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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Rowbottom-type properties and a cardinal arithmetic


Author: Jan Tryba
Journal: Proc. Amer. Math. Soc. 96 (1986), 661-667
MSC: Primary 03E10; Secondary 03C50, 03E05, 03E55
DOI: https://doi.org/10.1090/S0002-9939-1986-0826499-7
MathSciNet review: 826499
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Abstract: Assuming Rowbottom-type properties, we estimate the size of certain families of closed disjoint functions. We show that whenever $ k$ is Rowbottom and $ {2^\omega } < {\aleph _{{\omega _1}}}(k)$, then $ {2^{ < k}} = {2^\omega }$ or $ k$ is the strong limit cardinal. Next we notice that every strongly inaccessible Jónsson cardinal $ k$ is $ v$-Rowbottom for some $ v < k$. In turn, Shelah's method allows us to construct a Jónsson model of cardinality $ {k^ + }$ provided $ {k^{{\text{cf(}}k{\text{)}}}} = {k^ + }$. We include some additional remarks.


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DOI: https://doi.org/10.1090/S0002-9939-1986-0826499-7
Keywords: Closed disjoint functions, $ I$-functions, norm of a function, Chang's Conjecture, $ v$-Rowbottom cardinals, Jónsson cardinals, possible scale for a sequence of ordinals
Article copyright: © Copyright 1986 American Mathematical Society

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