Comment by ionfish

Lots. For example, that every set has a cardinality (is bijective with some aleph). That's pretty intuitive, too, as (I think) are the following.

* Let X and Y be sets. Then either they have the same cardinality, or one is smaller than the other.

* Let X be an infinite set. Then there is a bijection between X and the cartesian product of X with itself, X × X.

* Tychonoff's theorem: every product of compact topological spaces is compact.