Proceedings of the American Mathematical Society

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Diamond, GCH and weak square

Author: Martin Zeman
Journal: Proc. Amer. Math. Soc. 138 (2010), 1853-1859
MSC (2010): Primary 03E04, 03E05
Published electronically: January 14, 2010
MathSciNet review: 2587470
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Abstract: Shelah proved recently that if $ \kappa>\omega$ and $ S\subseteq\kappa^+$ is a stationary set of ordinals of cofinality different from $ \mathrm{cf}(\kappa)$, then $ 2^\kappa=\kappa^+$ implies $ \Diamond_{\kappa^+}(S)$. We show that for singular $ \kappa$, an elaboration on his argument allows us to derive $ \Diamond_{\kappa^+}(T)$ from $ 2^\kappa=\kappa^+ + \square^*_\kappa$ where $ T=\{\delta<\kappa^+ {\mathop{\vert}} \mathrm{cf}(\delta)=\mathrm{cf}(\kappa)\}$. This gives a strong restriction on the existence of saturated ideals on $ \kappa^+$.

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Additional Information

Martin Zeman
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697

Keywords: Diamond, weak square, generalized continuum hypothesis
Received by editor(s): February 3, 2009
Published electronically: January 14, 2010
Additional Notes: The author was supported in part by NSF grant DMS-0500799
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.