Infinite Set has Countably Infinite Subset/Proof 1
Let $S$ be an infinite set.
Suppose that there exists an injection $\psi: \N \to S$.
Let $T$ be the image of $\psi$.
Now, suppose that that there exists a surjection $\phi: \N \to S$.
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.