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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.

MSC:
03E35 Consistency and independence results
03E60 Determinacy principles
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[1] DOI: 10.1007/BFb0069296
[2] Set theory (1980)
[3] DOI: 10.1090/S0002-9947-1982-0656481-5
[4] DOI: 10.1007/BFb0069299
[5] Mathematical logic (1967) · Zbl 0149.24309
[6] Cabal Seminar 79–81 1019 (1983)
[7] Descriptive set theory (1980) · Zbl 0433.03025
[8] Fundamental Mathematicae 53 pp 205– (1963)
[9] DOI: 10.1090/S0273-0979-1982-15009-1 · Zbl 0509.03025
[10] DOI: 10.1090/S0002-9904-1968-11995-0 · Zbl 0165.31605
[11] Mathematical logic and foundations of set theory pp 24– (1970)
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