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A characterization of inaccessible cardinals

Published online by Cambridge University Press:  18 May 2009

G. B. Preston
G. B. Preston
Affiliation:
Royal Military College of Science, Shrivenham
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A cardinal number which is too large to be reached by some process is generally said to be inaccessible by that process. Many kinds of inaccessible cardinals have been discussed and for a general survey the book of H. Bachmann [1, Chapter 7] may be consulted. We consider here two inaccessibility properties. We shall denote the cardinal of a set X by |X|. The first inaccessibility property will be called regularity: the cardinal| X| will be said to be regular if there does not exist a disjoint cover {X1: i ε I} of X such that

(i)|X1|<|X|, for each i in I, and

(ii)|I|<|X|.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 4 , July 1962 , pp. 153 - 157
Copyright
Copyright © Glasgow Mathematical Journal Trust 1962

References

1.Bachmann, H., Transfinite Zahlen (Ergebnisse der Math. N.F. 1, Springer, 1955).CrossRefGoogle Scholar
2.Tarski, A., Über unerreichbare Kardinalzahlen, Fund. Math. 30 (1938), 68–89.CrossRefGoogle Scholar
3.Preston, G. B., Embedding any semigroup in a D-simple semigroup, Trans. Amer. Math. Soc. 93 (1959), 351–355.Google Scholar