The destruction of the axiom of determinacy by forcings on ℝ when Θ Is regular

Abstract

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