2013 seminar talk: Some problems related to Borel ideals: Towers, Luzin-type families, forcing (in)destructibility, and more

Talk held by Barnabas Farkas (KGRC) at the KGRC seminar on 2013-11-07.


I will discuss the possible existence of (maximal) towers in Borel ideals. I will prove the following results: Theorem 1. After adding uncountable many Cohen reals, there are towers in every tall Borel P-ideals. Theorem 2 (Brendle). It is consistent that there are no towers in any Borel P-ideals.

Related to Theorem 2, I will show that although we cannot expect more, namely "domination" from $\sigma$-centered forcing notions, the Localization forcing (which is $\sigma$-$n$-linked for every $n$) dominates every Borel P-ideals. Here the reverse implication is still an open problem. 

Theorem 2 will lead us to the next natural question, namely the possible existence of so-called idealized Luzin-type families of size $\omega_2$. To obtain such a family (very probably) we need some kind of iterated destruction of the ideal without Cohen reals (at least). I will show that the Random forcing works for the summable ideal but not for the density zero ideal.

Bottom menu

Kurt Gödel Research Center for Mathematical Logic. Währinger Straße 25, 1090 Wien, Austria. Phone +43-1-4277-50501. Last updated: 2010-12-16, 04:37.