What are some results that shook the foundations of one or more fields of mathematics? [closed]
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.
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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?
- Goedel's incompleteness theorems - forcing one to understand the limitations of formal languages.
- Skolem's theorems - forcing one to reconsider everything they new about first order logic.
- Robinson's infinitesimals - showing they really do exist.
- Cantor's set theory - creating a heaven for us all.
- The discovery of non-Euclidean spaces - settling Euclid's fifth, and opening the door to much of modern geometry.
- The existence of transcendental numbers - showing how shaky our initial understanding of the reals was.
- The insolvability of the general $n$-degree polynomial, $n\ge 5$, by radicals - for the result itself and the far reaching tools and techniques created.
Galois (and Abel) showed that one can only get so far in solving polynomials using radicals.
Cohen showing the continuum hypothesis is not provable, was a shocker and it changed set theory entirely. Arguably set theorists were close to adding $V=L$ to the "canonical" set theoretic axioms, but Cohen's work showed that there is a whole world outside of $L$!
And of course Cantor's work. Finding out that there are different infinities? Mind blowing.
Cauchy and the finitary definition of limits was a foundational cornerstone for modern analysis, which shook the foundations in a positive way.