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In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for class forcing. The proof uses Jensen Coding. Is this still true in ZFC${^-}$? I am particularly interested whether generic $\mathbf{\Sigma_3^1}$-absoluteness might be consistent for class forcings over models of Second Order Arithmetic (resp. ZFC$^-+$V=HC).

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    $\begingroup$ Welcome to MathOverflow! :-) $\endgroup$
    – Asaf Karagila ♦
    Commented Jun 17, 2015 at 8:58
  • $\begingroup$ Was there any progress on this? It puts me in mind of aFriedman-Gitman-Kanovei paper arxiv.org/abs/1808.04732 though there may be no real connection $\endgroup$
    – Ned
    Commented Mar 31, 2021 at 15:14

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