Countable OD sets of reals belong to the ground model

Archive for Mathematical Logic 57 (3-4):285-298 (2018)

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Abstract
It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension, that every countable ordinal-definable set of reals belongs to the ground universe. It is true in the Solovay collapse model that every non-empty OD countable set of sets of reals consists of \ elements.
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DOI 10.1007/s00153-017-0569-0
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References found in this work BETA

A Definable E 0 Class Containing No Definable Elements.Vladimir Kanovei & Vassily Lyubetsky - 2015 - Archive for Mathematical Logic 54 (5-6):711-723.
On Coding Uncountable Sets by Reals.Joan Bagaria & Vladimir Kanovei - 2010 - Mathematical Logic Quarterly 56 (4):409-424.
Unbounded and Dominating Reals in Hechler Extensions.Justin Palumbo - 2013 - Journal of Symbolic Logic 78 (1):275-289.

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