Peano Arithmetic and \(\Pi_1^0\)-Classes

  • Robert I. Soare
Part of the Theory and Applications of Computability book series (THEOAPPLCOM)


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


Computable Function Computability Theory Equivalent Property Axiomatizable Theory Peano Arithmetic 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Robert I. Soare
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

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