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The cardinality of reduced power set algebras


Author: Alan D. Taylor
Journal: Proc. Amer. Math. Soc. 103 (1988), 277-280
MSC: Primary 03E05; Secondary 06E05
DOI: https://doi.org/10.1090/S0002-9939-1988-0938683-4
MathSciNet review: 938683
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Abstract: We prove a general result on the cardinality of reduced powers of structures via filters that has several consequences including the following: if $ I$ is a uniform, countably complete ideal on the real line $ \mathcal{R}$ and $ \mathcal{B}$ is the Boolean algebra of subsets of $ \mathcal{R}$ modulo $ I$, then $ \left\vert \mathcal{B} \right\vert > {2^{{\aleph _0}}}$ and if $ {2^\nu } \leq {2^{{\aleph _0}}}$ for all $ \nu < {2^{{\aleph _0}}}$ then $ \left\vert \mathcal{B} \right\vert = {2^{{2^{{\aleph _0}}}}}$. This strengthens some results of Kunen and Pelc [7] and Prikry [8] obtained by Boolean ultrapower techniques. Our arguments are all combinatorial and some applications are included.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0938683-4
Article copyright: © Copyright 1988 American Mathematical Society

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