Axiom of Choice
I recently learnt about the Axiom of Choice.
Given any non-empty set, can you pick an item from the set?
My initial reaction was "Of course you can't", and was quite shocked that mathematicians were trying (in vain) to prove this axiom. In their shoes, I would have tried (in vain) to prove the axiom false.
Apparently, this axiom is neither provable, nor falsifiable, and leads to two self-consistent branches of set theory. (In the same way that Euclid's Parallel postulate leads to Euclidean geometry, and non-Euclidean geometries).