Comment by Xcelerate

Right, Hilbert’s goal was (loosely speaking) to “find a finitely describable formal system” sufficient to “capture all truths”. When Gödel showed that can’t be done, that shouldn’t imply we just stop with the best theory we have so far and call it a day—it means there are an infinite number of more powerful theories (with necessarily longer minimal descriptions) waiting to be discovered.

In fact, both Gödel and Turing worked on this problem quite a bit. Gödel thought we might be able to find some sort of “meta-principle” that could guide us toward discovering an ever increasing hierarchy of more powerful axioms, and Turing’s work on ordinal progressions followed exactly this line of thinking as well. Feferman’s completeness theorem even showed that all arithmetical truths could be discovered via an infinite process. (Now of course this process is not finitely axiomatizable, but one can certainly extract some useful finite axioms out of it — the strength of PA after all is equivalent to the recursive iteration up to ε_0 of ‘Q_{n+1} = Q_n + Q_n is consistent’ where Q_0 is Robinson arithmetic).