The Road to Gödel

Abstract

What does this title mean?1 Godel’s incompleteness theorem, as he originally presented it, seems to be an extraordinary and tricky magical construction. We can now, with modern recursion theory or computability theory,2 reduce the problem to the unsolvability of the halting problem, or the theorem that there is a recursively enumerable set that is not recursive (computably enumerable set that is not computable) — though that is not the way Godel presented it himself. A friend of mine of the highest distinction in this very field of recursion (computability) theory once said to me in informal conversation that all of us know how the original Godel statement was constructed, but no one really understands what it says. It is just an artificial product and has no intuitive content.