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

Vladimir Kanovei () and Vassily Lyubetsky ()
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Vladimir Kanovei: Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia
Vassily Lyubetsky: Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia

Mathematics, 2023, vol. 11, issue 15, 1-9

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 ? n 1 well-ordering of the reals, but ( B n ) no well-orderings of the reals (not necessarily good) exist in the previous class ? 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.

Keywords: forcing; projective well-orderings; projective classes; Peano arithmetic (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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