Full Idea
Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert's conviction that for every precise mathematical question there is a discoverable answer.
Gist of Idea
Gödel's Theorems did not refute the claim that all good mathematical questions have answers
Source
report of Kurt Gödel (works [1930]) by Peter Koellner - On the Question of Absolute Undecidability Intro
Book Reference
-: 'Philosophia Mathematica' [-], p.1
A Reaction
The normal simplistic view among philosophes is that Gödel did indeed decisively refute the optimistic claims of Hilbert. Roughly, whether Hilbert is right depends on which axioms of set theory you adopt.
Related Idea
Idea 17885 Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable [Gödel, by Koellner]