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# A result concerning cardinalities of ultraproducts1

## Extract

The cardinality problem for ultraproducts is as follows: Given an ultrafilter over a set I and cardinals αi, iI, what is the cardinality of the ultraproduct ? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter over a set I of power κ (uniform means that all elements of have power κ). It is open whether every countably incomplete has the property that, for all infinite α, the ultra-power has power ακ. However, it is shown in [4] that certain countably incomplete , namely the κ-regular , have this property.

This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the κ-good ultrafilters, do have this property. The κ-good ultrafilters are important in model theory because they are exactly the ultrafilters such that every ultraproduct modulo is κ-saturated (see [5]).

Let be an ultrafilter on a set I. Let αi, n, iI, n ∈ ω, be cardinals and αi, n, ≥ αi, m if n < m. Let

.

Then ρn are nonincreasing and therefore there is some m and ρ such that ρn = ρ if nm. We call ρ the eventual value (abbreviated ev val) of ρn.

## Footnotes

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1

The preparation of this paper was supported by NSF grants GP-27633 and GP-27964.

## References

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[1]Eklof, P. C., The structure of uhraproducts of Abelian groups (to appear).
[2]Frolik, A., Sums of ultrafilters, Bulletin of the American Mathematical Society, vol. 73 (1967), pp. 8791.
[3]Keisler, H. J., Good ideals infields of sets, Annals of Mathematics, vol. 79 (1964), pp. 338359.
[4]Keisler, H. J., On cardinalities of uhraproducts, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 644647.
[5]Keisler, H. J., A survey of ultraproducts, Proceedings of the 1964 International Congress, North-Holland, Amsterdam, 1965, pp. 112126.
[6]Keisler, H. J., Ideals with prescribed degree of goodness, Annals of Mathematics, vol. 81 (1965), pp. 112116.
[7]Keisler, H. J., Ultraproducts of finite sets, this Journal, vol. 32 (1967), pp. 4757.
[8]Kunen, K., Ultrafilters and independent sets, Transactions of the American Mathematical Society, vol. 172 (1972), pp. 299306.
[9]Frayne, T., Morel, A. and Scott, D., Reduced direct products, Fundamenta Mathematicae, vol. 51 (1962), pp. 195228.
[10]Shelah, S., Saturation of ultrapowers and Keisler's order, Annals of Mathematical Logic, vol. 4 (1972), pp. 75114.

# A result concerning cardinalities of ultraproducts1

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