Galvin's Question on non-$\sigma$-Well Ordered Linear Orders

TL;DR

This paper proves the consistency of the existence of minimal elements in a class of linear orders with specific uncountability and order-embedding properties, addressing a question posed by Galvin.

Contribution

It establishes the consistency of minimal elements in a class of non-$\sigma$-well ordered linear orders, answering an open question by Galvin.

Findings

Proves the consistency of minimal elements in the class $\mathcal{C}$

Shows that such linear orders can exist under certain set-theoretic assumptions

Addresses an old open problem in order theory

Abstract

Assume $\mathcal{C}$ is the class of all linear orders $L$ such that $L$ is not a countable union of well ordered sets, and every uncountable subset of $L$ contains a copy of $\omega_1$. We show it is consistent that $\mathcal{C}$ has minimal elements. This answers an old question due to Galvin.