How can I prove, without using priority argument, that Cof, the set of indices of cofinite c.e. sets, is $\Sigma^0_3$-complete?
I know an injury-free proof of Rec being $\Sigma^0_3$-complete, where Cof is the set of indices of recursive c.e. sets. I tried to reduce Rec to Cof to obtain an injury-free proof for Cof, but it seems nontrivial.
EDIT: in the first version I wrote Rec where I should have written Cof, and vice versa. Sorry about my confusion.