# Hamburg Set Theory Workshop, February 10, 2014

The Hamburg Set Theory Workshop 2014 is part of the ML Colloquium of the Arbeitsbereich Mathematische Logik und interdisziplinäre Anwendungen der Logik. Everyone is cordially invited to attend. There will be an informal lunch in the student restaurant (Mensa) and an informal dinner.

11:00-12:00 Andrew Brooke-Taylor (Bristol) Evasion of large cardinals LUNCH BREAK Merlin Carl (Konstanz) Infinite Time Algorithmic Randomness Break Yurii Khomskii (Vienna) Suslin proper forcing and regularity properties Break Wolfgang Wohofsky (Vienna) Strong measure zero sets and their characterization via translations

### Abstracts

Andrew Brooke-Taylor (Bristol)
Evasion of large cardinals

The evasion number e is a cardinal characteristic of the continuum introduced by Blass in connection with questions about homomorphisms of groups. However, the definition of e itself is very combinatorial, and generalises easily to larger cardinals. I will talk about lifting inequalities from the well-studied ω case up to a large cardinal κ, focusing especially on the consistency of e>b in this context. This is joint work with Jörg Brendle.

Merlin Carl (Konstanz)
Infinite Time Algorithmic Randomness

We consider algorithmic randomness for machine models of infinitary computations. We show that a theorem of Sacks, according to which a real x is computable from a randomly chosen real with positive probability iff it is recursive holds for many of these models, but is independent from ZFC for ordinal Turing machines. Furthermore, we define an analogue of ML-randomness for Infinite Time Register Machines and show that some classical results like van Lambalgen’s theorem continue to hold.

Yurii Khomskii (Vienna)
Suslin proper forcing and regularity properties

Recently, in joint work with Vera Fischer and Sy Friedman, we obtained a number of new results separating regularity properties on the Δ13, Σ13 and Δ14 levels of the projective hierarchy. In this talk I will present some of the methods we used to obtain these results, focusing on the concept of “Suslin/Suslin+ proper forcing”, a strengthening of Shelah’s concept of proper forcing which only works for easily definable forcing notions, and which was a crucial technical ingredient in our proofs.

Wolfgang Wohofsky (Vienna)
Strong measure zero sets and their characterization via translations

The Galvin-Mycielski-Solovay theorem confirms a conjecture of Prikry saying that a set of reals is strong measure zero if and only if it can be translated away from each meager set. This connection gives rise to a variety of new “notions of smallness”. It is possible to generalize the theorem to arbitrary locally compact Polish groups. However, some amount of compactness seems to be necessary: we show that the theorem consistently fails for the Baer-Specker group Zω. This is joint work with Michael Hrusak and Ondrej Zindulka.