Fallen Cardinals

Menachem Kojman; Saharon Shelah

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Fallen Cardinals

Menachem Kojman·Saharon Shelah

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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.

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BibTeX RIS

Menachem Kojman, Saharon Shelah. 2000-09-07. Fallen Cardinals. https://arxiv.org/abs/math/0009079

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