On projective ordinals

Alexander S. Kechris
1974 Journal of Symbolic Logic (JSL)  
We study in this paper the projective ordinals , where = sup{ξ: ξ is the length of a Δn 1 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 , 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 δ1 1 = ℵ1 and the result of
more » ... and the result of Martin that δ3 1 = ℵ ω+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 δn 1 with n ≥ 3 is a fixed point of the increasing enumeration of the uniform indiscernibles.
doi:10.2307/2272639 fatcat:qd3i5225yrdxhhlbzugn3y25xm