An Archimedean Field is an ordered field $F$ of characteristic zero such that for all $0 \leq a \leq b \ $ in $F$ exists a natural $n$ such that $b \leq na $.
Let $ \ L = <=,+,. , \leq, 0,1>$ be a language of ordered Rings.
I'm trying to prove that the notion of Archimedean Field cannot be First Order Axiomatizable, that is, there isn't a set $\Sigma$ of sentences of $L$ such that $F$ is archimedean if and only if $F \vDash \Sigma$.
I know that there are ordered fields that are not archimedean, but I dont know if it will help.
Any hints?
I would appeal to the compactness theorem. Add to your supposed first order axiomatization the countable set $\exists c\ c \gt 1, \exists c\ c \gt 1+1, \exists c\ c \gt 1+1+1,\ldots$ Any finite subset is satisfied by the Archimedean field because you have finitely many of the new axioms and you just need to take $c$ large enough. Then the whole set must have a model, but that model is non-Archimedean. This model satisfies the original axiomatization, so it does not select only the Archimedean ordered fields.