Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1974-0337617-8
MathSciNet review: 0337617
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489. MR 18, 458. MR 0081864 (18:458a)
  • [2] A. Lévy, Axiom schemata of strong infinity in axiomatic set theory, Pacific J. Math. 10 (1960), 223-238. MR 23 #A1522. MR 0124205 (23:A1522)
  • [3] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286.
  • [4] J. H. Schmerl, On hyperinaccessible-like models, Notices Amer. Math. Soc. 16 (1969), 843. Abstract #69T-E59.
  • [5] J. H. Schmerl and S. Shelah, On models with orderings, Notices Amer. Math. Soc. 16 (1969), 840. Abstract #69T-E50.
  • [6] J. H. Schmerl, On $ \kappa $-like models for inaccessible $ \kappa $, Doctoral Dissertation, University of California, Berkeley, Calif., 1971.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 02K35, 04A10, 04A20

Retrieve articles in all journals with MSC: 02K35, 04A10, 04A20


Additional Information

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

American Mathematical Society