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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$\beta$-recursion theory

Author: Sy D. Friedman
Journal: Trans. Amer. Math. Soc. 255 (1979), 173-200
MSC: Primary 03D60; Secondary 03E45
MathSciNet review: 542876
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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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Keywords: <IMG WIDTH="18" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img4.gif" ALT="$\beta$">-recursion theory, priority arguments, fine structure of <I>L</I>, <!– MATH ${\Sigma _1}$ –> <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="${\Sigma _1}$">-cofinality
Article copyright: © Copyright 1979 American Mathematical Society