P 0 1 \pi^0_1 -presentations of algebras

Archive for Mathematical Logic 45 (6):769-781 (2006)
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Abstract

In this paper we study the question as to which computable algebras are isomorphic to non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-presentations. On the other hand, many of this structures fail to have non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma_{1}^{0}$$\end{document}-presentation.

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Hiearchies of Boolean algebras.Lawrence Feiner - 1970 - Journal of Symbolic Logic 35 (3):365-374.
Stability among r.e. quotient algebras.John Love - 1993 - Annals of Pure and Applied Logic 59 (1):55-63.

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