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Forcing is a method first used to prove the continuum hypothesis is independent of the classical axioms of set theory

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Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

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