Completeness

A math system is complete if all true statements can be derived in it. That is, is there a way to prove all true propositions within a system of math?

Hilbert's first concern was the completeness of mathematics: is there a way to prove all true propositions within a system of math? Hilbert's second question was about the consistency of mathematics: is there a way to prove that no contradictory statements arise in a system of math? Hilbert's third question was about decidability: is there an algorithm that can always determine whether a statement follows from the axioms?

Gödel's incompleteness theorem means that truth and provability are not at all the same thing. There will always be some true statements about mathematics that cannot be proven.

Hilbert could console himself with the hope that at least we could still prove math's consistency, that is, that math is free of contradictions, but then Gödel published his second incompleteness theorem in which he showed that any consistent formal system of math cannot prove its own consistency.

So, taken together, Gödel's two incompleteness theorems say that the best one can hope for is a consistent, yet incomplete system of math. But such a system cannot prove its own consistency, so a contradiction could always arise, proving the system inconsistent. You might do your math work within a system that had been inconsistent all along.