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March 2021 The Ultrapower Axiom UA and the number of normal measures over $\aleph_1$ and $\aleph_2$
Arthur W. Apter
Tbilisi Math. J. 14(1): 49-53 (March 2021). DOI: 10.32513/tmj/1932200814


We show that assuming the consistency of certain large cardinals (namely a supercompact cardinal with a measurable cardinal above it of the appropriate Mitchell order) together with the Ultrapower Axiom UA introduced by Goldberg in [3], it is possible to force and construct choiceless universes of 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 $\mu_\omega = \{x \subseteq \aleph_1 \mid x$ contains a club set$\}$, and $\aleph_2$ will carry exactly $\tau$ normal measures, where $\tau = \aleph_n$ for $n=0,1,2$ or $\tau = n$ for $n \ge 1$ an integer (so in particular, $\tau \le \aleph_2$ is any nonzero finite or infinite cardinal). This complements the results of [1] in which $\tau \ge \aleph_3$ and contrasts with the well-known facts that assuming AD + DC, $\aleph_1$ 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 Grant 63505-00-51.


The author wishes to thank the referee for helpful suggestions and corrections which considerably improved the exposition.


Download Citation

Arthur W. Apter. "The Ultrapower Axiom UA and the number of normal measures over $\aleph_1$ and $\aleph_2$." Tbilisi Math. J. 14 (1) 49 - 53, March 2021.


Received: 6 June 2020; Accepted: 3 October 2020; Published: March 2021
First available in Project Euclid: 1 April 2021

Digital Object Identifier: 10.32513/tmj/1932200814

Primary: 03E35
Secondary: 03E45, 03E55

Rights: Copyright © 2021 Tbilisi Centre for Mathematical Sciences


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Vol.14 • No. 1 • March 2021
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