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For a paper I'm writing I need to use (as a blackbox) the following theorem: if there is a proper class of Woodin cardinals and $G$ is set-generic, then $L(\mathbb{R})$ and $L(\mathbb{R})^{V[G]}$ are elementarily equivalent. I have seen this result quoted in a number of places, but I can't seem to track down a reference. What is a good source to cite for this?

More generally, is there a single source where the basic generic absoluteness results are presented together? I'm not interested (at the moment) in consistency strengths but rather outright implications, if only because those are snappier to state for broader audiecnes.

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    $\begingroup$ Theorem 3.1.12 of Larson's stationary tower book or theorem 7.22 of Steel's Outline of Inner Model Theory. Larson's book and Steel's exposition on the derived model theorem contain many (but not all) of the famous generic absoluteness results. $\endgroup$
    – Sean Cody
    Commented Aug 4 at 1:44
  • $\begingroup$ @SeanCody Lovely! If you post that as an answer I'll accept it. Thanks, this is a tricky literature for me to navigate. $\endgroup$
    – Noah Schweber
    Commented Aug 4 at 1:55

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You can find this result as theorem 3.1.12 of Larson's The Stationary Tower monograph or theorem 7.22 of Steel's Outline of Inner Model Theory. More generally, Larson's book and Steel's exposition on the derived model theorem contain many (but not all) of the famous generic absoluteness results of this kind.

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  • $\begingroup$ Thanks! Am I right to infer that Woodin did not publish (yet) this result? $\endgroup$
    – Noah Schweber
    Commented Aug 6 at 3:19
  • $\begingroup$ @NoahSchweber No problem. That's correct. $\endgroup$
    – Sean Cody
    Commented Aug 6 at 5:59
  • $\begingroup$ @NoahSchweber There’s still time 😂 $\endgroup$
    – Monroe Eskew
    Commented Aug 26 at 20:12

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