Notre Dame Journal of Formal Logic

A Nonstandard Counterpart of WWKL

Stephen G. Simpson and Keita Yokoyama


In this paper, we introduce a system of nonstandard second-order arithmetic $\mathsf{ns}$-$\mathsf{WWKL_0}$ which consists of $\mathsf{ns}$-$\mathsf{BASIC}$ plus Loeb measure property. Then we show that $\mathsf{ns}$-$\mathsf{WWKL_0}$ is a conservative extension of $\mathsf{WWKL_0}$ and we do Reverse Mathematics for this system.

Article information

Notre Dame J. Formal Logic, Volume 52, Number 3 (2011), 229-243.

First available in Project Euclid: 28 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]

second-order arithmetic nonstandard analysis reverse mathematics weak weak Koeonig's lemma Martin-Loef random


Simpson, Stephen G.; Yokoyama, Keita. A Nonstandard Counterpart of WWKL. Notre Dame J. Formal Logic 52 (2011), no. 3, 229--243. doi:10.1215/00294527-1435429.

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  • [1] Brown, D. K., M. Giusto, and S. G. Simpson, "Vitali's theorem and WWKL", Archive for Mathematical Logic, vol. 41 (2002), pp. 191–206.
  • [2] Downey, R., D. R. Hirschfeldt, J. S. Miller, and A. Nies, "Relativizing Chaitin's halting probability", Journal of Mathematical Logic, vol. 5 (2005), pp. 167–92.
  • [3] Horihata, Y., and K. Yokoyama, "Nonstandard second-order arithmetic and Riemann's mapping theorem", in preparation.
  • [4] Keisler, H. J., "Nonstandard arithmetic and reverse mathematics", Bulletin of Symbolic Logic, vol. 12 (2006), pp. 100–125.
  • [5] Reimann, J., and T. A. Slaman, "Measures and their random reals", Available online at, 2008.
  • [6] Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic. Springer-Verlag, 1999. Second Edition, Perspectives in Logic, Association for Symbolic Logic, Cambridge University Press, 2009.
  • [7] Simpson, S. G., "Almost everywhere domination and superhighness", Mathematical Logic Quarterly, vol. 53 (2007), pp. 462–82.
  • [8] Simpson, S. G., K. Tanaka, and T. Yamazaki, "Some conservation results on weak König's lemma", Annals of Pure and Applied Logic, vol. 118 (2002), pp. 87–114.
  • [9] Tanaka, K., "The self-embedding theorem of WKL"$_0$ and a non-standard method. Fifth Asian Logic Conference (Singapore, 1993), Annals of Pure and Applied Logic, vol. 84 (1997), pp. 41–49.
  • [10] Yokoyama, K., Standard and Non-standard Analysis in Second Order Arithmetic, Ph.D. thesis, Tohoku University, 2007. Available as Tohoku Mathematical Publications 34, 2009.
  • [11] Yokoyama, K., "Formalizing non-standard arguments in second-order arithmetic", The Journal of Symbolic Logic, vol. 75 (2010), pp. 1199–1210.
  • [12] Yu, X., "Lebesgue convergence theorems and reverse mathematics", Mathematical Logic Quarterly, vol. 40 (1994), pp. 1–13.
  • [13] Yu, X., and S. G. Simpson, "Measure theory and weak König's lemma", Archive for Mathematical Logic, vol. 30 (1990), pp. 171–80.