Fixed points in Peano arithmetic with ordinals
type: Item
instance of: scholarly article
title: Fixed points in Peano arithmetic with ordinals
zbMATH Open document ID: 0776.03027
DOI: 10.1016/0168-0072(93)90039-G
author: Gerhard Jäger
published in: Annals of Pure and Applied Logic
publication date: 1993-05-16T00:00:00Z
review text: By modern proof theory this paper shows that some basic fixed point theory of finitely iterated approximations is conservative over Peano arithmetic PA. The finite iteration is hidden in the \(\Sigma^ \Omega\)- reflection axioms and vanishes when \(\leq_{0n}\)-relativization is substituted for \(<_{0n}\)-relativization: then by chancelling all ordinal quantifiers and replacing throughout \(P^ \alpha_ A\), \(<_{0n}\) and ordinal variables \(\alpha,\beta,\dots\) by \(P^ 0_ A\equiv\lambda xA(\bot,x)\), \(<_{\mathbb{N}}\), 0 respectively \textit{all} the systems considered become conservative over PA. The question now is how far conservativity over PA holds when ordinal terms \(\omega,\dots\) and appropriate axioms for them are added.
Mathematics Subject Classification ID: 03F03
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03F05
zbMATH DE Number: 169665
zbMATH Keywords: fixed point theory of finitely iterated approximations
zbMATH Keywords: Peano arithmetic
zbMATH Keywords: finite iteration
zbMATH Keywords: conservativity over PA
reviewed by: Horst Luckhardt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 169665