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A saturation property of ideals and weakly compact cardinals1

  • Joji Takahashi (a1)

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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].

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Corresponding author

Department of Mathematics, University of Papua New Guinea, Port Moresby, Papua New Guinea

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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.

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References

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[BTW]Baumgartner, J. E., Taylor, A. D. and Wagon, S., Structural properties of ideals, Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 197 (1982).
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[KT]Keisler, H. J. and Tarski, A., From accessible to inaccessible cardinals, Fundament a Mathematicae, vol. 53 (1964), pp. 225308.
[Ku]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.
[Pa]Paris, J. B., Boolean extensions and large cardinals, Ph.D. thesis, Manchester University, Manchester, 1969.
[Pr]Prikry, K., Changing measurable into accessible cardinals, Dissertationes Mathematicae (Rozprawy Matematycze), vol. 68 (1970).
[Si]Silver, J. H., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 45110.
[So]Solovay, R. M., Real-valued measurable cardinals, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 397428.
[ST]Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, ser. 2, vol. 94 (1971), pp. 201245.
[Ta]Takahashi, J., Partition and saturation properties of ideals, Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota, 1982.
[Tar]Tarski, A., Drei (Überdeckungssätze der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 30 (1938), pp. 132155.
[Ul]Ulam, S., Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 140150.

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A saturation property of ideals and weakly compact cardinals1

  • Joji Takahashi (a1)

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