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

• Published in: Mathematics (Basel) Vol. 11; no. 11; p. 2517
• Main Authors: Kanovei, Vladimir, Lyubetsky, Vassily
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 30-05-2023
• Subjects: Concrete construction; Consistency; forcing; Geometry, Projective; Hypotheses; Jensen’s forcing; Mathematics; Ordered sets; projective classes; projective well-orderings; Set theory; Tests, problems and exercises; Russia
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11112517

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-Δ31-real forcing is applied to define a model of set theory in which there exists a good Δn1 well-ordering of the reals, but there are no Δn−11 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 ω’, 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 PA2 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.