Basically, to show that archimedeanity is not a first order property, you want to obtain two equivalent structures, one archimedean and the other not.
Let's work with we want to find for instance in the language with an element such that for all
So consider the theory where is a new constant symbol ( is therefore a -theory )
1. Write as a set of first-order formulae.
2. Use compactness to prove that is consistent.