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<p>The Model Hypothesis (abbreviated MH) and Δ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1) MH implies the existence of <i>P</i>-points in <i>ω</i>* and is therefore not a theorem of ZFC; (2) MH also fails in the side-by-side Sacks models; (3) as Δ holds in these models, this implies Δ is strictly weaker than MH; (4) furthermore, Δ holds in a large class of forcing extensions in which it was not previously known to hold.</p>

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On Roitman’s principles MH and Δ

  • 期刊论文

  • 发表日期：

    2026年5月25日

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  • Hector Barriga-Acosta,
  • Will Brian,
  • Alan Dow

摘要

The Model Hypothesis (abbreviated MH) and Δ are set-theoretic axioms introduced by J. Roitman in her work on the box product problem. Answering some questions of Roitman and Williams on these two principles, we show (1) MH implies the existence of P-points in ω* and is therefore not a theorem of ZFC; (2) MH also fails in the side-by-side Sacks models; (3) as Δ holds in these models, this implies Δ is strictly weaker than MH; (4) furthermore, Δ holds in a large class of forcing extensions in which it was not previously known to hold.