A theory $T$ is a set of sentences. An axiomsystem $\Phi$ of $T$ has the charachteristic that $T=\{\psi:\Phi\models\psi\}$.
An expansion of $T$ is a set of sentences $T'$ which has the charachteristics that $T\subseteq T'$.
$T'$ is complete if either $\psi\in T'$ or $\neg\psi\in T'$.
(I always assume that an expansion of a theory is not necessarily complete - is this true?)
For example $T=\{\psi:\text{Groupaxioms}\models \psi\}$ is not complete because there are commutative groups and not commutative groups. The expansion $T'=\{\psi:\text{{Groupaxiomis}}\cup\text{{has one element}}\models\psi\}$ is a complete expansion.
(The last example is made by myself so maybe it is wrong)
Can someone give me a hint to solve the problem?
Let every complete expansion have a finite axiomsystem. By assumption I know that there are only finitely many complete expansions thus I look at the intersection over those axiomsystems which I assume must be an axiomsystem for the initial theory but I am not sure how to justify that.
For the other side of the proof I assume that the initial theory has a finite axiomsytem by assumption I also know that there are only finitely many complete expansions. If one of those expansions would have an infinite axiomsystem (does every theory always have an axiomsystem ? - I assume yes because the theory itself is an axiomsystem - correct me if I am wrong) then I always could add a sentence to my initial axiomsystem which is the infitie axiomsystem but not in the inital one and create infite expansions which would be a contradiction (I always assume that simply by adding a new sentence to an axiomsystem we always create a new expansion - but is this true?) if those expansions are also complete (but I am not sure if I can assume this - i.e. need help to show that there are indeed infinte complete expansions)
Those are my ideas please help