On projective ordinals

Abstract

We study in this paper the projective ordinals , where = sup{ξ: ξis the length of a Δn1prewellordering 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 , for all even 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 δ11 = ℵ1 and the result of Martin that δ31 = ℵω+1 by proving that , where λ2n+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 δn1 with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles.