# Solovay’s model

Solovay proved that if $$\kappa$$ is inaccessible, then if we adjoin a generic $$G \subseteq \mathrm{Col}(\omega,{<}\kappa)$$, then in the extension, every set of reals in $$L(\mathbb R)$$ is Baire- and Lebesgue-measurable and has the perfect subset property. In several publications, such as “Martin’s Maximum Part 1” by Foreman-Magidor-Shelah, this is applied to a situation in which a generic for something other than the Levy collapse is taken. A lemma is invoked that says the following:

Suppose $$G \subseteq \mathbb P$$ is generic a forcing that turns an inaccessible $$\kappa$$ into $$\omega_1$$, and every real in $$V[G]$$ lives in some $$V[H]$$, where $$H$$ is $$\mathbb Q$$-generic and $$|\mathbb Q| < \kappa$$. Then there is another extension $$V[G’]$$, where $$G’$$ is $$\mathrm{Col}(\omega,{<}\kappa)$$-generic, and $$\mathbb R^{V[G]} = \mathbb R^{V[G’]}$$. Thus the conclusion of Solovay’s Theorem holds in $$V[G]$$.

Question: Do we actually need the above lemma? Can we prove Solovay’s Theorem directly about a forcing $$\mathbb P$$ satisfying the hypotheses of the above lemma? Is there any place in the proof where we use a special feature of the Levy collapse such as its “continuity”?

The usual proof (at least the proof that I know) uses the fact that in $$V[H]$$, the forcing $$P/H$$ (or $$P:Q$$) is again equivalent to the Levy collapse, and in particular sufficiently homogeneous, so that every formula with parameters in $$V[H]$$ (in the forcing language of $$P/H$$) has Boolean value $$1$$ or $$0$$.