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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer-Verlag GmbH Germany, part of Springer Nature 2020
- ISSN
- 0933-5846
- eISSN
- 1432-0665
- DOI
- 10.1007/s00153-020-00732-9
- Publisher site
- See Article on Publisher Site
Abstract
In this note we study several topics related to the schema of local reflection Rfn(T)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf {Rfn} (T)$$\end{document} and its partial and relativized variants. Firstly, we introduce the principle of uniform reflection with Σn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varSigma _n$$\end{document}-definable parameters, establish its relationship with relativized local reflection principles and corresponding versions of induction with definable parameters. Using this schema we give a new model-theoretic proof of the Σn+2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varSigma _{n+2}$$\end{document}-conservativity of uniform Σn+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varSigma _{n+1}$$\end{document}-reflection over relativized local Σn+1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varSigma _{n+1}$$\end{document}-reflection. We also study the proof-theoretic strength of Feferman’s theorem, i.e., the assertion of 1-provability in S of the local reflection schema Rfn(S)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf {Rfn} (S)$$\end{document}, and its generalized versions. We relate this assertion to the uniform Σ2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varSigma _2$$\end{document}-reflection schema and, in particular, obtain an alternative axiomatization of IΣ1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textsf {I} \varSigma _1$$\end{document}.
Journal
Archive for Mathematical Logic – Springer Journals
Published: Mar 30, 2020