>>7811955

Well, consider what the Tarski-Vaught criterion implies.
Problems there?

Here's an elementary thing I like:

Prove that [math](\mathbb{N}, +, \cdot)[/math] is a definable substructure of [math](\mathbb{Z}, +, \cdot)[/math].

>>7811955
Where is this taken from?

Here's a more advanced thing I like:

Show that it is consistent with ZF that there is a countable collection A of 2-element sets such that A has no choice function.

This is as weak as choice can get, and even it can fail.