ZF + AD proves that for all nontrivial forcings $$\mathbb{P}$$ ℙ on a wellorderable set of cardinality less than Θ, $${1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ 1 ℙ ⊨ ℙ ¬ AD . ZF + AD + Θ is regular proves that for all nontrivial forcing $$\mathbb{P}$$ ℙ which is a surjective image of ℝ, $${1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ 1 ℙ ⊨ ℙ ¬ AD . In particular, ZF + AD + V = L(ℝ) proves that for every nontrivial forcing $$ \mathbb{P}\in {L_\Theta }\left(\mathbb{R}\right),{1_\mathbb{P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ ℙ ∈ L Θ ( ℝ ) , 1 ℙ ⊨ ℙ ¬ AD .

中文翻译：

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当 Θ 是正则时，通过对 ℝ 的强迫来破坏确定性公理

ZF + AD 证明对于所有非平凡强迫 $$\mathbb{P}$$ ℙ 在一个小于 Θ 的可排序基数集上， $${1_\mathbb{P}}{ \Vdash _\mathbb{P}}\ ,\neg {\rm{AD}}$$ 1 ℙ ⊨ ℙ ¬ AD . ZF + AD + Θ 是正则证明对于所有非平凡强迫 $$\mathbb{P}$$ ℙ 是 ℝ 的满射图像， $${1_\mathbb{P}}{ \Vdash _\mathbb{P} }\,\neg {\rm{AD}}$$ 1 ℙ ⊨ ℙ ¬ AD . 特别地，ZF + AD + V = L(ℝ) 证明对于每个非平凡的强迫 $$ \mathbb{P}\in {L_\Theta }\left(\mathbb{R}\right),{1_\mathbb{ P}}{ \Vdash _\mathbb{P}}\,\neg {\rm{AD}}$$ ℙ ∈ L Θ ( ℝ ) , 1 ℙ ⊨ ℙ ¬ AD .