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Saturated ideals and the singular cardinal hypothesis

  • Yo Matsubara (a1)

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The large cardinal-like properties of saturated ideals have been investigated by various authors, including Foreman [F], and Jech and Prikry [JP], among others. One of the most interesting consequences of a strongly compact cardinal is the following theorem of Solovay [So2]: if a strongly compact cardinal exists then the singular cardinal hypothesis holds above it. In this paper we discuss the question of relating the existence of saturated ideals and the singular cardinal hypothesis. We will show that the existence of “strongly” saturated ideals implies the singular cardinal hypothesis. As a biproduct we will present a proof of the above mentioned theorem of Solovay using generic ultrapowers. See Jech and Prikry [JP] for a nice exposition of generic ultrapowers. We owe a lot to the work of Foreman [F]. We would like to express our gratitude to Noa Goldring for many helpful comments and discussions.

Throughout this paper we assume that κ is a strongly inaccessible cardinal and λ is a cardinal >κ. By an ideal on κλ we mean a κ-complete fine ideal on Pκλ. For I an ideal on κλ let PI denote the poset of I-positive subsets of κλ.

Definition. Let I be an ideal on κλ. We say that I is a bounding ideal if 1 ⊩-PIδ(δ is regular cardinal ”.

We can show that if a normal ideal is “strongly” saturated then it is bounding.

Theorem 1. If 1 is an η-saturated normal ideal onκλ, where η is a cardinal <λsuch that there are fewer thanκmany cardinals betweenκand η (i.e. η < κ+κ), then I is bounding.

Proof. Let I be such an ideal on κλ. By the work of Foreman [F] and others, we know that every λ+-saturated normal ideal is precipitous. Suppose G is a generic filter for our PI. Let j: VM be the corresponding generic elementary embedding. By a theorem of Foreman [F, Lemma 10], we know that MλM in V[G]. By η-saturation, cofinalities ≥η are preserved; that is, if cfvαη, then cfvα = cfv[G]α. From jVκ being the identity on Vκ and M being λ-closed in V[G], we conclude that cofinalities <κ are preserved. Therefore if cfvα ≠ cfv[G]α then κ ≤ cfvα < η.

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References

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[CLP]Carr, D. M., Levinski, J.-P., and Pelletier, D. H., On the existence of strongly normal ideals over Pκλ, Archive for Mathematical Logic, vol. 30 (1990), pp. 5972.
[F]Foreman, M., Potent axioms, Transactions of the American Mathematical Society, vol. 294 (1986), pp. 128.
[JP]Jech, T. J. and Prikry, K., Ideals over uncountable sets: applications of almost disjoint functions and generic ultrapowers, Memoir no. 214, American Mathematical Society, Providence, Rhode Island, 1979.
[Si]Silver, J., On the singular cardinals problem, Proceedings of the international congress of mathematicians (Vancouver, 1974), Vol. 1, Canadian Mathematical Congress, Montreal, 1975, pp. 265268.
[So1]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.
[So2]Solovay, R. M., Strongly compact cardinals and the GCH, Proceedings of the Tarski symposium, Proceedings of Symposia in Pure Mathematics, vol. 25, American Mathematical Society, Providence, Rhode Island, 1974, pp. 365372.

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