Metamathematics of comparability
Harvey M. Friedman · Cambridge University Press eBooks · 2017
Abstract. A number of comparability theorems have been investigated from the viewpoint of reverse mathematics. Among these are various comparability theorems between countable well orderings ([2],[8]), and between closed sets in metric spaces ([3],[5]). Here we investigate the reverse mathematics of a comparability theorem for countable metric spaces, countable linear orderings, and sets of rationals. The previous work on closed sets used a strengthened notion of continuous embedding. The usual weaker notion of continuous embedding is used here. As a byproduct, we sharpen previous results of [3],[5]. 1. COMPARABILITY OF COUNTABLE WELL ORDERINGS. In this paper, we assume that the field of all linear orderings is a subset of w, and all linear orderings are reflexive. (This is an official convention that we break at