Fallen Cardinals

by Kojman and Shelah. [KjSh:720]
Annals Pure and Applied Logic, 2001
We prove that for every singular cardinal mu of cofinality omega, the complete Boolean algebra comp P_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.


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