# Sh:720

- Kojman, M., & Shelah, S. (2001).
*Fallen cardinals*. Ann. Pure Appl. Logic,**109**(1-2), 117–129. arXiv: math/0009079 DOI: 10.1016/S0168-0072(01)00045-8 MR: 1835242 -
Abstract:

We prove that for every singular cardinal \mu of cofinality \omega, the complete Boolean algebra {\rm comp}{\mathcal P}_\mu(\mu) contains as a complete subalgebra an isomorphic copy of the collapse algebra {\rm Comp}\;{\rm Col}(\omega_1,\mu^{\aleph_0}). Consequently, adding a generic filter to the quotient algebra {\mathcal P}_\mu(\mu)={\mathcal 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.

- published version (13p)

Bib entry

@article{Sh:720, author = {Kojman, Menachem and Shelah, Saharon}, title = {{Fallen cardinals}}, journal = {Ann. Pure Appl. Logic}, fjournal = {Annals of Pure and Applied Logic}, volume = {109}, number = {1-2}, year = {2001}, pages = {117--129}, issn = {0168-0072}, doi = {10.1016/S0168-0072(01)00045-8}, mrclass = {03G05 (03E04 03E40 54D80)}, mrnumber = {1835242}, mrreviewer = {Stefan Geschke}, doi = {10.1016/S0168-0072(01)00045-8}, note = {\href{https://arxiv.org/abs/math/0009079}{arXiv: math/0009079}}, arxiv_number = {math/0009079}, dedication = {Dedicated to Petr Vop\v{e}nka} }