There may be no minimal non $\sigma$-scattered linear orders

TL;DR

This paper explores the consistency of the non-existence of minimal non $\sigma$-scattered linear orders under certain set-theoretic assumptions, extending understanding of the structure of such orders.

Contribution

It demonstrates the consistency of no minimal non $\sigma$-scattered linear orders assuming a supercompact cardinal and analyzes implications of PFA${}^+$ on the structure of non $\sigma$-scattered orders.

Findings

No minimal non $\sigma$-scattered linear order exists under certain set-theoretic assumptions.

PFA${}^+$ implies all non $\sigma$-scattered orders contain specific types or structures.

CH is consistent with the non-existence of Aronszajn trees with a small base.

Abstract

In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $\sigma$-scattered. This shows that a theorem of Laver, which asserts that the class of $\sigma$-scattered linear orders is well quasi-ordered, is sharp. We also prove that PFA${}^+$ implies that every non $\sigma$-scattered linear order either contains a real type, an Aronszajn type, or a ladder system indexed by a stationary subset of $\omega_1$, equipped with either the lexicographic or reverse lexicographic order. Our work immediately implies that CH is consistent with "no Aronszajn tree has a base of cardinality $\aleph_1$." This gives an affirmative answer to a problem due to Baumgartner.