The Dirac delta function in two settings of reverse mathematics
type: Item
instance of: scholarly article
title: The Dirac delta function in two settings of reverse mathematics
zbMATH Open document ID: 1248.03023
author: Sam Sanders
author: Keita Yokoyama
published in: Archive for Mathematical Logic
publication date: 2012-02-10T00:00:00Z
full work available at URL: https://biblio.ugent.be/publication/1938292
review text: The basic aim of the program of reverse mathematics (see [\textit{S. Simpson}, Subsystems of second order arithmetic. 2nd ed. Cambridge: Cambridge University Press; Urbana, IL: Association for Symbolic Logic (ASL) (2009; Zbl 1181.03001)]), initiated by H. Friedman, is to determine the weakest axiom systems needed to prove particular mathematical statements. Quite often the statement turns out to be in fact equivalent (over a weak base theory) to the axiom system, and even more strikingly, classical reverse mathematics identifies five subsystems of second-order arithmetic which are enough to classify the bulk of the mathematical results considered; the weakest of these systems are \(\mathrm{RCA}_0\) and \(\mathrm{WKL}_0\). However, it turns out that some natural statements (mainly from calculus) are equivalent to another system, \(\mathrm{WWKL}_0\), intermediate in strength between \(\mathrm{RCA}_0\) and \(\mathrm{WKL}_0\). The purpose of this paper is to provide further examples of this phenomenon. First, it is shown that \(\mathrm{WWKL}_0\) is equivalent to the statement that every bounded continuous function on \([0,1]\) is Riemann integrable (whereas the statements that every continuous function on \([0,1]\) is bounded, or that every such function is Riemann integrable, are known to be equivalent to \(\mathrm{WKL}_0\)). However, the main focus of the paper is on the Dirac delta ``function''. The authors prove that \(\mathrm{WWKL}_0\) is equivalent to a natural formalization of the basic property of Dirac delta: \(\int_{-\infty}^{+\infty}f(x)\delta(x)\,dx=f(0)\) for any continuous function \(f\) with bounded support. In the second part of the paper, the authors consider reverse mathematics in the setting of nonstandard analysis, namely over the theory ERNA of \textit{P. Suppes} and \textit{R. Sommer} [``Finite models of elementary recursive nonstandard analysis'', Notas Soc. Mat. Chile 15, 73--95 (1996)]. It is shown that a suitable version of the basic property o…
reviewed by: Emil Jeřábek
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 26A42
Mathematics Subject Classification ID: 26E35
zbMATH DE Number: 6005022
zbMATH Keywords: reverse mathematics
zbMATH Keywords: Dirac delta
zbMATH Keywords: WWKL
zbMATH Keywords: nonstandard analysis
MaRDI profile type: Publication
OpenAlex ID: W2028509288
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DOI: 10.1007/S00153-011-0256-5
description: scientific article; zbMATH DE number 6005022