id: 06005022 dt: j an: 06005022 au: Sanders, Sam; Yokoyama, Keita ti: The Dirac delta function in two settings of reverse mathematics. so: Arch. Math. Logic 51, No. 1-2, 99-121 (2012). py: 2012 pu: Springer-Verlag, Berlin la: EN cc: ut: reverse mathematics; Dirac delta; WWKL; nonstandard analysis ci: Zbl 1181.03001; Zbl 1141.03032; Zbl 1231.03059 li: doi:10.1007/s00153-011-0256-5 ab: The basic aim of the program of reverse mathematics (see [{\it 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: $\int_{-\infty}^{+\infty}f(x)δ(x)\,dx=f(0)$ 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 {\it P. Suppes} and {\it 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 $Π_1$-transfer principle, studied in [{\it C. Impens} and {\it S. Sanders,} J. Symb. Log. 73, No. 2, 689‒710 (2008; Zbl 1141.03032)]. This is in spite of the fact that $Π_1$-transfer is also equivalent to nonstandard versions of various statements classically equivalent to $\mathrm{WKL}_0$ rather than $\mathrm{WWKL}_0$ [{\it S. Sanders,} J. Symb. Log. 76, No. 2, 637‒664 (2011; Zbl 1231.03059)]. rv: Emil Jeřábek (Praha)