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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 $\mathcal{F}$.

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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. –  Chris Eagle Feb 1 '13 at 16:06
    
Here are some more places to start: –  Alex Kruckman Feb 3 '13 at 19:34

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