Jumps and the classification of scattered linear orders

SEALS Conference, Gainesville, February 2020

Abstract: We begin by studying the classification problem for countable scattered linear orders. Letting $\cong_\alpha$ denote isomorphism of scattered orders of rank $\alpha$, we can define the “$\mathbb Z$-jump” of equivalence relations in such a way that the $\mathbb Z$-jump of $\cong_\alpha$ is $\cong_{\alpha+1}$. More generally, for any countable group $\Gamma$, we will define the $\Gamma$-jump of equivalence relations. After introducing the basic theory of these jump operators, we will discuss the question of when the $\Gamma$-jump is proper. In particular we will show the $\mathbb Z$-jump is proper, and hence the complexity of $\cong_\alpha$ increases properly with $\alpha$. This is joint work with John Clemens.