Full Idea
Hilbert's project was to establish the consistency of classical mathematics using just finitary means, to convince all parties that no contradictions will follow from employing the infinitary notions and reasoning.
Gist of Idea
Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions
Source
report of David Hilbert (works [1900]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
Book Reference
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.156
A Reaction
This is the project which was badly torpedoed by Gödel's Second Incompleteness Theorem.