my forty years on his shoulders - Department of Mathematics

7In particular, it is not yet clear how strongly and in what way the Gödelincompleteness phenomena will penetrate normal mathematical activity.Progress along these lines is steady but painfully slow. We are confident that amuch clearer assessment will be possible by the end of this century - and perhapsnot much earlier.In section 12, we take the opportunity to speculate far into the future.2. THE COMPLETENESS THEOREM.In his Ph.D. dissertation, (Gödel 1929), Gödel proved his celebratedcompleteness theorem for a standard version of the axioms and rules of firstorder predicate calculus with equality.This result of Gödel was anticipated, in various senses, by earlier work ofT. Skolem as discussed in detail in the Introductory notes in Vol. I of (Gödel1986-2003 44-59). These Introductory notes were written by Burton Dreben andJean van Heijenoort.On page 52, the following passage from a letter from Gödel to Hao Wang,is quoted (December 7, 1967):“The completeness theorem, mathematically, is indeed an almost trivialconsequence of Skolem 1923a. However, the fact is that, at the time, nobody(including Skolem himself) drew this conclusion (neither from Skolem 1923a nor,as I did, from similar considerations).”According to these Introductory Notes, page 52, the situation is properlysummarized as follows: