Weakly normal ideals ou PKl and the singular cardinal hypothesis

Yoshihiro Abe

Fundamenta Mathematicae (1993)

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

Abstract

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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 P κ λ carries a weakly normal ideal and λ is regular or cf λ ≤ κ. This is applied to solving the singular cardinal hypothesis.

How to cite

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Abe, 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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  1. [1] Y. Abe, Weakly normal filters and the closed unbounded filter on P κ λ , Proc. Amer. Math. Soc. 104 (1988), 1226-1234. Zbl0692.04004
  2. [2] Y. Abe, Saturated ideals and subtle properties of P κ λ , circulated. 
  3. [3] Y. Abe, Weakly normal filters and large cardinals, Tsukuba J. Math. 16 (1992), 487-494. Zbl0787.03042
  4. [4] D. M. Carr, The minimal normal filter on P κ λ , Proc. Amer. Math. Soc. 86 (1982), 316-320. Zbl0497.03037
  5. [5] M. Foreman, Potent axioms, Trans. Amer. Math. Soc. 294 (1986), 1-28. Zbl0614.03051
  6. [6] C. A. Johnson, On ideals and stationary reflection, J. Symbolic Logic 54 (1989), 568-575. Zbl0703.03030
  7. [7] A. Kanamori, Weakly normal filters and irregular ultrafilters, Trans. Amer. Math. Soc. 220 (1976), 393-399. Zbl0341.02058
  8. [8] Y. Matsubara, Menas's conjecture and generic ultrapowers, Ann. Pure Appl. Logic 36 (1987), 225-234. Zbl0632.03043
  9. [9] Y. Matsubara, private communication. 
  10. [10] R. Mignone, A direct weakening of normality for filters, preprint. Zbl0778.03014
  11. [11] M. Shioya, Weakly normal closures of filters on P κ λ , to appear. 
  12. [12] J. Silver, On the singular cardinals problem, in: Proc. Internat. Congress Math. Vancouver, 1974, 265-268. 
  13. [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. [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. [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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