On Π₁-automorphisms of recursive linear orders

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Abstract

If σ is the order type of a recursive linear order which has a nontrivial automorphism, we let denote the least complexity in the arithmetical hierarchy such that every recursive order of type σ has a nontrivial automorphism of complexity . In Chapter 16 of his book Linear orderings [R], Rosenstein discussed the problem of determining for certain order types σ . For example Rosenstein proved that , where ζ is the order type of the integers, by constructing a recursive linear order of type ζ which has no nontrivial Σ 1 -automorphism and showing that every recursive linear order of type ζ has a nontrivial Π 1 -automorphism. Rosenstein also considered linear orders of order type 2 · η , where 2 is the order type of a two-element chain and η is the order type of the rational numbers. It is easily seen that any recursive linear order of type 2 · η has a nontrivial ⊿ 2 -automorphism; he showed that there is a recursive linear order of type 2 · η that has no nontrivial Σ 1 -automorphism. This left the question, posed in [R] and also by Lerman and Rosenstein in [LR], of whether or ⊿ 2 . The main result of this article is that :

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