IDEALS OVER UNCOUNTABLE SETS Theorem 7.2.1. Assume that K-. carries a nonregular ultrafilter. Then a) If N, 2 u N then 2 = 2 . 1 w1 x X 0) b) If N is a strong limit then 2 K . H 0 Under the same assumption, Benda and Ketonen proved in [3] that 2 = H implies 2 = tf . Some of the results presented in this paper were announced in [8].

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