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

Published online by Cambridge University Press:  12 March 2014

Alexander S. Kechris
Alexander S. Kechris*
Affiliation:
California Institute of Technology, Pasadena, California 91125
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Abstract

We prove the following Main Theorem: ZF + AD + V ⇒ L(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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 1 , March 1984 , pp. 161 - 173
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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