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
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-31933-4_10
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31932-7
Online ISBN: 978-3-642-31933-4
eBook Packages: Computer ScienceComputer Science (R0)