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A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part III—The Case of Second-Order PA

Author

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  • Vladimir Kanovei

    (Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia
    These authors contributed equally to this work.)

  • Vassily Lyubetsky

    (Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia
    These authors contributed equally to this work.)

Abstract

A model of set theory ZFC is defined in our recent research, in which, for a given n ≥ 3 , ( A n ) there exists a good lightface Δ n 1 well-ordering of the reals, but ( B n ) no well-orderings of the reals (not necessarily good) exist in the previous class Δ n − 1 1 . Therefore, the conjunction ( A n ) ∧ ( B n ) is consistent, modulo the consistency of ZFC itself. In this paper, we significantly clarify and strengthen this result. We prove the consistency of the conjunction ( A n ) ∧ ( B n ) for any given n ≥ 3 on the basis of the consistency of PA 2 , second-order Peano arithmetic, which is a much weaker assumption than the consistency of ZFC used in the earlier result. This is a new result that may lead to further progress in studies of the projective hierarchy.

Suggested Citation

  • Vladimir Kanovei & Vassily Lyubetsky, 2023. "A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part III—The Case of Second-Order PA," Mathematics, MDPI, vol. 11(15), pages 1-9, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:15:p:3294-:d:1203424
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    1. Vladimir Kanovei & Vassily Lyubetsky, 2023. "A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II," Mathematics, MDPI, vol. 11(11), pages 1-38, May.
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