... In particular if X is a set of reals of cardinality ℵ1 , then X serves as a single -element basis for the class of uncountable separable linear orders PFA is a strengthening of the Baire Category ... C UNCOUNTABLE LINEAR ORDERS 671 ∗ Conjecture (PFA) [4] The orders X, ω1 , ω1 , C and C ∗ form a five element basis for the uncountable linear orders whenever X is a set of reals of cardinality ... 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 , ω1 , C, C ∗ where X is any...