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The Dirac delta function in two settings of Reverse Mathematics

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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.

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References

  1. Brown D.K., Giusto M., Simpson S.G.: Vitali’s theorem and WWKL. Arch. Math. Log. 41(2), 191–206 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dirac P.A.M.: The Principles of Quantum Mechanics, 1st edn. Clarendon Press, Oxford (1927)

    Google Scholar 

  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)

  5. Friedman H.: Systems of second order arithmetic with restricted induction, I & II (Abstracts). J. Symb. Log. 41, 557–559 (1976)

    Article  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hrbacek K.: Relative set theory: internal view. J. Log. Anal. 1(8), 1–108 (2009)

    MathSciNet  Google Scholar 

  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. Impens C., Sanders S.: Transfer and a supremum principle for ERNA. J. Symb. Log. 73, 689–710 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. Impens C., Sanders S.: Saturation and ∑2-transfer for ERNA. J. Symb. Log. 74, 901–913 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Péraire Y.: Théorie relative des ensembles internes. Osaka J. Math. 29(2), 267–297 (1992) (French)

    MATH  MathSciNet  Google Scholar 

  12. Sakamoto N., Yamazaki T.: Uniform versions of some axioms of second order arithmetic. MLQ Math. Log. Q. 50(6), 587–593 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sanders S.: More infinity for a better finitism. Ann. Pure Appl. Log. 121, 1525–1540 (2010)

    Article  Google Scholar 

  14. Sanders S.: ERNA and Friedman’s Reverse Mathematics. J. Symb. Log. 76(2), 637–664 (2011)

    Article  MATH  Google Scholar 

  15. Schwartz L.: Théorie des Distributions. Hermann, Paris (1951)

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  17. Simpson, S.G., Yokoyama, K.: A nonstandard counterpart of WWKL (preprint)

  18. Simpson S.G.: Subsystems of Second Order Arithmetic, 2nd edn, Perspectives in Logic. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  19. Simpson, S.G. (ed): Reverse mathematics 2001. Lecture Notes in Logic, vol. 21, Association for Symbolic Logic. La Jolla, CA (2005)

  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. 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)

    MathSciNet  Google Scholar 

  22. Yokoyama, K.: Standard and non-standard analysis in second order arithmetic. PhD thesis, Tohoku University, Sendai (2007). Available as Tohoku Mathematical Publications 34, 2009

  23. Yu X., Simpson S.G.: Measure theory and weak König’s lemma. Arch. Math. Log. 30(3), 171–180 (1990)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    Sam Sanders

  2. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Keita Yokoyama

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  1. Sam Sanders
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  2. Keita Yokoyama
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Correspondence to Sam Sanders.

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Sanders, S., Yokoyama, K. The Dirac delta function in two settings of Reverse Mathematics. Arch. Math. Logic 51, 99–121 (2012). https://doi.org/10.1007/s00153-011-0256-5

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