Hope someone can help me on this:
Prove: " Any recursively axiomatizable true theory with equality has undecidable sentence (i.e. incomplete)? "
[Definition.:A sentence S is undecidable sentence in T if its neither provable nor disprovable in T] [Theorem: If the above T is undecidable then T is incomplete]
I started by the following: Let T be a recursively axiomatizable true theory, then we can find a theory T1 which has a recursive axiom set with all theorems in T. So, to show that T is incomplete it suffices to show that T1 is incomplete! ???
Note: This ia a problem#3.57 in "Introduction to mathematical logic" By Elliott Mendelson, http://books.google.com/books?id=ZO1p4QGspoYC&pg=PT231&dq=any++recursively+axiomatizable+true+theory+has+undecidable+sentence&hl=en&ei=V1q0TYj9EMrz0gHbhNWDCQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC8Q6AEwAA#v=onepage&q=any%20%20recursively%20axiomatizable%20true%20theory%20has%20undecidable%20sentence&f=false