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Asmodeus

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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.