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 \(\text {OD}\) elements.
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The study was supported by the Russian Scientific Fund (Grant No. 14-50-00150). The work of Vladimir Kanovei was also supported by RFBR Grant 17-01-00705 and by ESI at Vienna, during the visit in December 2016.
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Kanovei, V., Lyubetsky, V. Countable OD sets of reals belong to the ground model. Arch. Math. Logic 57, 285–298 (2018). https://doi.org/10.1007/s00153-017-0569-0
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DOI: https://doi.org/10.1007/s00153-017-0569-0