A partition property characterizing cardinals hyperinaccessible of finite type
HTML articles powered by AMS MathViewer
- by James H. Schmerl PDF
- Trans. Amer. Math. Soc. 188 (1974), 281-291 Request permission
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
- P. Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427–489. MR 81864, DOI 10.1090/S0002-9904-1956-10036-0
- Azriel Lévy, Axiom schemata of strong infinity in axiomatic set theory, Pacific J. Math. 10 (1960), 223–238. MR 124205, DOI 10.2140/pjm.1960.10.223 F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264-286. J. H. Schmerl, On hyperinaccessible-like models, Notices Amer. Math. Soc. 16 (1969), 843. Abstract #69T-E59. J. H. Schmerl and S. Shelah, On models with orderings, Notices Amer. Math. Soc. 16 (1969), 840. Abstract #69T-E50. J. H. Schmerl, On $\kappa$-like models for inaccessible $\kappa$, Doctoral Dissertation, University of California, Berkeley, Calif., 1971.
Additional Information
- © Copyright 1974 American Mathematical Society
- 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