The hope of Hilbert was that once correctly formalized, it would be possible to show that mathematics was sound and that any statement has a proof. Godel's incompleteness results put an end to that hope, at least for recursively enumerable theories strong enough to recover arithmetic.
A large part of Euclid's Elements, namely the geometric part, does not assume arithmetic but can actually be built on first order logic. Several axiomatics were devised circa 1900 to dispel a few gaps in the treatise of Euclid, mostly due to a lack of treatment of order on lines, corresponding to a point being between two other points. Amongst the many axiomatics developped at that time, the one introduced in the "Grundlagen der geometrie" by Hilbert is the most famous. The big result was yet to come.
Alfred Tarski gave a slightly different axiomatic for elementary geometry, of a more logical nature, and relying only on first order logic. He was then able to show that his theory of elementary geometry is
-- consistent: a well formed statement is either true or false, and cannot be both.
-- complete: a proposition (or its negation) stated in the language of first order logic always has a proof.
-- algorithmically decidable: there is an explicit algorithm that decides if a statement is true or false.
So, Alfred Tarski realized Hilbert's program in the context of elementary geometry and showed what mathematicians have anticipated for more than two millenia: the logical perfection of Euclid's geometry.