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On the Reals Which Cannot Be Random

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Book cover Computability and Complexity

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10010))

Abstract

We investigate which reals can never be L-random. That is to give a description of the reals which are always belong to some \(L[\lambda ]\)-null set for any continuous measure \(\lambda \). Among other things, we prove that \(NCR_L\) is an L-cofinal subset of \(Q_3\) under \(ZFC+PD\).

Yu was partially supported by National Natural Science Fund of China grant 11322112 and Humboldt foundation. Both authors thank Professor Ambos-Spies from Heidelberg University and Professor Schindler from University of Münster for their hospitality.

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Notes

  1. 1.

    The function j was introduced in [14]. We use \(\kappa \) to denote \(\aleph _1\) in V to avoid any confusion.

  2. 2.

    This is slightly different from [20, Definition 6.11]. In our situation, b needs not be wellfounded.

  3. 3.

    i.e. \(E_{\alpha }^{\mathcal {T}}\) might be empty, in which case we do nothing and put \(\mathcal {M}_{\alpha +1}^{\mathcal {T}} = \mathcal {M}_{\alpha }^{\mathcal {T}}\), and similarly for \(\mathcal {U}\).

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Authors and Affiliations

  1. Institute of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, 210093, People’s Republic of China

    Liang Yu

  2. Institut für Mathematische Logik, Universität Münster, Münster, Germany

    Yizheng Zhu

Authors
  1. Liang Yu
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  2. Yizheng Zhu
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Correspondence to Liang Yu .

Editor information

Editors and Affiliations

  1. Victoria University of Wellington, Wellington, New Zealand

    Adam Day

  2. University of Bergen, Bergen, Norway

    Michael Fellows

  3. Victoria University of Wellington, Wellington, New Zealand

    Noam Greenberg

  4. University of Auckland, Auckland, New Zealand

    Bakhadyr Khoussainov

  5. Massey University, Auckland, New Zealand

    Alexander Melnikov

  6. University of Bergen, Bergen, Norway

    Frances Rosamond

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Yu, L., Zhu, Y. (2017). On the Reals Which Cannot Be Random. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_36

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  • DOI: https://doi.org/10.1007/978-3-319-50062-1_36

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50061-4

  • Online ISBN: 978-3-319-50062-1

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