This one is interesting also in the sense of having an interesting history: In his writings, Cantor used a principle "every set can be well-ordered." As far as I understand, he claimed that it was obvious.
In 1905, König proposed and published an alleged proof that $\mathbb{R}$ cannot be well-ordered:
Suppose that $\mathbb{R}$ is well-ordered with an ordering relation $\preceq$. Since there are uncountably many reals, there is an undefinable one (say, undefinable even in the language with the symbol $\preceq$). Since $\preceq$ is a well-ordering, there exists the least one $x_0$ which is not definable. But we have just defined it. A contradiction.
In the same year, a paper by Zermelo ''A proof of the principle that every set can be well-ordered'' was published. The author reduced the contrversial principle to several reasonably looking statements which became a basis of what is known as Zermelo--Fränkel set theory.
On the other hand, what became known as König paradox has to wait a bit to be resolved until better understanding of truth predicates was obtained.
