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Given a model , let , the *standard system* of , be the set of those coded by elements of , where codes iff . Thus is the class of finite sets. We showed that if is nonstandard, contains all recursive sets, and that for any non-recursive there is a nonstandard such that .

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Due Tuesday March 18 at 1pm.

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We proved this result under the assumption that exponentiation is Diophantine. This is the key result, and will be dealt with in the following lecture.

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Due Tuesday March 11 **at the beginning of lecture**.

**Important update:** In problem 4.(a), recall that is a universal predicate for unary formulas, so if is the Gödel number of a formula in one free variable , then holds iff holds. Hence, asking that is finite is the same as asking that is finite. Actually, this is a serious typo:

is -complete. The set is -complete.

Sorry about this. Either ignore 4.(a), or try to show (for extra credit) that the set is -complete, or (for a much more challenging problem) that the corresponding set with “cofinite” is -complete.

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We also showed that there incomparable Turing degrees below .

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

We proved the –– (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 and .

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We defined machines with oracles and stated the corresponding result: Given an oracle , a function is computable by a Turing machine with oracle iff is recursive in (meaning that it is built up from and the basic functions by composition, recursion and minimalization) iff has a graph definable in the structure . We then defined the partial order among subsets of which holds iff is recursive in .

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Due Tuesday March 4 at the beginning of lecture.

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