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I'm trying to refresh my knowledge about mathematical logic and I'm still unsatisfied with my insight of Gödel's Completeness Theorem.

I have read Gödel's original paper (1930 - reprinted into J.van Heijenoort (editor), From Frege to Gödel, 1967) but I'm still unsatisfied with my understanding of it. I've tried also with Wikipedia's entry about Gödel's proof but I'm still struggling with it.

I've also tried with some "modern translations" : J.L.Bell & A.B.Slomson, Models and Ultraproducts (1969), Ch.12.1: Gödel's completeness proof (pag.233-on) and with S.C.Kleene, Introduction to Metamathematics (1952), para.72 : Gödel's completeness theorem (pag.389-on), without a good mastering of the details.

So this are my first question :

Are there available detailed comments/explanations of Gödel's original proof, with all the needed details explicitely supplied ?

I've studied Henkin's version and I think I've mastered it. Some textbooks (e.g.Ian Chiswell & Wilfrid Hodges, Mathematical Logic (2007), introduce Henkin's construction with sentential logic, I think in order to "introduce" the student to the construction "testing" it in a simplified environment. My second question is :

Make it sense trying to adapt Gödel's original proof to sentential calculus in order to understand the specific details necessary to prove it for f-o logic ?

Mauro ALLEGRANZA
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  • Try to find a copy of Elliott Mendelson's Introduction to Mathematical Logic in your closest library. The first two chapters cover what you want. – user10444 Jan 08 '14 at 20:55
  • The proof goes differently than what Godel did,but the way it is done gives a deeper understanding. – user10444 Jan 08 '14 at 21:30

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