A Hierarchy of Chains Embeddable into the Lexicographic Power $\boldsymbol{({\mathbb{R}}^\omega,\prec_{\rm lex})}$

Abstract

A hierarchy of chains is a transfinite sequence of linear orderings such that each chain in the sequence order-embeds into all chains following it but not in those preceding it. We construct a c + -long hierarchy of chains that order-embed into the lexicographic power $({\mathbb{R}}^\omega,\prec_{\rm lex})$ . Each linear ordering L in this hierarchy is such that there exists a tree representation of L, which is an ℝ-branching tree with no infinite branches. The existence of such a hierarchy sheds some light on the hidden complexity of $({\mathbb{R}}^\omega,\prec_{\rm lex})$ .