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<p>We show that assuming ZF + AD<sup>+</sup>+“<i>V</i> = L(℘(ℝ))”, any poset which increases Θ does not preserve the truth of AD. We also show that in ZF+AD, any non-trivial poset on ℝ does not preserve the truth of AD. This answers the question of Chan and Jackson [2, Question 5.7]. Furthermore, we show that under the assumptions ZF + AD<sup>+</sup>+“<i>V</i> = L(℘(ℝ))” + “Θ is regular”, there is a poset on Θ which adds a new subset of Θ while preserving the truth of AD. This answers the question of Cunningham [3, Section 5].</p>

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Preservation of AD via forcings

  • 期刊论文

  • 发表日期：

    2025年5月9日

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  • Daisuke Ikegami,
  • Nam Trang

摘要

We show that assuming ZF + AD++“V = L(℘(ℝ))”, any poset which increases Θ does not preserve the truth of AD. We also show that in ZF+AD, any non-trivial poset on ℝ does not preserve the truth of AD. This answers the question of Chan and Jackson [2, Question 5.7]. Furthermore, we show that under the assumptions ZF + AD++“V = L(℘(ℝ))” + “Θ is regular”, there is a poset on Θ which adds a new subset of Θ while preserving the truth of AD. This answers the question of Cunningham [3, Section 5].