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<p>We show that the forcing axiom for countably compact, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\omega _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah [<CitationRef CitationID="CR9">9</CitationRef>] and sets a new limit to the generalization of Martin’s Axiom to the stage of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\omega _2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

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Limits on forcing axioms at \(\omega _2\) compatible with the continuum hypothesis

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Stevo Todorcevic,
  • Shihao Xiong

摘要

We show that the forcing axiom for countably compact, \(\omega _2\) ω 2 -Knaster, well-met posets is inconsistent. This is supplemental to an inconsistency result of Shelah [9] and sets a new limit to the generalization of Martin’s Axiom to the stage of \(\omega _2\) ω 2 .