Prove that the ordering on the real number is not isomorphic to an ultrapower of the ordering of the rationals. Hint: an ultrapower of $\mathbb{Q} $ is not complete.
I cannot understand the hint; I mean, $\mathbb{Q} $ is an elementary substructure of $\mathbb{R}$ and then they should be elementary equivalent, then the completeness shouldn't be first order aziomatizable. Moreover by Los's theorem and ultrapower of $\mathbb{Q} $ should be elementary equivalent to $\mathbb{Q} $. Hence the only way I can see to prove the statement is by mean of cardinality, but in this case the hint wouldn't help me at all. Where am I wrong?
It is indeed true that as ordered sets, any ultrapower of $\mathbb{Q}$ and $\mathbb{R}$ are elementarily equivalent.
However, any ultrapower of $\mathbb{Q}$ can be given an ordered field structure in which the sentence $\forall x\lnot (x\cdot x=1+1)$ is true. But the reals as an ordered set cannot be given such a field structure.