# Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive OrdinalsOrdinal Numbers

Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of... [Gentzen’s representation of ordinal numbers less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}, which is based on decimal numbers syntactically, is given. Systems σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document} which provide information about the ordering of the numbers in Gentzen’s notation are introduced. The relationship between Gentzen’s representation and Cantor normal form is analysed with the help of systems σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document} and a recursive algorithm for translating Gentzen’s notation to Cantor normal form is defined. Furthermore, correctness of this algorithm is proved and some easy examples of how it works are presented.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of Constructive OrdinalsOrdinal Numbers

Part of the SpringerBriefs in Philosophy Book Series
12 pages      /lp/springer-journals/where-is-the-g-del-point-hiding-gentzen-s-consistency-proof-of-1936-l9TjZlqYlb
Publisher
Springer International Publishing
ISBN
978-3-319-02170-6
Pages
29 –40
DOI
10.1007/978-3-319-02171-3_3
Publisher site
See Chapter on Publisher Site

### Abstract

[Gentzen’s representation of ordinal numbers less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}, which is based on decimal numbers syntactically, is given. Systems σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document} which provide information about the ordering of the numbers in Gentzen’s notation are introduced. The relationship between Gentzen’s representation and Cantor normal form is analysed with the help of systems σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document} and a recursive algorithm for translating Gentzen’s notation to Cantor normal form is defined. Furthermore, correctness of this algorithm is proved and some easy examples of how it works are presented.]

Published: Oct 23, 2013

Keywords: Ordinals; Constructive ordinals; Ordinal numbers; Ordinal numbers less than ε0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document}; Gentzen’s representation of ordinal numbers; Cantor normal form; Well-ordering; Transfinite induction