The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-σ-scattered. Specifically, we will show that Jensen's principle ♢ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-σ-scattered linear orders of cardinality greater than ℵ1, as given a successor cardinal κ+, we obtain such linear orderings of cardinality κ+ with the additional property that their square is the union of κ-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-σ-scattered linear orders of cardinality κ+ exist for every cardinal κ in Gödel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal μ of Mitchell order μ++. [-]

We say that a real closed field is an IPA-real closed field if it admits an integer part (IP) which is a model of Peano Arithmetic (PA). In [2] we prove that the value group of an IPA-real closed field must satisfy very restrictive conditions (i.e. must be an exponential group in the residue field, in the sense of [4]). Combined with the main result of [1] on recursively saturated real closed fields, we obtain a valuation theoretic characterization of countable IPA-real closed fields. Expanding on [3], we conclude the talk by considering recursively saturated o-minimal expansions of real closed fields and their IPs.[-]