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The axiom of determinacy implies dependent choices in L(\({\mathbb{R}})\). (English) Zbl 0584.03037
We prove the following Main Theorem: \(ZF+AD+V=L({\mathbb{R}})\Rightarrow DC\). As a corollary we have that \(Con(ZF+AD)\Rightarrow Con(ZF+AD+DC)\). Combined with the result of Woodin that \(Con(ZF+AD)\Rightarrow Con(ZF+AD+\neg AC^{\omega})\) it follows that DC (as well as \(AC^{\omega})\) is independent relative to \(ZF+AD\). It is finally shown (jointly with H. Woodin) that \(ZF+AD+\neg DC_{{\mathbb{R}}}\), where \(DC_{{\mathbb{R}}}\) is DC restricted to reals, implies the consistency of \(ZF+AD+DC\), in fact implies \({\mathbb{R}}^{\#}\) (i.e. the sharp of L(\({\mathbb{R}}))\) exists.

03E35 Consistency and independence results
03E60 Determinacy principles
Full Text: DOI
[1] DOI: 10.1007/BFb0069296
[2] Set theory (1980)
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