Oberseminar mathematische Logik
Time and location
Tuesdays 16-18 in room N0.003, or by appointment (see below), Endenicher Allee 60.
- 08 November (10.30 -12.00, Room SR 0.007): Ralf Schindler (Münster): MM++ implies (∗)
For many years, forcing axioms and Woodin's Pmax axiom (∗) were competing natural hypotheses which both decided the value of the continuum to be ℵ2. The relationship of, say, Martin's Maximum with (∗) was a bit of a mystery. We show that Martin's Maximum++ implies Woodin's axiom (∗). This is joint work with David Asperó.
- 10 December: Wolfgang Wohofsky (Wien): A Sacks amoeba forcing preserving distributivity of P(omega) / Fin
In my talk, I would like to present joint work with Otmar Spinas, in which we show that it is consistent that h (the distributivity number of P(omega)/fin, in other words, the least number of maximal almost disjoint families without a common refinement) is strictly smaller than add(s_0) (i.e., the least number of Marczewski null sets whose union is not Marczewski null, where a set is Marczewski null if each perfect set has a perfect subset disjoint from it). More explicitly, we show that this relation between h and add(s_0) holds in the model obtained by a countable support iteration of length omega_2 of a specific kind of Sacks amoeba forcing which happens to have the pure decision and the Laver property, and therefore does not add Cohen reals. The model actually satisfies h=cov(M).
- 14 January: Sandra Müller (Wien): Infinite decreasing chains in the Mitchell order
It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. We make a first step in understanding this case by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length ω1. This is joint work with Omer Ben-Neria.