Fallen cardinals

✍ Scribed by Menachem Kojman; Saharon Shelah

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 137 KB
• Volume: 109
• Category: Article
• ISSN: 0168-0072
• DOI: 10.1016/s0168-0072(01)00045-8

✦ Synopsis

We prove that for every singular cardinal of coÿnality !; the complete Boolean algebra Comp P ( ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(!1; ℵ 0 ). Consequently, adding a generic ÿlter to the quotient algebra P ( ) = P( )=[ ] ¡ collapses ℵ 0 to ℵ1. Another corollary is that the Baire number of the space U ( ) of all uniform ultraÿlters over is equal to !2. The corollaries a rm two conjectures of Balcar and Simon. The proof uses pcf theory.