Logic Seminar

Justin MooreCornell University
A virtual five element basis for uncountable linear orders, part 1

Monday, October 20, 2025 - 2:55pm
Malott 205

We prove that if $X$ is an uncountable set of reals and $C$ is a Countryman line, then for any uncountable linear order $L$, there is a proper forcing extension in which $L$ contains a copy of $X$, $\omega_1$, $\omega_1^*$, $C$, or $C^*$. In particular, if there is an inaccessible cardinal, then there is a proper forcing extension in which the uncountable linear orders have a five element basis consisting of $X$, $\omega_1$, $\omega_1^*$, $C$, and $C^*$. This also answers several related questions related to the consistency strength of a five element basis for the uncountable linear orders and which hypotheses imply it. Central to our proof is a preservation lemma concerning how subtrees of $\omega_1$-trees can be added in countable support iterations. This is joint work with John Krueger.