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<p>We prove characterizations of bounded forcing axioms and their strengthenings in terms of weak genericity. These characterizations resemble those due to Jensen and Woodin, used to characterize the full forcing axioms and their strengthenings. The weak genericity characterizations allow us to quickly deduce various other characterizations in terms of generic <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Σ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>-absoluteness. We also provide a limiting result on the extent of such generic absoluteness type characterizations of certain strengthenings of bounded forcing axioms.</p>

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Bounded forcing axioms and weak genericity

  • 期刊论文

  • 发表日期：

    2025年10月13日

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  • Gunter Fuchs,
  • Benjamin Goodman

摘要

We prove characterizations of bounded forcing axioms and their strengthenings in terms of weak genericity. These characterizations resemble those due to Jensen and Woodin, used to characterize the full forcing axioms and their strengthenings. The weak genericity characterizations allow us to quickly deduce various other characterizations in terms of generic \(\Sigma _1\) Σ 1 -absoluteness. We also provide a limiting result on the extent of such generic absoluteness type characterizations of certain strengthenings of bounded forcing axioms.