Jean van Heijenoort's contributions to proof theory and its history
type: Item
instance of: scholarly article
title: Jean van Heijenoort's contributions to proof theory and its history
zbMATH Open document ID: 1272.03001
author: Irving H. Anellis
published in: Logica Universalis
publication date: 2013-03-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rml/1204834854
review text: Every logic student knows \textit{J. van Heijenoort} through [From Frege to Gödel. Cambridge, Mass.: Harvard University Press (1967; Zbl 0183.00601)]. But, his work on model-theoretic proof theory, the falsifiability tree method in particular, is not widely known and is mostly unpublished, according to the author. In this article, he presents a survey of this aspect and a complete list of van Heijenoort's unpublished writings. The author provides a wide scope of background material, beginning with Frege's \textit{Begriffsschrift} and Hilbert's \textit{Beweistheorie} -- their aims and differences, etc. Then comes Löwenheim-Skolem's work on non-characterizability and Herbrand's work as a proof-theoretic counterpart to this. Indeed, Herbrand expansion can be viewed in the context of treating quantifiers as infinitary conjunctions and disjunctions. (Van Heijenoort has done a lot of work on Herbrand. He edited [\textit{J. Herbrand}, Écrits logiques. Paris: Presses Universitaires de France (1968; Zbl 0194.30305)], a complete work). The author gives `a suspense story', so to speak, about ``Herbrand errors'' -- who discovered what, when, how to correct, who said what to whom, when, and so forth. About the tree method, the author begins with a ``brief history'' including C. L. Dodgson (a.k.a. Lewis Carroll), Beth, and Smullyan. After giving a short explanation of the method and samples of theorems, he reports that van Heijenoort proved its consistency and completeness for classical, intuitionistic, modal, 3-valued, and higher-order logics, through many unpublished papers. This article carries an extensive bibliography (208 items). Internal cross reference is wrong here and there; for instance, [86] is not a translation of [96], but of [85].
Mathematics Subject Classification ID: 03-03
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 03C30
zbMATH DE Number: 6145483
zbMATH Keywords: van Heijenoort's work
zbMATH Keywords: falsifiability tree method
zbMATH Keywords: theories of quantification
zbMATH Keywords: tableaux methods
zbMATH Keywords: history of logic
zbMATH Keywords: model-theoretic proof theory
MaRDI profile type: Publication
OpenAlex ID: W1986344372
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