I'm in Mathematical Logic course. I just wanted to tie a few loose ends. In our logic text, ("A Friendly Introduction to Mathematical Logic" -C. Leary) he uses the Henkin axioms to prove Godel's Completeness theorem. I understand the whole proof, except the last part that requires equivalence classes. Can anyone spell this out for me. Further, can someone explain the self-reference lemma to me? For instance, our text uses v=4. Why is this so? Is it because 4 = 2^2, which is not a godel number? What is the precise difference between the completeness in the first sense, and (in)completeness in the second sense? Any explanations or good links would be greatly appreciated.