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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A partition property characterizing cardinals hyperinaccessible of finite type


Author: James H. Schmerl
Journal: Trans. Amer. Math. Soc. 188 (1974), 281-291
MSC: Primary 02K35; Secondary 04A10, 04A20
MathSciNet review: 0337617
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Abstract: Let $ {\mathbf{P}}(n,\alpha )$ be the class of infinite cardinals which have the following property: Suppose for each $ \nu < \kappa $ that $ {C_\nu }$ is a partition of $ {[\kappa ]^n}$ and card $ ({C_\nu }) < \kappa $; then there is $ X \subset \kappa $ of length $ \alpha $ such that for each $ \nu < \kappa $, the set $ X - (\nu + 1)$ is $ {C_\nu }$-homogeneous. In this paper the classes $ {\mathbf{P}}(n,\alpha )$ are studied and a nearly complete characterization of them is given. A principal result is that $ {\mathbf{P}}(n + 2,n + 5)$ is the class of cardinals which are hyperinaccessible of type n.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0337617-8
Keywords: Partition, inaccessible cardinal, hyperinaccessible cardinal
Article copyright: © Copyright 1974 American Mathematical Society