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An algebraic characterization of indistinguishable cardinals

Published online by Cambridge University Press:  12 March 2014

A. B. Slomson
A. B. Slomson*
Affiliation:
University of Leeds
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Extract

Two cardinals are said to be indistinguishable if there is no sentence of second order logic which discriminates between them. This notion, which is defined precisely below, is closely related to that of characterizable cardinals, introduced and studied by Garland in [3]. In this paper we give an algebraic criterion for two cardinals to be indistinguishable. As a consequence we obtain a straightforward proof of an interesting theorem about characterizable cardinals due to Zykov [6].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 35 , Issue 1 , March 1970 , pp. 97 - 104
Copyright
Copyright © Association for Symbolic Logic 1970

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Footnotes

1

Most of the results given here were presented at the British Mathematical Colloquium, Birmingham, March 1969.

References

[1] Bell, J. L. and Slomson, A. B., Models and ultraproducts: An introduction, North-Holland, Amsterdam, 1969.Google Scholar
[2] Church, A., Introduction to mathematical logic, Princeton University Press, Princeton, N.J., 1956.Google Scholar
[3] Garland, S. J., Second order cardinal characterlability, Summer Institute for Axiomatic Set Theory, Los Angeles, Calif., Lecture Notes, 1967.Google Scholar
[4] Kochen, S. B., Ultraproducts in the theory of models, Annals of mathematics, vol. 74 (1962), pp. 221–261.CrossRefGoogle Scholar
[S] Löwenheim, L., über Moglichkeiten im Relativkalkül, Mathematische Annalen, vol. 76 (1915), pp. 447–470.CrossRefGoogle Scholar
[6] Zykov, A. A., The spectrum problem in extended predicate calculus, American Mathematical Society Translations, (2), vol. 3 (1956), pp. 1–14.Google Scholar