Algebraic Closure of Field is Unique
Let $F$ be a field.
Let $K$ and $L$ be algebraic closures of $F$.
Then $K$ and $L$ are $F$-isomorphic.
Axiom of Choice
This theorem depends on the Axiom of Choice.
Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.
However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.