What are some results that shook the foundations of one or more fields of mathematics? An example would be the proof that $\sqrt{2}$ is not rational, which was a violation of some fundamental assumptions that mathematicians at the time made about numbers. Another would be Russell's paradox, which proved that naive set theory contains contradictions. I'm looking for a list of examples of similar results which forced mathematicians to change widely held beliefs and rebuild the foundations of their field. How did mathematicians at the time rearrange their foundations to account for newly proven results?
There must be lots of examples.
EDIT
The question was put on hold as too broad. Hopefully this narrows it a bit. I am looking for examples where most mathematicians believed that a certain foundational set of propositions was true, and then a result was proven that showed that it was false. Most theorems do not fit the bill. Wiles' proof of Fermat's last theorem was a historic achievement, but as far as I am aware, it did not force Wiles' contemporaries to reevaluate their fundamental assumptions or beliefs about mathematics. Gödel's incompleteness theorems, on the other hand, showed that Hilbert's program was unattainable, causing a fundamental shift in our understanding of notions like provability.
EDIT
I don't understand why this keeps being closed. It seems to be generating quality answers and the answerers seem to be able to tell what is being asked. Maybe some commenter could let me know?

Gödel's incompleteness theorem -- any formal system with a reasonable level of descriptive power contains statements that are both true and unprovable. And these statements can be constructed. 
The construction of hyperbolic geometry -- satisfies all of Euclidean axioms except the parallel postulate. 
