The Journal of Symbolic Logic

Research Article

The axiom of determinacy implies dependent choices in L(R)

Alexander S. Kechris

California Institute of Technology, Pasadena, California 91125


We prove the following Main Theorem: ZF + AD + VL(R) ⇒ DC. As a corollary we have that Con(ZF + AD) ⇒ Con(ZF + AD + DC). Combined with the result of Woodin that Con(ZF + AD) ⇒ Con(ZF + AD + ¬ ACω) it follows that DC (as well as ACω) is independent relative to ZF + AD. It is finally shown (jointly with H. Woodin) that ZF + AD + ¬DCR, where DCR is DC restricted to reals, implies the consistency of ZF + AD + DC, in fact implies R# (i.e. the sharp of L(R)) exists.

(Received April 29 1982)


1   Research partially supported by NSF Grant No. MCS-8117804. The author is an A. P. Sloan Foundation Fellow.