Automorphisms ofη-like computable linear orderings and Kierstead's conjecture

Abstract

Abstract. We develop an approach to the longstanding conjecture of H.A. Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering B, such that B has no interval of order type η, and such that the order type of B is determined by a 0 ′ -limitwise monotonic maximal block function, there exists computable L ∼ = B such that L has no nontrivial Π 0 1 automorphism.

On limitwise monotonicity and maximal block functions

“…This is the fact that computable approximations to 0 -limitwise monotonic functions have similar properties to those of Π 0 2 functions, and that this facilitates the use of strategy tree constructions to build η-like linear orderings whose order type τ is determined by a 0 -limitwise monotonic maximal block function. The latter is exploited in [HLC14] to provide a generalisation-to the class of all such order types τ -of Kierstead's result [Kie87] that there exists computable L of order type 2 · η such that L has no nontrivial Π 0 1 automorphism. Section 5, which concludes the paper, is a preliminary investigation of the question of finding, for given n ≥ 1, an η-like computable linear ordering A n which is ∆ 0 n+1 categorical but not ∆ 0 n categorical.…”