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A saturation property of ideals and weakly compact cardinals1

Published online by Cambridge University Press:  12 March 2014

Joji Takahashi*
Affiliation:
Department of Mathematics, Bayero University, Kano, Nigeria
*
Department of Mathematics, University of Papua New Guinea, Port Moresby, Papua New Guinea

Extract

Suppose κ is a regular uncountable cardinal, λ is a cardinal > 1, and is a κ-complete uniform ideal on κ. This paper deals with a saturation property Sat(κ, λ, ) of , which is a weakening of usual λ-saturatedness. Roughly speaking, Sat(κ, λ, ) means that can be densely extended to λ-saturated ideals on small fields of subsets of κ. We will show that some consequences of the existence of a λ-saturated ideal on κ follow from weaker ∃: Sat(κ, λ, ), and that ∃: Sat(κ, λ, ) is connected with weak compactness and complete ineffability of κ in much the same way as the existence of a saturated ideal on κ is connected with measurability of κ.

In §2, we define Sat(κ, λ, ), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(κ, λ) is also defined and shown to be equivalent to ∃: Sat(κ, λ, ) under a certain condition.

In §3, we show that ∃: Sat(κ, κ, ) implies that κ is highly Mahlo, improving results in [KT] and [So].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1986

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Footnotes

1

The results in this paper are from the author's Ph.D. thesis [Ta] written under the supervision of Karel Prikry, to whom the author is grateful.

References

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