The Dirac delta function in two settings of reverse mathematics
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 of Dirac delta as above is over ERNA equivalent to the \(\Pi_1\)-transfer principle, studied in [\textit{C. Impens} and \textit{S. Sanders,} J.\ Symb.\ Log.\ 73, No. 2, 689--710 (2008; Zbl 1141.03032)]. This is in spite of the fact that \(\Pi_1\)-transfer is also equivalent to nonstandard versions of various statements classically equivalent to \(\mathrm{WKL}_0\) rather than \(\mathrm{WWKL}_0\) [\textit{S. Sanders,} J.\ Symb.\ Log.\ 76, No. 2, 637--664 (2011; Zbl 1231.03059)].
scientific article; zbMATH DE number 6005022
scientific article