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

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 Δ n 1 well-ordering of the reals, but (B n) no well-orderings of the reals (not necessarily good) exist in the previous class Δ 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. [ABSTRACT FROM AUTHOR]