Intersection of axiomatizable sets

[ky2345]

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.

 

[mighty2000]

Any suggestions?

Dear fellow scientist,

After doing some research and thinking about this problem, I have come to the conclusion that T\cap F may not necessarily be axiomatizable. My reasoning is as follows:

Firstly, we know that T and F are both axiomatizable, which means that they are both decidable sets of sentences. This also implies that T\cap F is a decidable set, since it is a subset of both T and F.

However, just because T\cap F is decidable does not necessarily mean that it is axiomatizable. A theory is axiomatizable if it consists of all consequences of a decidable set of sentences. In other words, every sentence in the theory can be derived from a finite set of sentences. But in the case of T\cap F, we cannot guarantee that every sentence in this theory can be derived from a finite set of sentences, since it is possible that there are sentences in T\cap F that cannot be derived from either T or F alone. Therefore, T\cap F may not be axiomatizable.

To illustrate this point, let's consider the following scenario: T is the theory of natural numbers and F is the theory of real numbers. Both T and F are axiomatizable, but their intersection, T\cap F, is not axiomatizable since there are sentences in T\cap F that cannot be derived from either T or F alone (e.g. the sentence "there exists a real number that is not a natural number").

In conclusion, while T\cap F is a theory and it is consistent, it may not necessarily be axiomatizable. I hope this helps in your understanding of this problem.