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[Gentzen’s consistency proof of 1936 is explained in full detail. The chapter begins with the method of assigning ordinal numbers in Gentzen’s representation to derivations. The second part introduces reduction steps for derivations whose endsequent is not in endform and shows, at the same time, that every reduction lowers the ordinal number of the given derivation. The proof consists in analysing cases according to the last inference rule that is used in the derivation to reduce. We obtain the consistency of arithmetic in this way as the derivation of →0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \rightarrow 0=1$$\end{document} would be reduced infinitely many times. Hence, we would construct an infinite decreasing sequence of ordinal numbers which is not possible.]
Published: Oct 23, 2013
Keywords: Consistency of arithmetic; Consistency proof of 1936; Gentzen’s consistency proofs; Reduction steps; Reduction steps for derivations
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