First-order Compactness exercises

Math 406

2004.11.22

Problem 1. Show that every Archimedean ordered field is elementarily equivalent to some countable, non-Archimedean ordered field.

Problem 2. Show that every non-Archimedean ordered field contains infinitesimal elements, that is, positive elements a that are less than every positive rational number.

Problem 3. Find an example of a non-Archimedean ordered field.

Problem 4. The order of an element g of a group is the size of the subgroup {g^{n} : n ∈ ℤ} that g generates. In a periodic group, all elements have finite order. Suppose G is a periodic group in which there is no finite upper bound on the orders of elements. Show that G ≡ H for some non-periodic group H.

Problem 5. Suppose (X, <) is an infinite total order in which X is well-ordered by <. Show that there is a total order (X^{∗}, <^{∗}) such that

(X, <) ≡ (X^{∗}, <^{∗}),

but X^{∗} is not well-ordered by <^{∗}.