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On Projective Ordinals

Kechris, Alexander S. (1974) On Projective Ordinals. Journal of Symbolic Logic, 39 (2). pp. 269-282. ISSN 0022-4812.

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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 < δ^1_n 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 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.

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Additional Information:© 1974 Association for Symbolic Logic. Received August 5, 1972. The results in this paper are included in the author's doctoral dissertation submitted to the University of California, Los Angeles, in June 1972. The author would like to express his sincerest thanks to his thesis advisor, Professor Yiannis N. Moschovakis, both for creating his interest in descriptive set theory and for his guidance and encouragement. The preparation of the paper was partially supported by NSF grant GP-27964.
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MathSciNet ReviewMR0398833
Zentralblatt MATH Identifier0292.02056
Issue or Number:2
Record Number:CaltechAUTHORS:20130529-103426984
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Official Citation:On Projective Ordinals Alexander S. Kechris; 269-282
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:38704
Deposited By: Ruth Sustaita
Deposited On:30 May 2013 15:39
Last Modified:03 Oct 2019 04:59

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