Resumen de A five element basis for the uncountable linear orders

Justin Tatch Moore

• 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, ¥ø1, ¥ø.1, C, C. where X is any suborder of the reals of cardinality .1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.