# Axiomatizable classes

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 $\mathcal{F}$.

• I'd have thought this would be easier using compactness, but I suppose you can use ultraproducts if you want. Anyway: the first step is to notice that some of these are obviously axiomatizable and write down axioms for them. Feb 1, 2013 at 16:06

The class of finite sets is axiomatizable

False. A non-trivial ultraproduct of a family of finite sets of increasing size is infinite.

The class of infinite sets is axiomatizable

True. Just write down "there are at least $$n$$ distinct elements" for all $$n$$.

The class of infinite sets is finitely-axiomatizable

False. If it were finitely axiomatizable, its complement would be axiomatizable.

Now repeat!

The class of fields of characteristic $$0$$ is axiomatizable

True. Just write down "$$\underbrace{1+\dots+ 1}_{n\text{ times}}\neq 0$$ for all $$n$$.

The class of fields of characteristic $$\neq 0$$ is axiomatizable

False. A non-trivial ultraproduct of a family of fields of increasing characteristic has characteristic $$0$$.

The class of fields of characteristic $$0$$ is finitely-axiomatizable

False. If it were finitely axiomatizable, its complement would be axiomatizable.