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Flexibility in Fragments of Peano Arithmetic

Chapter in book
Authors Rasmus Blanck
Published in Studies in Weak Arithmetics, Volume 3/ eds. Patrick Cégielski, Ali Enayat, Roman Kossak
Pages 1-20
ISBN 978-1-57586-953-7
Publisher CSLI Publications
Place of publication Stanford
Publication year 2016
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 1-20
Language en
Keywords arithmetic, incompleteness, flexible formulae, independent formulae
Subject categories Logic, Mathematical logic

Abstract

This paper concerns flexible formulae of arithmetic: formulae whose "extensions as sets are left undetermined by the formal system". Formally, this means that a formula γ(x) is flexible for a class of formulae X if, for each ξ(x) ϵ X, the theory T + ∀x(γ(x) ↔ ξ(x)) is consistent. We compare different kinds of flexibility results, and gauge the amount of induction needed for their proofs. By formalising these arguments, we are also able to derive their model-theoretic counterparts, assuring the existence of certain kinds of end-extensions of models of fragments of Peano arithmetic.

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