Orders on computable rings

Mathematical Logic Quarterly 66 (2):126-135 (2020)
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Abstract

The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to over. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to over. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We then prove that “pre‐orders on rings not necessarily commutative extend to orders” is equivalent to.

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References found in this work

Reverse Mathematics and Fully Ordered Groups.Reed Solomon - 1998 - Notre Dame Journal of Formal Logic 39 (2):157-189.
Recursion theory and ordered groups.R. G. Downey & Stuart A. Kurtz - 1986 - Annals of Pure and Applied Logic 32:137-151.
Countable algebra and set existence axioms.Harvey M. Friedman - 1983 - Annals of Pure and Applied Logic 25 (2):141.
Π10 classes and orderable groups.Reed Solomon - 2002 - Annals of Pure and Applied Logic 115 (1-3):279-302.
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.

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