On Projective Ordinals

We study in this paper the projective ordinals δ^1_n, where δ^1_n = sup{ξ: ξ is the length of ɑ Δ^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the "definable length" of the continuum. We prove first in §2 that projective determinacy implies δ^1_n 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in §3) the classical fact that δ^1_1 ℵ_l and the result of Martin that δ^1_3 = ℵ_(ω + 1) by proving that δ^1_(n2+1) = λ^+_(2n+1), where λ_(2n+1) is a cardinal of cofinality ω. Finally we discuss in §4 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that ∀α (α^# exists) implies that every δ^1_n with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles

Countable ordinals and the analytical hierarchy, I

The following results are proved, using the axiom of Projective Determinacy: (i) For n ≥ 1, every II(1/2n+1) set of countable ordinals contains a Δ(1/2n+1) ordinal, (ii) For n ≥ 1, the set of reals Δ(1/2n) in an ordinal is equal to the largest countable Σ(1/2n) set and (iii) Every real is Δ(1/n) inside some transitive model of set theory if and only if n ≥ 4

The bounded proper forcing axiom and well orderings of the reals

We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(ω_1) which is Δ_1 definable with parameter a subset of ω_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of ω_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the Härtig quantifier is not lightface projective

Classes of structures with no intermediate isomorphism problems

We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle on a cone