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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:
 3e+35 Consistency and independence results 3e+60 Determinacy principles
##### Keywords:
infinite game; inner model; sharp
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##### References:
 [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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