1. ## first-order theory

Recall that the ordering on Q resp. on R is archimedean, i.e. for every x in Q resp. x in R there is some n in N with -n<x<n. Use the compactness theorem to prove that archimedeanity is not a first-order property.

Could anyone please give me some hints how to handle this problem? Any help is appreciated!

2. Hi

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 $\mathbb{R},$ we want to find for instance $A\equiv\mathbb{R}$ in the language $\{+,\times ,0,1,\leq\}$ with an element $a\in A$ such that for all $n\in\mathbb{N},$ $n

So consider the theory $T=Th(\mathbb{R})\cup\{c\ \text{is strictly greater than any natural integer}\}$ where $c$ is a new constant symbol ( $T$ is therefore a $\{+,\times ,0,1,\leq,c\}$-theory )

1. Write $\{c\ \text{is strictly greater than any natural integer}\}$ as a set of first-order formulae.

2. Use compactness to prove that $T$ is consistent.

3.Conclude.