From Dedekind to Zermelo versus Peano to Gödel
type: Item
instance of: scholarly article
title: From Dedekind to Zermelo versus Peano to Gödel
zbMATH Open document ID: 1395.01020
author: Reinhard Kahle
published in: Mathematische Semesterberichte
publication date: 2018-02-06T00:00:00Z
review text: The author compares Dedekind's and Peano's approach to the characterization of the natural numbers. He points out the problems emerging from the set theoretical frame of Dedekind's theory and states also that the mathematical arguments don't depend on that problematic set theoretical frame. After sketching Peano's axiomatic approach, the author refers to the decisive difference: Peano's axioms can be restricted to the first-order logic but Dedekind's theory of natural numbers can't. In the following, the author discusses the relation between the first-order logic and Gödel's incompleteness theorem, non-standard models of the (first-order) Peano arithmetic, Dedekind's refusal of the possibility to characterize his approach in first-order logic, and some works of Zermelo in which he followed the ``tradition of Dedekind''.
reviewed by: Karl Heinz Schlote
Mathematics Subject Classification ID: 01A55
zbMATH DE Number: 6835167
zbMATH Keywords: characterization of natural numbers
zbMATH Keywords: non-standard models of the (first-order) Peano arithmetic
MaRDI profile type: Publication
full work available at URL: s00591-017-0193-4
OpenAlex ID: W2675760575
cites work: Completeness and Categoricity. Part I: Nineteenth-century Axiomatics to Twentieth-century Metalogic
cites work: Q3414548
cites work: The induction principle
cites work: Q5573940
cites work: Gödel's incompleteness theorems. On the occasion of Kurt Gödel's 100th anniversary on April 28, 2006
cites work: Q4001935
cites work: Q3866062
cites work: Q5275447
cites work: Logic in the 1930s: type theory and model theory
cites work: Q5236417
cites work: Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzählbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen
cites work: On the limit of the finite. The Hilbert programme in the context of formalism and finitism
cites work: Zermelo and the Skolem Paradox
cites work: On the sources of my book \textit{Moderne Algebra}
cites work: Q4357731
cites work: Collected works. Volume I: Set theory, miscellanea. Edited by Heinz-Dieter Ebbinghaus and Akihiro Kanamori
DOI: 10.1007/S00591-017-0193-4
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description: scientific article; zbMATH DE number 6835167