# The Solovay model(s) and ordinal definability

forcingset-theory

Across the literature, there are (a priori) different things that are commonly called the Solovay model. More precisely, one forces with $$\text{Col}(\omega,<\kappa)$$, where $$\kappa$$ is inaccessible, and then passes to one of the following inner models:

• $$L(\mathbb R)$$
• $$\text{HOD}(\mathbb R)$$
• $$\text{HOD}(\text{ON}^\omega)$$

The proofs that the resulting models satisfy ZF + Every set of reals is Lebesgue measurable and has the BP/PSP are all more are less the same, and the proof of DC is slightly more complicated for $$L(\mathbb R)$$.

I know that one can have $$\text{HOD}(\mathbb R)\neq L(\mathbb R)$$ (and that equality is also consistent), but do we always have $$\text{HOD}(\mathbb R)= \text{HOD}(\text{ON}^\omega)$$? Things come down to whether a set that is definable from some countable sequence of ordinals is also definable from an ordinal and a real. The answer is obviously yes if the sequence has range contained in the inaccessible that is being collapsed, but I’m unsure about happens above that.

For a less precise question, are there any properties of the reals on which two of the three models above disagree?

Let's start with the general case. Of course you can have that $$\rm HOD(\Bbb R)\neq HOD(Ord^\omega)$$. For example, if $$\sf CH$$ holds or $$\kappa$$ is measurable, then there is a forcing adding a new $$\omega$$ sequence of ordinals without adding ordinals (Namba forcing or Prikry forcing).
This means that also in the case of Solovay's model over an arbitrary model of $$\sf ZFC+\kappa$$ is inaccessible these models can be different. Because maybe we started with some $$\lambda>\kappa$$ which was measurable, did a Prikry forcing, which does not add reals, and then did the Solovay construction. Who's to say what the ground model was and what properties hold there?
However, as far as regularity properties of the reals, those are essentially the same across all three models. Specifically because these properties are mostly concerned with measurability with respect to ideals on the Borel sets (or their completions to the full power set). And as such we can code a lot of the data into the reals, at least assuming $$\sf AC_\omega(\Bbb R)$$, which we get through $$\sf DC$$.
Just to note, if we start with $$V=L$$ and an inaccessible cardinal, as Solovay did, then the three models are in fact the same.