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

  • Alexander S. Kechris (a1)
Abstract
Abstract

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.

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[K] A. S. Kechris , AD and projective ordinals, Cabal Seminar 76–77 ( A. S. Kechris and Y. N. Moschovakis , editors), Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin, 1978, pp. 91132.

[KMM] A. S. Kechris , D. A. Martin and Y. N. Moschovakis (editors), Cabal Seminar 79–81, Lecture Notes in Mathematics, Vol. 1019, Springer-Verlag, Berlin, 1983.

[Ma] D. A. Martin , The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.

[MMS] A. D. Martin , Y. N. Moschovakis and J. R. Steel , The extent of definable scales, Bulletin (New Series) of the American Mathematical Society, vol. 6 (1982), pp. 435440.

[Mo3] Y. N. Moschovakis , Determinacy and prewellorderings of the continuum, Mathematical logic and foundations of set theory ( Y. Bar-Hillel , editor), North-Holland, Amsterdam, 1970, pp. 2462.

[So] R. M. Solovay , The independence of DC from AD, Cabal Seminar 76–77 ( A. S. Kechris and Y. N. Moschovakis , editors), Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin, 1978, pp. 171184.

[St-VW] J. R. Steel and R. Van Wesep , TWO consequences of determinacy consistent with choice, Transactions of the American Mathematical Society, vol. 272 (1982), pp. 6787.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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