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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”?

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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 $0$.

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  • $\begingroup$ Thank you Martin, this is spot-on and very helpful. $\endgroup$ – Monroe Eskew Sep 25 '19 at 19:49

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