Fallen Cardinals
Abstract
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compPmu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega1,mualeph0). Consequently, adding a generic filter to the quotient algebra Pmu(mu)=P(mu)/[mu]<mu collapses mualeph0 to aleph1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.
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