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

Primarily AI-generated textHuman understanding: some partsmath.LO — Logic

Contributed by Elliot Glazer ↗

SubmitterElliot Glazer

Version 1 / Sep 30, 2026 / CC BY 4.0

Abstract

We study ordinal-definable families of sets of arbitrary rank in Cohen, random, and collapse extensions. Over LL, 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 κ\kappa is singular strong limit and has uncountable cofinality, adding κ\kappa Cohen reals makes every member of a short-parameter definable family of size at most κ\kappa definable from a short parameter. Adding κ\kappa random reals gives a single short-parameter definable enumeration for every such family of size strictly below κ\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 2c2^{\mathfrak c} classes modulo null sets consists of measurable sets; the corresponding category assertion also holds. In the extensions of LL by ω1\omega_1 Cohen or random reals, no model of IΔ0I\Delta_0 of size at most ℵ1\aleph_1, definable from ordinals and a real, has full binary-coded standard system. Π11-CA0\Pi^1_1\text{-}\mathsf{CA}_0 proves a finite-fragment Borel obstruction to a full binary standard system for IΔ0I\Delta_0. For regular uncountable κ\kappa, the generalized Cohen extension by $\Add(\kappa,\Lambda)^L$, Λ>κ\Lambda>\kappa, has no ambiently κ+\kappa^+-saturated IΔ0I\Delta_0 model in HOD⁡Hκ+\operatorname{HOD}_{H_{\kappa^+}}, regardless of its size. Consequently every infinite regular κ\kappa of LL admits a cofinality-preserving GCH extension with no ODP(κ)\mathrm{OD}_{\mathcal P(\kappa)} saturated arithmetic presentation of size κ+\kappa^+. ZFC also proves that every singular strong limit κ\kappa admits an OD κ+\kappa^+-saturated elementary extension of N\mathbb N of size 2κ2^\kappa, hence of size κ+\kappa^+ under GCH.

Provenance statement

The applications of the definability results to the Feferman-Levy model and definability of models of arithmetic were discovered by me, and the latter has been verified end-to-end in Lean (see the linked Palomar repo). The Boolean algebra analyses remain unchecked.
FormalizationsPalomar

Tools used

OpenAI
CodexVersion Astra

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Version history

  1. v1Initial depositCurrentSep 30, 2026