This question was motivated by this [recent question by Ricky Demer](https://mathoverflow.net/questions/55944/computable-completion-of-zfc).


In his paper [$\Pi^0\_1$ classes and Boolean combinations of recursively enumerable sets](http://www.ams.org/mathscinet-getitem?mr=344094), Carl Jockusch showed that there is no completion of Peano Arithmetic which is a Boolean combination of computably enumerable (c.e.) sets. This was generalized to any essentially undecidable theory by James Schmerl in his paper [Undecidable theories and reverse mathematics](http://www.ams.org/mathscinet-getitem?mr=2185445).


Are there similar restrictions on the type of sets which can appear as *axiomatizations* of completions of Peano Arithmetic or other computably axiomatizable and essentially undecidable theories? (Other than the trivial fact that such axiomatizations cannot be c.e.) Specifically, does every computably axiomatizable theory have a completion which is axiomatizable by a co-c.e. set?



Axiomatizations of complete theories
Yes. Every consistent c.e. theory has a $\Sigma^0\_2$ (even low $\Delta^0\_2$) completion, and therefore a completion axiomatized by a $\Pi^0\_1$ set.


In more detail, given a c.e. theory $T$, one can construct its completion by the following procedure: let $\{\phi\_n\mid n\in\omega\}$ be a computable enumeration of all sentences in the language of $T$, and put


$$\psi\_n=\begin{cases}\phi\_n&\text{if }T+\{\psi\_i\mid i<n\}+\phi\_n\text{ is consistent,}\\\neg\phi\_n&\text{otherwise.}\end{cases}$$


The branching condition is $\Pi^0\_1$, hence the whole thing can be implemented by an algorithm with an oracle for the halting problem. Thus $S=\{\psi\_n\mid n\in\omega\}$ is a $\Delta^0\_2$ completion of $T$. This completion can be $\Pi^0\_1$-axiomatized using Craig's trick: if $S=\{\phi\mid\exists n\,P(n,\phi)\}$, where $P$ is $\Pi^0\_1$, then $S$ is axiomatizable by $\{\underbrace{\phi\land\cdots\land\phi}\_{n\text{ times}}\mid P(n,\phi)\}$.


