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 \ elements.

It is known that every Borel hypersmooth but non-smooth equivalence relation is Borel bi-reducible to E1. We prove a ROD version of this result in the Solovay model.

A generic extension of the constructible universe by reals is defined, in which the union of ‐classes of x and y is a lightface set, but neither of these two ‐classes is separately ordinal‐definable.

We prove that if ≼ is an analytic partial order then either ≼ can be extended to a Δ 2 1 linear order similar to an antichain in 2 ω 1 , ordered lexicographically, or a certain Borel partial order ⩽ 0 embeds in ≼. Similar linearization results are presented, for κ -bi-Souslin partial orders and real-ordinal definable orders in the Solovay model. A corollary for analytic equivalence relations says that any Σ 1 1 equivalence relation E , such that (...) E 0 does not embed in E , is fully determined by intersections with E -invariant Borel sets coded in L. (shrink)