Π10 classes and orderable groups

Annals of Pure and Applied Logic 115 (1-3):279-302 (2002)

Abstract
It is known that the spaces of orders on orderable computable fields can represent all Π10 classes up to Turing degree. We show that the spaces of orders on orderable computable abelian and nilpotent groups cannot represent Π10 classes in even a weak manner. Next, we consider presentations of ordered abelian groups, and we show that there is a computable ordered abelian group for which no computable presentation admits a computable set of representatives for its Archimedean classes
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DOI 10.1016/s0168-0072(01)00097-5
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References found in this work BETA

Subsystems of Second-Order Arithmetic.Stephen G. Simpson - 2004 - Studia Logica 77 (1):129-129.
Computability Theory and Linear Orders.Rod Downey - 1998 - In I͡Uriĭ Leonidovich Ershov (ed.), Handbook of Recursive Mathematics. Elsevier. pp. 138--823.
Reverse Mathematics and Fully Ordered Groups.Reed Solomon - 1998 - Notre Dame Journal of Formal Logic 39 (2):157-189.
Hiearchies of Boolean Algebras.Lawrence Feiner - 1970 - Journal of Symbolic Logic 35 (3):365-374.
Recursion Theory and Ordered Groups.R. G. Downey & Stuart A. Kurtz - 1986 - Annals of Pure and Applied Logic 32 (2):137-151.

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Citations of this work BETA

Degrees of Orders on Torsion-Free Abelian Groups.Asher M. Kach, Karen Lange & Reed Solomon - 2013 - Annals of Pure and Applied Logic 164 (7-8):822-836.
Left-orderable computable groups.Matthew Harrison-Trainor - 2018 - Journal of Symbolic Logic 83 (1):237-255.

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