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An ideal characterization of Mahlo cardinals

Published online by Cambridge University Press:  12 March 2014

Qi Feng
Qi Feng*
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
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Abstract

We show that a cardinal κ is a (strongly) Mahlo cardinal if and only if there exists a nontrivial κ-complete κ-normal ideal on κ. Also we show that if κ is Mahlo and λ ≧ κ and λ<κ = λ then there is a nontrivial κ-complete κ-normal fine ideal on Pκ(λ). If κ is the successor of a cardinal, we consider weak κ-normality and prove that if κ = μ+ and μ is a regular cardinal then (1) μ< μ = μ if and only if there is a nontrivial κ-complete weakly κ-normal ideal on κ, and (2) if μ< μ = μ < λ<μ = λ then there is a nontrivial κ-complete weakly κ-normal fine ideal on Pκ(λ).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 2 , June 1989 , pp. 467 - 473
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1]Baumgartner, J., Taylor, A., and Wagon, S., On splitting stationary subsets of large cardinals, this Journal, vol. 42 (1977), pp. 203–214.Google Scholar
[2]Jech, T., Stationary subsets of inaccessible cardinals, Axiomatic set theory (Baumgartner, J., editor), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 115–142.CrossRefGoogle Scholar
[3]Jech, T., Set theory, Academic Press, New York, 1978.Google Scholar
[4]Jech, T., Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165–198.CrossRefGoogle Scholar
[5]Mahlo, P., Über lineare trasfinite Mengen, Berichte über die Verhandlungen der Könglich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 63 (1911), pp. 187–225.Google Scholar
[6]Mahlo, P., Z̃ur Theorie und Andwendung der ρ 0-Z̃ahlen, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 64 (1912), pp. 108–112.Google Scholar
[7]Mahlo, P., Z̃ur Theorie und Andwendung der ρ 0-Z̃ahlen. II, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, vol. 65 (1913), pp. 268–282.Google Scholar