Recursive linear orders with incomplete successivities

Abstract

A recursive linear order is said to have intrinsically complete successivities if, in every recursive copy, the successivities form a complete set. We show (Theorem 1) that there is a recursive linear order with intrinsically complete successivities but (Theorem 2) that this cannot be a discrete linear oder. We investigate the related issues of intrinsically non-low and non-semilow successivities in discrete linear orders. We show also (Theorem 3) that no recursive linear order has intrinsically <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w t t"> <mml:semantics> <mml:mrow> <mml:mi>w</mml:mi> <mml:mi>t</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">wtt</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -complete successivities.