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UNIVERSALLY BAIRE SETS AND GENERIC ABSOLUTENESS

Published online by Cambridge University Press:  09 January 2018

TREVOR M. WILSON
TREVOR M. WILSON*
Affiliation:
DEPARTMENT OF MATHEMATICS MIAMI UNIVERSITY OXFORD, OH45056, USAE-mail:twilson@miamioh.edu
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Abstract

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.

Keywords

Primary 03E57Secondary 03E1503E55generic absolutenessuniversally BaireWoodin cardinal
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 4 , December 2017 , pp. 1229 - 1251
Copyright
Copyright © The Association for Symbolic Logic 2017 

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