–Kleene ordinal ${\displaystyle \omega _{1}^{x}}$  is the supremum of the x-computable ordinals.

${\displaystyle \omega _{\omega }^{\mathsf {CK}}}$ , first defined by Stephen G. Simpson and dubbed the "Great Church–Kleene ordinal" is an extension of the Church–Kleene ordinal. This is the smallest limit of admissible ordinals, yet this ordinal is not admissible. Alternatively, his is the smallest α such that ${\displaystyle L_{\alpha }\cap {\mathsf {P}}(\omega )}$  is a model of ${\displaystyle \Pi _{1}^{1}}$ -comprehension.

## Recursively ordinals

Recursively x ordinals, not to be confused with recursive ordinals, are kinds of nonrecursive ordinals.

An ordinal α is called recursively inaccessible if it is admissible and limit of admissibles (α is the αth admissible). Alternatively, it is recursively inaccessible if ${\displaystyle L_{\alpha }\models {\mathsf {KPi}}}$ , an extension of Kripke–Platek set theory based on an inaccessible cardinal; or, lastly, on the arithmetical side, such that ${\displaystyle L_{\alpha }\cap {\mathsf {P}}(\omega )}$  is a model of ${\displaystyle \Delta _{2}^{1}}$  - comprehension.

An ordinal α is called recursively hyperinaccessible if it is recursively inaccessible and a limit of recursively inaccessibles (α is the αth recursively inaccessible).

An ordinal α is called recursively Mahlo if it is admissible and for any α-recursive function f: αα there is an admissible β < α such that {f(γ) : γ ∈ β} ⊆ β (that is, β is closed under f).

An ordinal α is called recursively weakly compact if it is ${\displaystyle \Pi _{3}}$ -reflecting (2-admissible).

## Weakenings of stable ordinals

Stable ordinals are some of the largest nonrecursive ordinals. There are various weakenings of stables:

• A countable ordinal ${\displaystyle \alpha }$  is called ${\displaystyle (+1)}$ -stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\alpha +1}}$ . The smallest ${\displaystyle (+1)}$ -stable ordinal is larger than the smallest recursively weakly compact ordinal.
• In general, a countable ordinal ${\displaystyle \alpha }$  is called ${\displaystyle (+\beta )}$ -stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\alpha +\beta }}$ .
• A countable ordinal ${\displaystyle \alpha }$  is called ${\displaystyle (^{+})}$ -stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\alpha ^{+}}}$ , where ${\displaystyle \beta ^{+}}$  is the smallest admissible ordinal ${\displaystyle >\beta }$ . The smallest ${\displaystyle (^{+})}$ -stable ordinal is larger than the smallest ${\displaystyle (+1)}$ -stable.
• A countable ordinal ${\displaystyle \alpha }$  is called ${\displaystyle (^{++})}$ -stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\alpha ^{++}}}$ , where ${\displaystyle \beta ^{+}}$  and ${\displaystyle \beta ^{++}}$  are the two smallest admissible ordinals ${\displaystyle >\beta }$ . The smallest ${\displaystyle (^{++})}$ -stable ordinal is larger than the smallest ${\displaystyle \Sigma _{1}^{1}}$ -reflecting.
• A countable ordinal ${\displaystyle \alpha }$  is called inaccessibly-stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\beta }}$ , where ${\displaystyle \beta }$  is the smallest recursively inaccessible ordinal ${\displaystyle >\alpha }$ . The smallest inaccessibly-stable ordinal is larger than the smallest ${\displaystyle (^{++})}$ -stable.
• A countable ordinal ${\displaystyle \alpha }$  is called Mahlo-stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\beta }}$ , where ${\displaystyle \beta }$  is the smallest recursively Mahlo ordinal ${\displaystyle >\alpha }$ . The smallest Mahlo-stable ordinal is larger than the smallest inaccessibly-stable.
• A countable ordinal ${\displaystyle \alpha }$  is called doubly ${\displaystyle (+1)}$ -stable iff ${\displaystyle L_{\alpha }\preceq _{1}L_{\beta }\preceq _{1}L_{\beta +1}}$ . The smallest doubly ${\displaystyle (+1)}$ -stable ordinal is larger than the smallest Mahlo-stable.

## Largest recursive ordinals

• An ordinal ${\displaystyle \alpha }$  is semi-nonprojectible[citation needed] if ${\displaystyle L_{\alpha }\preceq _{1}L_{\beta }}$  where β is the smallest nonprojectible ordinal.
• An ordinal ${\displaystyle \alpha }$  is nonprojectible if α is a limit of α-stable ordinals, or; if the set ${\displaystyle X=\{\beta <\alpha |L_{\beta }\prec _{\Sigma _{1}}L_{\alpha }\}}$  is unbounded in α.
• The ordinal of ramified analysis, often written as ${\displaystyle \beta _{0}}$ , is the smallest ${\displaystyle \beta }$  such that ${\displaystyle L_{\beta }\cap {\mathsf {P}}(\omega )}$  is a model of second-order comprehension, and ${\displaystyle L_{\beta }\models ZFC^{-}}$ .
• The "Devlin–Jech ordinal"[citation needed] is the smallest ordinal α such that ${\displaystyle L_{\alpha }\models {\mathsf {KP}}+'\omega _{1}exists'}$ .
• The smallest ordinal α such that ${\displaystyle L_{\alpha }\models {\mathsf {ZFC^{-}}}+'\omega _{1}exists'}$ .
• The smallest stable ordinal. A countable ordinal is called stable iff ${\displaystyle L_{\alpha }\prec _{\Sigma _{1}}L}$ .

## References

• Church, Alonzo; Kleene, S. C. (1937), "Formal definitions in the theory of ordinal numbers.", Fundamenta mathematicae, Warszawa, 28: 11–21, JFM 63.0029.02
• Church, Alonzo (1938), "The constructive second number class", Bull. Amer. Math. Soc., 44 (4): 224–232, doi:10.1090/S0002-9904-1938-06720-1
• Kleene, S. C. (1938), "On Notation for Ordinal Numbers", Journal of Symbolic Logic, Vol. 3, No. 4, 3 (4): 150–155, doi:10.2307/2267778, JSTOR 2267778
• Rogers, Hartley (1987) [1967], The Theory of Recursive Functions and Effective Computability, First MIT press paperback edition, ISBN 978-0-262-68052-3
• Madore, David (2017), A zoo of ordinals, p. 3
• Simpson, Stephen G. (2009) [1999], Subsystems of Second-Order Arithmetic, Perspectives in Logic, 2, Cambridge University Press, pp. 246, 267, 292–293, ISBN 978-0-521-88439-6
• Richter, Wayne; Aczel, Peter (1974), Inductive Definitions and Reflecting Properties of Admissible Ordinals, pp. 312–313, 333, ISBN 0-7204-2276-0