Abstract

We provide partial answers to the following problem: Is the class of Borel linear orders well-quasi-ordered under embeddability? We show that it is indeed the case for those Borel orders which are embeddable in R ω , with the lexicographic ordering. For Borel orders embeddable in R 2 , our proof works in ZFC, but it uses projective determinacy for Borel orders embeddable in some $\mathbf{R}^n, n , and hyperprojective determinacy for the general case