Geodesic
A five element basis for the uncountable linear orders
Justin Tatch Moore1
1 Department of Mathematics, Boise State University, Boise, ID 83725, United States
Annals of mathematics, Tome 163 (2006) no. 2, pp. 669-688.

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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.
DOI : 10.4007/annals.2006.163.669

Justin Tatch Moore 1

1 Department of Mathematics, Boise State University, Boise, ID 83725, United States 
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Justin Tatch Moore. A five element basis for the uncountable linear orders. Annals of mathematics, Tome 163 (2006) no. 2, pp. 669-688. doi : 10.4007/annals.2006.163.669. http://geodesic.mathdoc.fr/articles/10.4007/annals.2006.163.669/

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