Abstract
One of the earliest purposes of computability theory was the study of logical systems and theories. We consider theories in a computable language: one which is countable, and whose function, relation, and constant symbols and their arities are effectively given. We also assume that languages come equipped with an effective coding for formulas and sentences in the languages, i.e., a Gödel numbering, and identify sets of formulas with the corresponding set of Gödel numbers. We can then speak of the Turing degree of a theory in a computable language
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© 2016 Springer-Verlag Berlin Heidelberg
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Soare, R.I. (2016). Peano Arithmetic and \(\Pi_1^0\)-Classes. In: Turing Computability. Theory and Applications of Computability. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31933-4_10
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DOI: https://doi.org/10.1007/978-3-642-31933-4_10
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31932-7
Online ISBN: 978-3-642-31933-4
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