Mathematicians have been dealing with this problem in concrete ways for longer than implied. For instance, the necessity of the parallel postulate for Euclidia

Mathematicians have been dealing with this problem in concrete ways for longer than implied. For instance, the necessity of the parallel postulate for Euclidian geometry and the existence of non-Euclidean geometries where it doesn't hold were demonstrated in the 19th century. The question of how many and which axioms are required as the basis of a field of mathematics has long been known to be a valid, meaningful question. What Gödel proved is that in the formal systems we use to model mathematical propositions and proofs, there is no finite number of axioms that enable you to prove or disprove all of the propositions that can be formulated. For someone who grew up in the post-Gödel world, this isn't shocking at all and looks like a natural progression from previous results. You need a certain number of postulates. To get Euclidean geometry, for example, you need the parallel postulate. It's natural to ask how many postulates you need to decide all interesting mathematical questions, and there's no reason to assume the answer to that question is finite. Apparently, that attitude reflects a lot of hindsight bias, but there is no way for us to experience the shock of Gödel's work the way it was experienced by mathematicians at the time, since the books we learn from now are written by mathematicians who have already digested Gödel's discoveries.