# Weakly normal ideals ou PKl and the singular cardinal hypothesis

Fundamenta Mathematicae (1993)

- Volume: 143, Issue: 2, page 97-106
- ISSN: 0016-2736

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topAbe, Yoshihiro. "Weakly normal ideals ou PKl and the singular cardinal hypothesis." Fundamenta Mathematicae 143.2 (1993): 97-106. <http://eudml.org/doc/212002>.

@article{Abe1993,

abstract = {In §1, we observe that a weakly normal ideal has a saturation property; we also show that the existence of certain precipitous ideals is sufficient for the existence of weakly normal ideals. In §2, generalizing Solovay’s theorem concerning strongly compact cardinals, we show that $λ^\{<κ\}$ is decided if $P_κλ$ carries a weakly normal ideal and λ is regular or cf λ ≤ κ. This is applied to solving the singular cardinal hypothesis.},

author = {Abe, Yoshihiro},

journal = {Fundamenta Mathematicae},

keywords = {weakly normal ideal; saturation; precipitous ideals; singular cardinal hypothesis},

language = {eng},

number = {2},

pages = {97-106},

title = {Weakly normal ideals ou PKl and the singular cardinal hypothesis},

url = {http://eudml.org/doc/212002},

volume = {143},

year = {1993},

}

TY - JOUR

AU - Abe, Yoshihiro

TI - Weakly normal ideals ou PKl and the singular cardinal hypothesis

JO - Fundamenta Mathematicae

PY - 1993

VL - 143

IS - 2

SP - 97

EP - 106

AB - In §1, we observe that a weakly normal ideal has a saturation property; we also show that the existence of certain precipitous ideals is sufficient for the existence of weakly normal ideals. In §2, generalizing Solovay’s theorem concerning strongly compact cardinals, we show that $λ^{<κ}$ is decided if $P_κλ$ carries a weakly normal ideal and λ is regular or cf λ ≤ κ. This is applied to solving the singular cardinal hypothesis.

LA - eng

KW - weakly normal ideal; saturation; precipitous ideals; singular cardinal hypothesis

UR - http://eudml.org/doc/212002

ER -

## References

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- [9] Y. Matsubara, private communication.
- [10] R. Mignone, A direct weakening of normality for filters, preprint. Zbl0778.03014
- [11] M. Shioya, Weakly normal closures of filters on ${P}_{\kappa}\lambda $, to appear.
- [12] J. Silver, On the singular cardinals problem, in: Proc. Internat. Congress Math. Vancouver, 1974, 265-268.
- [13] R. M. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory, Proc. Sympos. Pure Math. 13 I, D. Scott (ed.), Amer. Math. Soc., Providence, R.I., 1971, 397-428.
- [14] R. M. Solovay, Strongly compact cardinals and the GCH, in: Proc. Tarski Symposium, Proc. Sympos. Pure Math. 25, Amer. Math. Soc., Providence, R.I., 1974, 365-372.
- [15] R. M. Solovay, W. N. Reinhardt and A. Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), 73-116. Zbl0376.02055

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