The Destruction of the Axiom of Determinacy by Forcings on ℝ when Θ is Regular

2019/03/16 by Chan, William, Jackson, Stephen
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1903.07005

Abstract

ZF + AD proves that for all nontrivial forcings ℙ on a wellorderable set of cardinality less than Θ, 1_ℙ \Vdash_ℙ \negAD. ZF + AD + Θ is regular proves that for all nontrivial forcing ℙ which is a surjective image of ℝ, 1_ℙ \Vdash_ℙ \negAD. In particular, ZF + AD + V = L(ℝ) proves that for every nontrivial forcing ℙ ∈ L_Θ(ℝ), 1_ℙ \Vdash_ℙ \negAD.

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