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In §1, we observe that a weakly normal ideal has a saturation property; we also show that the existence of certain precipitous ideals is sufficient for the existence of weakly normal ideals. In §2, generalizing Solovay’s theorem concerning strongly compact cardinals, we show that ${\lambda }^{<\kappa }$ is decided if $P_{\kappa }\lambda$ carries a weakly normal ideal and λ is regular or cf λ ≤ κ. This is applied to solving the singular cardinal hypothesis.

We explore the possibility of forcing nonreflecting stationary sets of $P_{\kappa }\lambda$. We also present a $P_{\kappa }\lambda$ generalization of Kanamori’s weakly normal filters, which induces stationary reflection.