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index set theorem (Theorem)

Index Set Theorem: If $ A$ is an index set and $ A\neq\varnothing, \omega$, then either $ K\leq_1 A$ or $ K\leq_1 A^{\complement}$.

In the statement of the theorem, $ K$ is the halting set $ \{x : \varphi_x(x)\: converges \}$, $ \leq_1$ is the one-one reducibility (or 1-reducibility) relation symbol, and $ A^{\complement}$ stands for the complement of the set $ A$ (relative to $ \omega$).



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Cross-references: complement, relation symbol, index set

This is version 2 of index set theorem, born on 2008-06-27, modified 2008-07-09.
Object id is 10724, canonical name is IndexSetTheorem.
Accessed 176 times total.

Classification:
AMS MSC: 03D25 (Mathematical logic and foundations :: Computability and recursion theory :: Recursively enumerable sets and degrees)
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