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Chang's conjecture and powers of singular cardinals

  • Menachem Magidor (a1)

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In [2] Galvin and Hajnal showed, as a corollary to a more general result, that if , is a strong limit cardinal, then . They established similar bounds for powers of singular cardinals of cofinality greater than ω. Jech and Prikry in [3] showed that the Galvin-Hajnal bound can be improved if we assume that ω1 carries an ω2 saturated ω1 complete, nontrivial ideal. (See [7] for definitions), namely: under the given assumption provided is a strong limit cardinal.

In this paper we show that the same conclusion can be derived from Chang's Conjecture (see below) which is, at least consistencywise, a weaker assumption than the existence of an ω2 saturated ideal on ω1. We do not know if assumptions like these are necessary for obtaining the result.

Our notations and terminology should be understood by any reader acquainted with set theory. Chang's Conjecture is the following model theoretic assumption introduced by C. C. Chang:

which is deciphered as follows: Every structure 〈A, R,…〉 in a countable type where ∣A∣ = ω2, RA, ∣R∣ = ω1 has an elementary substructure: 〈A′,R′,…〉 where ∣A′∣ = ω1 and ∣R′∣ = ω0. The consistency of Chang's Conjecture modulo the existence of Ramsey cardinals is claimed in [5].

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References

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[1]Fodor, G., Stationary subsets and regressive functions, Acta Scientiarum Mathematicarum, (Szeged), vol. 27 (1966), pp. 105110.
[2]Galvin, F. and Hajnal, A., Inequalities for cardinal powers, Annals of Mathematics, vol. 101 (1975), pp. 491498.
[3]Jech, T. and Prikry, K., On ideals of sets and the power set operation (to appear).
[4]Shidenfield, J., Unramified forcing, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13 (1971), Providence, R.I., pp. 351382.
[5]Silver, J., The dependence of Kurepa Conjecture and two Cardinal's Conjectures in model theory, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13 (1971), Providence, R.I., pp. 383390.
[6]Silver, J., On the singular cardinals problem, Proceedings of the International Congress of Mathematics, Vancouver, 1974, Vol. I, pp. 265268.
[7]Solsvay, R. M., Real valued measurable cardinals, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13 (1971), Providence, R.I., pp. 397428.

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