A five element basis for the uncountable linear orders

Abstract

In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are $X$, $\omega_1$, $\omega_1^*$, $C$, $C^*$ where $X$ is any suborder of the reals of cardinality $\aleph_1$ and $C$ is any Countryman line. This confirms a longstanding conjecture of Shelah.