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On partial disjunction properties of theories containing Peano arithmetic

  • Taishi Kurahashi
Article

Abstract

Let \(\varGamma \) be a class of formulas. We say that a theory T in classical logic has the \(\varGamma \)-disjunction property if for any \(\varGamma \) sentences \(\varphi \) and \(\psi \), either \(T \vdash \varphi \) or \(T \vdash \psi \) whenever \(T \vdash \varphi \vee \psi \). First, we characterize the \(\varGamma \)-disjunction property in terms of the notion of partial conservativity. Secondly, we prove a model theoretic characterization result for \(\varSigma _n\)-disjunction property. Thirdly, we investigate relationships between partial disjunction properties and several other properties of theories containing Peano arithmetic. Finally, we investigate unprovability of formalized partial disjunction properties.

Keywords

Partial disjunction properties Partial existence properties Formal arithmetic Gödel’s incompleteness theorems 

Mathematics Subject Classification

03F30 03F40 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Institute of Technology, Kisarazu CollegeKisarazuJapan

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