On Categoricity of Scattered Linear Orders Of constructive Ranks

Abstract

In this article we investigated the complexity of isomorphisms be-tween scattered linear orders of constructive ranks. We gave the general upperbound and proved that this bound is sharp. Also, we constructed examples show-ing that the categoricity level of a given scattered linear order can be an arbitraryordinal from 3 to the upper bound, except for the case when the ordinal is thesuccessor of a limit ordinal. The existence question of the scattered linear orderswhose categoricity level equals to the successor of a limit ordinal is still open.