Are the statements below true or false:
- The class of finite sets is axiomatizable
- The class of infinite sets is axiomatizable
The class of infinite sets is finitely-axiomatizable
The class of fields of characteristic 0 is axiomatizable
- The class of fields of characteristic $\neq$ 0 is axiomatizable
- The class of fields of characteristic 0 is finitely-axiomatizable
I think I need to use the following criterion: Let $\mathcal{F}$ be a non-trivial ultrafilter on $\mathcal{P}(\mathbb{N})$ and $\mathcal{K}$ a class of structures of the same (countable) signature. Then $\mathcal{K}$ axiomatizable iff $\mathcal{K}$ closed under elementary equivalence and closed under ultraproducts modulo $\mathcal{F}$.