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

Abstract

A model of set theory ZFC is defined in our recent research, in which, for a given n≥3, (An) there exists a good lightface Δn1 well-ordering of the reals, but (Bn) no well-orderings of the reals (not necessarily good) exist in the previous class Δn−11. Therefore, the conjunction (An)∧(Bn) 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 (An)∧(Bn) for any given n≥3 on the basis of the consistency of PA2, 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