A five element basis for the uncountable linear orders | Annals of Mathematics

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, ω1, ω∗1, C, C∗ where X is any suborder of the reals of cardinality ℵ1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.

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数学年刊(Annals of Mathematics)

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