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Σ2-induction and the construction of a high degree

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Recursion Theory Week

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1432))

Abstract

We show that the subsystem of Peano Arithmetic containing only induction for Σ2 formulas suffices to prove the existence of a recursively enumerable set of high, incomplete degree. By a result of Mytilinaios and Slaman, bounding for Σ2 formulas does not suffice to prove that such a set exists. In contrast, by a result of Groszek and Slaman, induction for Σ1 formulas suffices to carry out any Friedberg-style finite injury priority argument.

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References

  1. Enderton, H. B., A Mathematical Introduction to Logic, Academic Press, New York, 1972.

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Author information

Authors and Affiliations

  1. Dept. of Math. and C.S., Dartmouth College, Bradley Hall, 03755, Hanover, NH, USA

    Marcia Groszek

  2. Dept. of Math., University of Crete, P.O. Box 470, 71409, Iraklion, Greece

    Michael Mytilinaios

Authors
  1. Marcia Groszek
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  2. Michael Mytilinaios
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Editor information

Klaus Ambos-Spies Gert H. Müller Gerald E. Sacks

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© 1990 Springer-Verlag

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Groszek, M., Mytilinaios, M. (1990). Σ2-induction and the construction of a high degree. In: Ambos-Spies, K., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1432. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086119

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  • DOI: https://doi.org/10.1007/BFb0086119

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52772-5

  • Online ISBN: 978-3-540-47142-4

  • eBook Packages: Springer Book Archive

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