The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.

The complexity of aII 4 set of natural numbers is encoded into a linear order to show that the finite condensation of a recursive linear order can beII 2–II 1. A priority argument establishes the same result, and is extended to a complete classification of finite condensations iterated finitely many times.

The properties of antisymmetry and linearity are easily seen to be sufficient for a recursively enumerable binary relation to be recursively isomorphic to a recursive relation. Removing either condition allows for the existence of a structure where no recursive isomorph exists, and natural examples of such structures are surveyed.