116b- Lecture 16

(Covered by Todor Tsankov)

We defined the analog K_X of the halting problem for any oracle X and showed that any set r.e. in X is many-to-one reducible to K_X (in particular, it is recursive in K_X). Hence, K_X is a complete \Sigma_1(X) set and no such set is recursive in X.

We proved the Smn (or index function) theorem and Kleene’s recursion (or fixed point) theorem. Finally, we introduced the notion of an index set and proved Rice’s theorem that the only recursive index sets are {\mathbb N} and \emptyset.

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