Department of Mathematics
A five element basis for the uncountable linear orders
Justin Moore
Boise State University
I will present a recent result in set theory: the class of uncountable linear orders consistently has a five elements basis. That is, there are five uncountable linear orders such that (consistently) any other uncountable linear order contains an isomorphic copy of one of these five. The list has long known to be minimal; it is provable from the usual axioms of mathematics that any basis must have at least five elements. It is not possible to prove such a result without appealing to additional axioms since, for instance, the Continuum Hypothesis implies that any basis must have as many elements as there are subsets of the reals. The talk will present each of the elements of the basis and discuss some of their properties. Some general remarks will also be made on the method of proof which can be considered as an infinitary version of Erdos's probabilistic method.
All interested persons are welcome.
The talk will be accessible to upper class students.