Lexicographic preference representation: Intrinsic length of linear orders on infinite sets

Abstract

The intrinsic length of a linear order is the minimum of all ordinals δ such that there is a binary-criteria lexicographic representation of the linear order in {0,1}δ. Assuming the Generalized Continuum Hypothesis, we show that, for each ordinal γ and infinite set X with cardinality κ, there exist a linear order on X such that γ is the intrinsic length of that linear order if and only if logκ≤γ≤κ. This intrinsic-length partition imposes a structure on the profusion of linear orders on an infinite set.