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A five element basis for the uncountable linear orders. (English) Zbl 1143.03026
Summary: In this paper I 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 long-standing conjecture of Shelah.

##### MSC:
 3e+35 Consistency and independence results 300000 Other combinatorial set theory 3e+40 Other aspects of forcing and Boolean-valued models 3e+55 Large cardinals
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