Gödel’s and Some Other Examples of Problem Transmutation

Abstract

I am much struck by the historical fact that in attempting to prove the (relative) consistency of analysis, Gödel was led to certain propositions undecidable in segments of set theory. He then brought the propositions into an elegant arithmetic form and, independently of this improvement, realized that his result also implied a negative reply to the initial problem of proving consistency (not only of analysis but even of number theory). For lack of a suitable term, I have chosen to call this phenomenon “problem transmutation,” by which I mean to cover an indefinite and wide range of related phenomena.