Journal of Symbolic Logic

The Axiom of Determinancy Implies Dependent Choices in $L(\mathbf{R})$

Alexander S. Kechris

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

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J. Symbolic Logic, Volume 49, Issue 1 (1984), 161-173.

First available in Project Euclid: 6 July 2007

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Kechris, Alexander S. The Axiom of Determinancy Implies Dependent Choices in $L(\mathbf{R})$. J. Symbolic Logic 49 (1984), no. 1, 161--173.

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