# User:IssaRice/Computability and logic/List of possibilities for completeness and decidability

Yes Yes No Smith's example of $T_1$ with just $\neg p$ as an axiom inside a propositional logic with propositional atoms $p,q,r$.
Yes No Yes Assuming the theory is effectively axiomatized and consistent, this is not possible. To see why, suppose the theory is negation-complete. We could then computably enumerate all the theorems (since the theory is effectively axiomatized), and after a finite amount of time we will find exactly one of $\phi$ or $\neg \phi$, which gives us an algorithm for deciding whether or not $\phi$ is a theorem.
No Yes Yes First-order logic with two binary predicates $R$ and $S$ with the axiom $\forall x \forall y (R(x,y)\wedge S(x,y))$.
No No Yes Assuming the theory is effectively axiomatized and consistent, this is not possible. To see why, suppose the theory is negation-complete. We could then computably enumerate all the theorems (since the theory is effectively axiomatized), and after a finite amount of time we will find exactly one of $\phi$ or $\neg \phi$, which gives us an algorithm for deciding whether or not $\phi$ is a theorem.