Advertisement

Archive for Mathematical Logic

, Volume 51, Issue 1–2, pp 99–121 | Cite as

The Dirac delta function in two settings of Reverse Mathematics

  • Sam Sanders
  • Keita Yokoyama
Article

Abstract

The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property \({\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}\) of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak König’s Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL0 as one of the ‘Big’ systems of Reverse Mathematics. In the context of ERNA’s Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA’s Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA’s Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.

Keywords

Nonstandard analysis Reverse mathematics WWKL ERNA Dirac delta 

Mathematics Subject Classification (2000)

03H05 03B30 03F35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brown D.K., Giusto M., Simpson S.G.: Vitali’s theorem and WWKL. Arch. Math. Log. 41(2), 191–206 (2002)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Chuaqui R., Suppes P.: Free-variable axiomatic foundations of infinitesimal analysis: a fragment with finitary consistency proof. J. Symb. Log. 60, 122–159 (1995)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Dirac P.A.M.: The Principles of Quantum Mechanics, 1st edn. Clarendon Press, Oxford (1927)Google Scholar
  4. 4.
    Friedman, H.: Some systems of second order arithmetic and their use. In: Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), vol. 1, pp. 235–242. Canadian Mathematical Congress, Montreal, Que (1975)Google Scholar
  5. 5.
    Friedman H.: Systems of second order arithmetic with restricted induction, I & II (Abstracts). J. Symb. Log. 41, 557–559 (1976)CrossRefGoogle Scholar
  6. 6.
    Friedman H.M., Simpson S.G., Smith R.L.: Countable algebra and set existence axioms. Ann. Pure Appl. Log. 25(2), 141–181 (1983)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Hrbacek K.: Relative set theory: internal view. J. Log. Anal. 1(8), 1–108 (2009)MathSciNetGoogle Scholar
  8. 8.
    Impens, C., Sanders, S.: The strength of nonstandard analysis. In: Van den Berg, I., Neves, V. (eds.) ERNA at Work. Springer, Wien New York Vienna (2007)Google Scholar
  9. 9.
    Impens C., Sanders S.: Transfer and a supremum principle for ERNA. J. Symb. Log. 73, 689–710 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Impens C., Sanders S.: Saturation and ∑2-transfer for ERNA. J. Symb. Log. 74, 901–913 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Péraire Y.: Théorie relative des ensembles internes. Osaka J. Math. 29(2), 267–297 (1992) (French)MATHMathSciNetGoogle Scholar
  12. 12.
    Sakamoto N., Yamazaki T.: Uniform versions of some axioms of second order arithmetic. MLQ Math. Log. Q. 50(6), 587–593 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Sanders S.: More infinity for a better finitism. Ann. Pure Appl. Log. 121, 1525–1540 (2010)CrossRefGoogle Scholar
  14. 14.
    Sanders S.: ERNA and Friedman’s Reverse Mathematics. J. Symb. Log. 76(2), 637–664 (2011)CrossRefMATHGoogle Scholar
  15. 15.
    Schwartz L.: Théorie des Distributions. Hermann, Paris (1951)MATHGoogle Scholar
  16. 16.
    Simpson S.G.: Which set existence axioms are needed to prove the Cauchy/Peano theorem for ordinary differential equations?. J. Symb. Log. 49(3), 783–802 (1984)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Simpson, S.G., Yokoyama, K.: A nonstandard counterpart of WWKL (preprint)Google Scholar
  18. 18.
    Simpson S.G.: Subsystems of Second Order Arithmetic, 2nd edn, Perspectives in Logic. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  19. 19.
    Simpson, S.G. (ed): Reverse mathematics 2001. Lecture Notes in Logic, vol. 21, Association for Symbolic Logic. La Jolla, CA (2005)Google Scholar
  20. 20.
    Sommer R., Suppes P.: Finite models of elementary recursive nonstandard analysis. Notas de la Sociedad Mathematica de Chile 15, 73–95 (1996)Google Scholar
  21. 21.
    Suppes P., Chuaqui R.: A finitarily consistent free-variable positive fragment of infinitesimal analysis. Proc. IXth Latin-Am. Symp. Math. Logic, Notas de Logica Mathematica 38, 1–59 (1993)MathSciNetGoogle Scholar
  22. 22.
    Yokoyama, K.: Standard and non-standard analysis in second order arithmetic. PhD thesis, Tohoku University, Sendai (2007). Available as Tohoku Mathematical Publications 34, 2009Google Scholar
  23. 23.
    Yu X., Simpson S.G.: Measure theory and weak König’s lemma. Arch. Math. Log. 30(3), 171–180 (1990)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGentBelgium
  2. 2.Mathematical InstituteTohoku UniversitySendaiJapan

Personalised recommendations