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

Where is the Gödel-point hiding: Gentzen’s Consistency Proof of 1936 and His Representation of... [This chapter deals with brief history of Gentzen’s consistency proofs and his work on proof systems which were actually byproducts of his ambition to prove the consistency of arithmetic. There is a plan in his handwritten thesis manuscript according to which he aimed to prove the consistency with the help of normalization for natural deduction. As this did not work, he developed a special semantic explanation of correctness in arithmetic. The proof based on this explanation was criticized, so Gentzen returned to an earlier idea of transfinite induction. The last proof shows directly that although the transfinite induction up to 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} can be formalized in arithmetic, it cannot be proved there. Further, this chapter deals with non-technical parts of his 1936 article.] 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 OrdinalsIntroduction

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Publisher
Springer International Publishing
Copyright
© The Author(s) 2014
ISBN
978-3-319-02170-6
Pages
1 –9
DOI
10.1007/978-3-319-02171-3_1
Publisher site
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Abstract

[This chapter deals with brief history of Gentzen’s consistency proofs and his work on proof systems which were actually byproducts of his ambition to prove the consistency of arithmetic. There is a plan in his handwritten thesis manuscript according to which he aimed to prove the consistency with the help of normalization for natural deduction. As this did not work, he developed a special semantic explanation of correctness in arithmetic. The proof based on this explanation was criticized, so Gentzen returned to an earlier idea of transfinite induction. The last proof shows directly that although the transfinite induction up to 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} can be formalized in arithmetic, it cannot be proved there. Further, this chapter deals with non-technical parts of his 1936 article.]

Published: Oct 23, 2013

Keywords: Gentzen; Gentzen’s thesis; Proof systems; Consistency proofs; Consistency proofs of arithmetic; History of proof systems; History of consistency proofs; Consistency of arithmetic; Hilbert’s program

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