And the reason for Hilbert's program? The problem of "Russel's Paradox" which is a contradiction in naive set theory - https://en.wikipedia.org/wiki/Russell%27s_paradox (Note that there were other paradoxes too).

Hilbert's idea was that by completely formalizing mathematics on a axiomatic/deductive basis, one can mechanically derive proofs so that you don't run into paradoxes/contradictions.

But then Godel showed such a formal system applied to basic mathematics can never be complete (if consistent) and never prove its own consistency.