On limitwise monotonicity and maximal block functions

Abstract

We prove the existence of a limitwise monotonic function g : N → N \{ 0} such that, for any � 0 function f : N → N \{ 0} ,R anf �= Ran g. Relativising this result we deduce the existence of an η-like computable linear ordering A such that, for any � 0 function F : Q → N \{ 0} ,a ndη-like B of order type { F( q)| q ∈ Q}, BA. We prove directly that, for any computable A which is either (i) strongly η-like or (ii) η-like with no strongly η-like interval, there exists 0 � -limitwise monotonic G : Q → N \{ 0} such that A has order type {G(q) | q ∈ Q}. In so doing we provide an alternative proof to the fact that, for every η-like computable linear ordering A with no strongly η-like interval, there exists computable B ∼ A with � 0 block relation. We also use our results to prove the existence of an η-like computable linear ordering which is � 0 categorical but not � 0 categorical.