# Proving that the set of sentences that are true using the symbols $+,<,=$ is the same over all ordered fields

I am interested in whether the set of formulas that one can prove true for a concrete ordered field using the symbols $$+,<$$ and $$=$$, depends on the field. In particular, I am interested in the set of formulas with existential prefix.

## 1 Answer

Let $$R$$ be an ordered field. Then the reduct of $$R$$ to the language $$L_{\mathrm{og}} = \{<,+\}$$ is an ordered abelian group. Moreover, $$R$$ is divisible: Every ordered field has characteristic $$0$$, hence contains an isomorphic copy of $$\mathbb{N}$$, and for all $$r\in R$$ and $$n\in \mathbb{N}$$ with $$n>0$$, we have $$n\cdot (n^{-1}r) = r$$.

Now the theory of ordered divisible abelian groups is well-known to be complete. See, for example, Corollary 3.1.17 in Marker's book Model Theory: An Introduction. So if $$R$$ and $$R'$$ are ordered fields, then their reducts are elementarily equivalent: $$R|_{L_{\mathrm{og}}}\equiv R'|_{L_{\mathrm{og}}}$$.