A direct proof of the five element basis theorem

TL;DR

This paper provides a straightforward proof confirming the consistency of a five-element basis for uncountable linear orders, simplifying previous complex proofs using saturation of Aronszajn trees.

Contribution

It offers a direct and simplified proof of the five element basis theorem for uncountable linear orders, building on the saturation approach of Aronszajn trees.

Findings

Confirmed the consistency of a five-element basis for uncountable linear orders

Simplified the proof using saturation of Aronszajn trees

Built upon the approach of Koenig, Larson, Moore, and Velickovic

Abstract

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of notion of saturation of Aronszajn trees considered by Koenig, Larson, Moore and Velickovic and simplifies the original proof of Moore.