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

Abstract

The problem of the existence of analytically definable well-orderings at a given level of the projective hierarchy is considered. This problem is important as a part of the general problem of the study of the projective hierarchy in the ongoing development of descriptive set theory. We make use of a finite support product of the Jensen-type forcing notions to define a model of set theory ZFC in which, for a given n>2, there exists a good Δn1 well-ordering of the reals but there are no such well-orderings in the class Δn−11. Therefore the existence of a well-ordering of the reals at a certain level n>2 of the projective hierarchy does not imply the existence of such a well-ordering at the previous level n−1. This is a new result in such a generality (with n>2 arbitrary), and it may lead to further progress in studies of the projective hierarchy.