Ordinal-definable families in Cohen, random, and collapse extensions
We study ordinal-definable families of sets of arbitrary rank in Cohen, random, and collapse extensions. Over $L$, countable OD families have OD enumerations after adding one Cohen or one random real. A single generalized Cohen subset gives the corresponding sharp theorem at every regular uncountable cardinal. When $\kappa$ is singular strong limit and has uncountable cofinality, adding $\kappa$ Cohen reals makes every member of a short-parameter definable family of size at most $\kappa$ definable from a short parameter. Adding $\kappa$ random reals gives a single short-parameter definable…
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