Decidability of the consistency for complete finitely axiomatized theories?

Question

Let $\Phi$ be a finite set of first order formulas over a signature $S$. Assume that (we can prove that) $\Phi$ is complete, i.e. for each first order formula $\phi$ over $S$, we have $\Phi \vdash \phi$ or $\Phi \vdash \neg\phi$ or both.

Is the question of the consistency of $\Phi$ decidable in this case, i.e. can it be decided whether the case that both $\Phi \vdash \phi$ and $\Phi \vdash \neg\phi$ occurs?

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It was remarked that I need to specify in more details how the problem is presented to an algorithm which should decide about the consistency. Because $\Phi$ is a finite set of formulas, it is equivalent to a single formula $\varphi$. Hence $\Phi$ can be presented (to the algorithm which should decide its consistency) by the sequence of symbols which make up the first order formula $\varphi$. The "(we can prove that)" part can be interpreted as the existence of a proof in $\operatorname{RCA}_0$ that $\Phi$ is complete. That proof could also be appended to the input, but if such a proof is known to exist, then the algorithm can also find it himself, hence there is no need to append it to the input. However, if we don't fix $\operatorname{RCA}_0$, then the information in which formal system the completeness of $\Phi$ can be proved should be part of the input.

Answer 1

A minimal answer to the question as originally posed. Consider the signature $S$ with no names, no predicates (not even identity), no function symbols, and just a single propositional variable $P$. So all wffs in this signature are (equivalent to) truth-functions of $P$, equivalent to either $P$ or $\neg P$ or $P \lor P$ or $P \land \neg P$.

Let $\Phi$ be a finite set of wffs including at least one wff that is equivalent to either $P$ or $\neg P$. Then trivially it entails $\varphi$ or $\neg \varphi$ for every available $\varphi$. So we have proved that $\Phi$ is complete.

But I haven't told you exactly what's in $\Phi$ so you can't decide whether it is consistent.

Answer 2

As noted previously, a finite set $\Phi$ of formulas is equivalent to a single formula $\varphi$. The inconsistent first order formulas are sufficiently well understood (computably enumerable, but not decidable). Hence the answer depends only on the properties of the consistent complete finitely axiomatized theories. There are complete finitely axiomatized theories with countable models, take the dense linear orders without endpoints as example. Probably enough is known about finitely axiomatized theories to provide a negative answer to the question without the "(we can prove that)" part, but I probably won't take the pain to dig through that material. Below is a proof that decidability follows if the "(we can prove that)" part would give rise to a computably enumerable set of "candidate" consistent complete theories (for example if the proof would be a consistent and complete theory whose axioms could be derived from $\Phi$).

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1. The inconsistent first order formulas over a signature $S$ are computably enumerable, because being inconsistent means that $\bot$ can be derived in a finite number of steps.

2. Iff the consistent (provably) complete finitely axiomatized theories $\mathcal T$ are computably enumerable, then the consistency of $\Phi$ from the question is decidable.

The algorithm is as follows: Check whether $\Phi$ is inconsistent. Simultaneously also check whether $\Phi$ corresponds to a theory $T \in \mathcal T$. Because each $T\in \mathcal T$ is complete, it is decidable whether $\Phi$ corresponds to $T$. So this algorithm will terminate by either finding a consistent theory $T$ which corresponds to $\Phi$, or otherwise finding a proof that $\Phi$ is inconsistent.

For the "only if" part, consider that the finite sets $\Phi$ of first order formulas over a signature $S$ which can be proved to be complete in a given formal system like $\operatorname{RCA}_0$ are computably enumerable, because any proof in a formal system has a finite number of steps. Hence decidability of consistency would make the set of the consistent (provably) complete theories computably enumerable.