math.LO — Logic

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

Contributed by Elliot Glazer

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.