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)