Metamathematics I Problem Set 4

Due Friday October 12

1) We say that a group (G, +, <) is archimedian if for all x, y ∈ G with x, y > 0 there is an integer m such that |x| < m|y|. Show that there are non-archimedian fields elementarily equivalent to the field of real numbers.

2) Show that there is no formula ϕ(v) in the language of groups such that for all groups G if g ∈ G, G ⊧ ϕ(g) if and only if g has finite order.

3) Let Ł be the language with one binary relation symbol <. Let T be an Ł-theory extending the theory of linear orders such that T has infinite models. Show that there is M ⊧ T and an order preserving embedding σ : Q → M of the rational numbers into M. [Hint: Use the Compactness Theorem.]

For example if T is the full theory of the (Z, <), there is M ≡ Z in which the rational order embedds.

4) Show that every torsion free abelian group (G, +) can be linearly ordered such that (a < b ∧ c ≤ d) → a + c < b + d. [Hint: First show this for finitely generated groups. Then use compactness.]