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Adding a closed unbounded set

Published online by Cambridge University Press:  12 March 2014

J. E. Baumgartner ,
L. A. Harrington  and
E. M. Kleinberg
J. E. Baumgartner
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
L. A. Harrington
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
E. M. Kleinberg
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Extract

The extreme interest of set theorists in the notion of “closed unbounded set” is epitomized in the following well-known theorem:

Theorem A. For any regular cardinal κ > ω, the intersection of any two closed unbounded subsets of κ is closed and unbounded.

The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal κ the intersection of fewer than κ many closed unbounded sets is closed and unbounded. Thus, if, for κ a regular uncountable cardinal, we let denote {A ⊆ κ ∣ A contains a closed unbounded subset}, then, for any such κ, is a κ-additive nonprincipal filter on κ.

Now what about the possibility of being an ultrafilterκ It is routine to see that this is impossible for κ > ℵ1. However, for κ = ℵ1 the situation is different. If were an ultrafilter, ℵ1 would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + “there exists a measurable cardinal” is consistent, then so is ZF + “ℵ1 is a measurable cardinal” [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) can be proven to be an ultrafilter. (See [3] and [5].)

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 2 , June 1976 , pp. 481 - 482
Copyright
Copyright © Association for Symbolic Logic 1976

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References

REFERENCES

[1] Friedman, H., On closed sets of ordinals, Proceedings of the American Mathematical Society, vol. 43 (1974), pp. 190–192.CrossRefGoogle Scholar
[2] Jech, T., ω1 can be measurable, Israel Journal of Mathematics, vol. 6 (1968), pp. 363–367.CrossRefGoogle Scholar
[3] Kleinberg, E. M., Strong partition properties for infinite cardinals, this Journal, vol. 35 (1970), pp. 410–428.Google Scholar
[4] Sacks, G. E. and Simpson, S. G., The α-finite injury method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343–367.CrossRefGoogle Scholar
[5] Solovay, R. M., Measurable cardinals and the Axiom of Determinateness, Lecture notes prepared in connection with the Summer Institute of Axiomatic Set Theory held at U.C.L.A., Summer 1967.Google Scholar