Σ3-Absoluteness in Forcing Extensions
نویسنده
چکیده
We investigate the consistency strength of the forcing axiom for Σ3 formulas, for various classes of forcings. We review that the consistency strength of Σ3-absoluteness for all set forcing or even just for ω1-preserving forcing is that of a reflecting cardinal. To get the same strength from the forcing axiom restricted to proper forcing, one can add the hypotheses that ω1 is inaccessible to reals. Then we investigate the strength of the forcing axiom restricted to ccc forcing notions under this additional hypothesis; to gauge it we introduce a weak version of a weak compact cardinal, namely, a lightface Σ2-indescribable cardinal.
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