Fallen Cardinals

Abstract

We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra compP_mu(mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra Comp Col(omega_1,mu^{aleph_0}). Consequently, adding a generic filter to the quotient algebra P_mu(mu)=P(mu)/[mu]^{<mu} collapses mu^{aleph_0} to aleph_1. Another corollary is that the Baire number of the space U(mu) of all uniform ultrafilters over mu is equal to omega_2. The corollaries affirm two conjectures by Balcar and Simon. The proof uses pcf theory.