index set theorem

Index Set Theorem: If $A$ is an index set and $A\mathrm{\ne }\mathrm{\varnothing }\mathrm{,}\omega$, then either $K{\mathrm{\le }}_{\mathrm{1}}A$ or $K{\mathrm{\le }}_{\mathrm{1}}{A}^{\mathrm{\complement }}$.

In the statement of the theorem, $K$ is the halting set $\{x:{\phi }_x(x)converges\}$, ${\le }_1$ is the one-one reducibility (or 1-reducibility) relation symbol, and $A^{\mathrm{\complement }}$ stands for the complement of the set $A$ (relative to $\omega$).