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2008 On the number of normal measures $\aleph_1$ and $\aleph_2$ can carry
Arthur W. Apter
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Tbilisi Math. J. 1: 9-14 (2008). DOI: 10.32513/tbilisi/1528768821


We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it), it is possible to force and construct choiceless universes of $\mathsf{ZF}$ in which the first two uncountable cardinals $\aleph_1$ and $\aleph_2$ are both measurable and carry certain fixed numbers of normal measures. Specifically, in the models constructed, $\aleph_1$ will carry exactly one normal measure, namely $\mu_w = \{x \subseteq \aleph_1 \mid x$ contains a club set$\}$, and $\aleph_2$ will carry exactly $\tau$ normal measures, where $\tau \ge \aleph_3$ is any regular cardinal. This contrasts with the well-known facts that assuming $\mathsf{AD} + \mathsf{AC}$, $\aleph_1$ is measurable and carries exactly one normal measure, and $\aleph_2$ is measurable and carries exactly two normal measures.

Funding Statement

The author's research was partially supported by PSC-CUNY Grants and CUNY Collaborative Incentive Grants. In addition, the author wishes to thank the referees for helpful comments and suggestions which have been incorporated into the current version of the paper.


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Arthur W. Apter. "On the number of normal measures $\aleph_1$ and $\aleph_2$ can carry." Tbilisi Math. J. 1 9 - 14, 2008.


Received: 12 April 2007; Revised: 13 August 2007; Accepted: 21 November 2007; Published: 2008
First available in Project Euclid: 12 June 2018

zbMATH: 1158.03032
MathSciNet: MR2434434
Digital Object Identifier: 10.32513/tbilisi/1528768821

Primary: 03E35
Secondary: 03E55

Rights: Copyright © 2008 Tbilisi Centre for Mathematical Sciences


Vol.1 • 2008
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