There are some models of $\sf ZF$ without the Axiom of choice, where some paradoxical statements hold that are not possible in $\sf ZFC$ (we do not require that all those statements necessarily hold in the same model). Paradoxical means that the statement may seem counterintuitive if one tries to identify the corresponding model with the intended universe of sets as usually conceived (this is, of course, subjective, or may have no sense at all for a pure formalist).
Here are some examples:
- There is a collection of non-empty sets whose Cartesian product is empty.
- There is a set that can be partitioned into disjoint nonempty parts, such that the number of parts exceeds the number of elements of the set.
- There is an infinite set without a countable infinite subset.
- A countable union of countable sets may be uncountable.
- The set of reals $\mathbb R$ is a countable union of countable sets.
- There is a pair of sets such that none of them is equinumerous with a subset of the other.
(some of these, and others are also listed here)
I'm interested in seeing more such examples, particularly those accessible to most people with only basic understanding of naïve set theory, and not too specific to some area like topology or group theory.