What evidence is there that \(2^{\land}65536\) is a natural number?
type: Item
instance of: scholarly article
title: What evidence is there that \(2^{\land}65536\) is a natural number?
zbMATH Open document ID: 0795.03088
author: David Isles
published in: Notre Dame Journal of Formal Logic
publication date: 1993-12-06T00:00:00Z
review text: At the outset the author warns that the title will appear strange to most mathematicians. By traditions that have evolved, actually, in most professions, in such cases the article will be treated with more or less benevolent neglect (as being sterile). Granted the statistical soundness of this rule of thumb, it turns out that in the present case closer attention leads, at least, to some salutary reminders. First, taken literally (but tempered by note 6), -- what is called on p. 472 -- the central assertion of the paper (a) (mis)represents Tarski's semantics: as interpreting derivations to refer to particular structures. In fact, Tarski interpreted (sets of) formulae, and thus all -- the infinitely many -- derivations from those formulae. (b) The assertion neglects -- not just a, but for an alert reader -- the principal (cl)aim of Hilbert's \(\varepsilon\)-notation, and especially his ideas on substitution methods. Here any (privileged) ranges of the bound variables in \(\varepsilon\)- terms are disregarded, and only the necessarily finitely many relations between such terms (in the finite proof considered) are respected. After more than 70 years of research on Hilbert's bright idea too much is known about blind spots in his expectations to dwell on the author's tinkering with his examples on pp. 467-478. (They do not even mention Hilbert's idea, let alone, the discovered blind spots.) A suitable salutary reminder here is two-fold. As far as logical (cl)aims are concerned the greater flexibility of considering models for single derivations -- than for all deductions from a (single) formula in (a) above -- has not found an effective use. But (later) experience in number theory provides a successful parallel; tacitly, for those with a suitable education: Hasse's local/global principle exploits the greater manageability of the finite fields \(\mathbb{Z}_ p\) (than of the ring \(\mathbb{Z}\) itself). Secondly, as to the -- to the author, dramatic, to those with (scientif…
Mathematics Subject Classification ID: 03F99
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 11U99
zbMATH DE Number: 440194
zbMATH Keywords: dinitary foundations
zbMATH Keywords: number theory
reviewed by: Georg Kreisel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093634481
OpenAlex ID: W2129658725
DOI: 10.1305/NDJFL/1093634481
description: scientific article; zbMATH DE number 440194