Thread: 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!

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 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.

3.Conclude.