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UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 4 / December 2017
- Published online by Cambridge University Press:
- 09 January 2018, pp. 1229-1251
- Print publication:
- December 2017
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- Article
- Export citation
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We prove several equivalences and relative consistency results regarding generic absoluteness beyond Woodin’s
${\left( {{\bf{\Sigma }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ generic absoluteness result for a limit of Woodin cardinals λ. In particular, we prove that two-step 
$\exists ^ℝ \left( {{\rm{\Pi }}_1^2 } \right)^{{\rm{uB}}_\lambda } $ generic absoluteness below a measurable limit of Woodin cardinals has high consistency strength and is equivalent, modulo small forcing, to the existence of trees for 
${\left( {{\bf{\Pi }}_1^2} \right)^{{\rm{u}}{{\rm{B}}_\lambda }}}$ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many “failures of covering” for the models 
$L\left( {T,{V_\alpha }} \right)$ for α below a measurable cardinal.
