Weakly normal ideals ou PKl and the singular cardinal hypothesis
Abe, Yoshihiro
Fundamenta Mathematicae, Tome 142 (1993), p. 97-106 / Harvested from The Polish Digital Mathematics Library
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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 Pκλ carries a weakly normal ideal and λ is regular or cf λ ≤ κ. This is applied to solving the singular cardinal hypothesis.

Publié le : 1993-01-01
EUDML-ID : urn:eudml:doc:212002 
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Abe, Yoshihiro. Weakly normal ideals ou PKl and the singular cardinal hypothesis. Fundamenta Mathematicae, Tome 142 (1993) pp. 97-106. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv143i2p97bwm/

[00000] [1] Y. Abe, Weakly normal filters and the closed unbounded filter on Pκλ, Proc. Amer. Math. Soc. 104 (1988), 1226-1234. | Zbl 0692.04004

[00001] [2] Y. Abe, Saturated ideals and subtle properties of Pκλ, circulated.

[00002] [3] Y. Abe, Weakly normal filters and large cardinals, Tsukuba J. Math. 16 (1992), 487-494. | Zbl 0787.03042

[00003] [4] D. M. Carr, The minimal normal filter on Pκλ, Proc. Amer. Math. Soc. 86 (1982), 316-320. | Zbl 0497.03037

[00004] [5] M. Foreman, Potent axioms, Trans. Amer. Math. Soc. 294 (1986), 1-28. | Zbl 0614.03051

[00005] [6] C. A. Johnson, On ideals and stationary reflection, J. Symbolic Logic 54 (1989), 568-575. | Zbl 0703.03030

[00006] [7] A. Kanamori, Weakly normal filters and irregular ultrafilters, Trans. Amer. Math. Soc. 220 (1976), 393-399. | Zbl 0341.02058

[00007] [8] Y. Matsubara, Menas's conjecture and generic ultrapowers, Ann. Pure Appl. Logic 36 (1987), 225-234. | Zbl 0632.03043

[00008] [9] Y. Matsubara, private communication.

[00009] [10] R. Mignone, A direct weakening of normality for filters, preprint. | Zbl 0778.03014

[00010] [11] M. Shioya, Weakly normal closures of filters on Pκλ, to appear.

[00011] [12] J. Silver, On the singular cardinals problem, in: Proc. Internat. Congress Math. Vancouver, 1974, 265-268.

[00012] [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.

[00013] [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.

[00014] [15] R. M. Solovay, W. N. Reinhardt and A. Kanamori, Strong axioms of infinity and elementary embeddings, Ann. Math. Logic 13 (1978), 73-116. | Zbl 0376.02055