Intersection of axiomatizable sets

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Homework Statement



Suppose that T and F are both axiomatizable, complete, consistent theories. Is T\cap F axiomatizable?

Homework Equations


A theory T is a set of sentences such that if yo can deduce a sentence a from T, then a is in T.

I have already proved that T\cap F is a theory.

A complete theory is one where either a or \lnot a are in the theory for every sentence a.

An inconsistent theory is one where a is in the theory and \lnot a is in the theory

An axiomatizable theory is a theory that consists of all consequences of a decidable set of sentences (CnB= the consequences of a set B)

If a theory is axioatizable and complete, then is is decidable (not sure if this is relevent)

I have already proved that T\cap F is consistent.

The Attempt at a Solution



Since both T and F are axiomatizable, there are decidable sets G and H such that T=CnG and F=CnH. Now, we know from a previous result that G\cap H is decidable. Thus, Cn(G\cap H) describes an axiomatizable set. Now, if a is in T\cap F, then it is in both CnG and CnH, so that G\vDash a and H\vDash a. Now, let A be a model of G\cap H. Then, it satisfies...

I have a feeling that the answer might be no, since I seem to be getting nowhere with my proof, but I am having trouble thinking of a counterexample.
 

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