In the 1930s, Kurt Gödel and Gerhard Gentzen proved results that cast new light on the problem.
... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.
As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that Peano arithmetic is consistent.
There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as Zermelo–Fraenkel set theory.
Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent.
Gentzen's result shows that a consistency proof can be obtained in a system that is much weaker than set theory.
Simpson (1988) argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories.