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

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-13-real forcing is applied to define a model of set theory in which there exists a good Δ1n of the reals, but there are no Δ1n−1s 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.

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

“…Given n ≥ 3 as in Theorem 1, a generic extension of L was defined in [1], in which (A n ) and (B n ) of Theorem 1 hold. This model involves a product forcing notion P ∈ L, defined in L, the constructible universe, as the finite support product of ℵ 1 -many forcing notions similar to Jensen's "minimal Π 1 2 singleton" forcing as in [5]. This forcing notion P depends on the index n in Theorems 1 and 2, of course, but we suppress this dependence, assuming that n ≥ 3 is fixed once and for all.…”

“…The following theorem, devoted to the problem of "effective" well-orderability of the real line R, was established in our recent paper [1].…”

“…Here ∆ 1 n are lightface projective classes. We refer to [1] for introductory material and references to earlier research on this topic. Generally we refer to Moschovakis' monograph [2] in matters of modern notation in descriptive set theory.…”

“…Generally we refer to Moschovakis' monograph [2] in matters of modern notation in descriptive set theory. We recall that a ∆ 1 n -good well-ordering is any ∆ 1 n well-ordering such that the class ∆ 1 n is closed under -bounded quantification, so that if P(y, x) is a binary ∆ 1 n relation on the reals, then the relations Q(z, x) := ∃ y x P(z, y) and R(z, x) := ∀ y x P(z, y)…”

“…belong to ∆ 1 n as well. Gödel [3] proved that the axiom of constructibility V = L implies the existence of a ∆ 1 2 -good well-ordering of R, and such well-orderings are behind some crucial applications of constructibility in set theory, see Section 5A in [2].…”