Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics.
In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system.
The main goal of Hilbert's program was to provide secure foundations for all mathematics.
In particular, this should include: Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve, at least if interpreted in the most obvious way.
This presents a challenge to Hilbert's program: Many current lines of research in mathematical logic, such as proof theory and reverse mathematics, can be viewed as natural continuations of Hilbert's original program.