A Model in Which Well-Orderings of the Reals First Appear at a Given Projective Level, Part II

Vladimir Kanovei () and Vassily Lyubetsky ()
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Vladimir Kanovei: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia
Vassily Lyubetsky: Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), 127051 Moscow, Russia

Mathematics, 2023, vol. 11, issue 11, 1-38

Abstract: We consider the problem of the existence of well-orderings of the reals, definable at a certain level of the projective hierarchy. This research is motivated by the modern development of descriptive set theory. Given n ? 3 , a finite support product of forcing notions similar to Jensen’s minimal- ? 3 1 -real forcing is applied to define a model of set theory in which there exists a good ? n 1 well-ordering of the reals, but there are no ? n ? 1 1 well-orderings of the reals (not necessarily good). We conclude that the existence of a good well-ordering of the reals at a certain level n ? 3 of the projective hierarchy is strictly weaker than the existence of a such well-ordering at the previous level n ? 1 . This is our first main result. We also demonstrate that this independence theorem can be obtained on the basis of the consistency of ZFC ? (that is, a version of ZFC without the Power Set axiom) plus ‘there exists the power set of ? ’, which is a much weaker assumption than the consistency of ZFC usually assumed in such independence results obtained by the forcing method. This is our second main result. Further reduction to the consistency of second-order Peano arithmetic PA 2 is discussed. These are new results in such a generality (with n ? 3 arbitrary), and valuable improvements upon earlier results. We expect that these results will lead to further advances in descriptive set theory of projective classes.

Keywords: forcing; projective well-orderings; projective classes; Jensen’s forcing (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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