# If a negation-complete first-order theory has a categorical-in-some-cardinal axiomatization, are all axiomatizations categorical in that cardinal?

Let $T$ be a set of sentences in some first-order language $L$, and assume $T$ is negation-complete, so that for any sentence of $\phi\in L$, either $T\vdash\phi$ or $T\vdash\lnot\phi$. Assume as well that $T$ has a categorical-in-cardinality-$\kappa$ axiomatization $\Gamma$, i.e. assume there exists a set of sentences $\Gamma$ expressed in $L$ such that (1) the deductive closure of $\Gamma$ is $T$, and (2) all models of $\Gamma$ with cardinality $\kappa$ are isomorphic.

Let $\Omega$ be an arbitrary axiomatization of $T$. Does it follow that $\Omega$ must also be categorical in $\kappa$?

I don't know much logic beyond the definitions, so I don't have a supply of examples of $\kappa$-categorical theories to test the conjecture on. And I don't see immediately why it would be impossible for one axiomatization to be $\kappa$-categorical without all of them being so.

## 2 Answers

If T and S are theories with the same deductive closure, then they have the same models. In particular if one is k-categorical then so is the other.

Note that completeness is irrelevant.

• Ah, duh. Thanks. – symplectomorphic May 12 '17 at 23:32

If $\Gamma$ is an axiomatization of a theory $T$, then models of $\Gamma$ are exactly the same thing as models of $T$. Indeed, if a model $M$ satisfies every statement in $\Gamma$, then it satisfies every statement in $T$, since every statement in $T$ is a consequence of the statements in $\Gamma$. The converse is trivial, since $\Gamma\subseteq T$.

So what models a theory has doesn't depend on how you axiomatize it. As a result, neither does categoricity of a theory, since that's just a statement about what models the theory has.