### 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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