Ordinal-definable families in Cohen, random, and collapse extensions

Abstract — v1

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 enumeration for every such family of size strictly below $\kappa$. The random bound is sharp. The proofs use coordinate amalgamation and small-index arguments. We also obtain arbitrary-rank descent for countable families after collapsing any infinite cardinal, with applications to choice in the relative Feferman–Levy model. In the full Solovay collapse extension of an arbitrary ZFC ground, every family definable from a real and ordinals that represents fewer than $2^{\mathfrak c}$ classes modulo null sets consists of measurable sets; the corresponding category assertion also holds. In the extensions of $L$ by $\omega_1$ Cohen or random reals, no model of $I\Delta_0$ of size at most $\aleph_1$, definable from ordinals and a real, has full binary-coded standard system. $\Pi^1_1\text{-}\mathsf{CA}_0$ proves a finite-fragment Borel obstruction to a full binary standard system for $I\Delta_0$. For regular uncountable $\kappa$, the generalized Cohen extension by \Add(\kappa,\Lambda)^L, $\Lambda>\kappa$, has no ambiently $\kappa^+$-saturated $I\Delta_0$ model in $\operatorname{HOD}_{H_{\kappa^+}}$, regardless of its size. Consequently every infinite regular $\kappa$ of $L$ admits a cofinality-preserving GCH extension with no $\mathrm{OD}_{\mathcal P(\kappa)}$ saturated arithmetic presentation of size $\kappa^+$. ZFC also proves that every singular strong limit $\kappa$ admits an OD $\kappa^+$-saturated elementary extension of $\mathbb N$ of size $2^\kappa$, hence of size $\kappa^+$ under GCH.