The axiom of determinancy implies dependent choices in l(r)

Journal of Symbolic Logic 49 (1):161 - 173 (1984)
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 ω ) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬ DC R , where DC R is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R # (i.e. the sharp of L(R)) exists
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DOI 10.2307/2274099
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Kai Hauser (1995). The Consistency Strength of Projective Absoluteness. Annals of Pure and Applied Logic 74 (3):245-295.
Daniel W. Cunningham (1995). The Real Core Model and its Scales. Annals of Pure and Applied Logic 72 (3):213-289.
Daniel W. Cunningham (2002). A Covering Lemma for L(ℝ). Archive for Mathematical Logic 41 (1):49-54.
Daniel W. Cunningham (1998). Is There a Set of Reals Not in K? Annals of Pure and Applied Logic 92 (2):161-210.

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