## A partition property characterizing cardinals hyperinaccessible of finite type

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- 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