Axiomatizable classes

Question

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

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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 $\mathcal{F}$.

Answer 1

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.

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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.