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

07.11.2013 15:00 - 16:30

B. Farkas (U Wien)

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.

Organiser:

KGRC

Location:
SR 101, 2. St., Währinger Str. 25