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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Characterizing $ E\sb{3}$ (the largest countable $ \Pi \sp{1}\sb{3}$ set)


Authors: David Guaspari and Leo Harrington
Journal: Proc. Amer. Math. Soc. 57 (1976), 127-129
MSC: Primary 02K30
DOI: https://doi.org/10.1090/S0002-9939-1976-0401476-2
MathSciNet review: 0401476
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Abstract: Assume projective determinacy. For any real $ \alpha $, let $ \lambda _3^\alpha = \sup \{ \xi \vert\xi $ is the type of a prewellordering of the reals which is $ \Delta _3^1$ in $ \alpha \} $. Then, $ {\mathcal{C}_3}$, the largest countable $ \Pi _3^1$ set of reals, is equal to $ \{ \alpha \vert\forall \beta (\lambda _3^\alpha \leqslant \lambda _3^\beta \Rightarrow \alpha $ is $ \Delta _3^1$ in $ \beta )\} $. This result, which is true for all odd levels and generalizes a previously known characterization of $ {\mathcal{C}_1}$, answers a question of Kechris.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0401476-2
Article copyright: © Copyright 1976 American Mathematical Society