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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A measurable cardinal with a closed unbounded set of inaccessibles from $o(\kappa)=\kappa$


Author: William Mitchell
Journal: Trans. Amer. Math. Soc. 353 (2001), 4863-4897
MSC (2000): Primary 03E35, 03E45, 03E55
Published electronically: July 17, 2001
MathSciNet review: 1852085
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Abstract:

We prove that $o(\kappa)=\kappa$ is sufficient to construct a model $V[C]$in which $\kappa$ is measurable and $C$ is a closed and unbounded subset of $\kappa$ containing only inaccessible cardinals of $V$. Gitik proved that $o(\kappa)=\kappa$ is necessary.

We also calculate the consistency strength of the existence of such a set $C$ together with the assumption that $\kappa$ is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set $C$ generate the closed unbounded ultrafilter of $V$ while $\kappa$ remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.


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Additional Information

William Mitchell
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: mitchell@math.ufl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02853-7
PII: S 0002-9947(01)02853-7
Received by editor(s): March 30, 2001
Published electronically: July 17, 2001
Additional Notes: This work was partially supported by grant number DMS-962-6143 from the National Science Foundation.
Article copyright: © Copyright 2001 American Mathematical Society