𝛽-recursion theory

Friedman, Sy D.

doi:10.1090/S0002-9947-1979-0542876-7

$\beta$-recursion theory
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Trans. Amer. Math. Soc. 255 (1979), 173-200 Request permission

Abstract:

We define recursion theory on arbitrary limit ordinals using the J-hierarchy for L. This generalizes $\alpha$-recursion theory, where the ordinal is assumed to be ${\Sigma _1}$-admissible. The notion of tameness for a recursively enumerable set is defined and the degrees of tame r.e. sets are studied. Post’s Problem is solved when ${\Sigma _1}\operatorname {cf} \beta \beta {\ast }$. Lastly, simple sets are constructed for all $\beta$ with the aid of a $\beta$-recursive version of Fodor’s Theorem.

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