Gödel’s incompleteness theorems are bombs exploding at the very center of the Hilbert program, decisively and entirely refuting it. The incompleteness theorems show, first, that we cannot in principle enumerate a complete axiomatization of the truths of elementary mathematics, even in the context of arithmetic, and second, no sufficient axiomatization can prove its own consistency, let alone the consistency of a much stronger system. Hilbert’s world is a mirage.
