We develop a basic translation procedure that translates a hod mouse to an equivalent mouse. Unlike the translation procedure used by Steel and Zhu, our procedure works in a coarse setting without assuming AD+. Nevertheless, the procedure resembles the one developed by Steel in [8]. We use the translation procedures to answer a question of Trevor Wilson. Namely, we show that if there is a stati...

The omitting types theorem of infinitary logic is used to prove that every small II set of analysis or any small 2. set of set theory is constructible. In what follows we could use either the omitting types theorem for infinitary logic or the same theorem for what Grilliot[2] calls (eA)-logic. I find the latter more appealing. Suppose i_ is a finitary logical language containing the symbols of ...

In these notes we go through the very basic results from descriptive set theory. In our approach to the topic, the use of games is more explicit than what is usual in the literature. We do not assume any previous knowledge on set theory nor on first-order logic. For the history of the topic, see [Mo] (Borel* sets are due to D. Blackwell and Vaught codes are due to R. Vaught). For further readin...

ly, we can describe a two-player game with positional winning conditions by a directed game graph G = (V, V0, V1, E), with a partioning V = V0 ∪ V1 of the nodes into positions where Player 0 moves and positions where Player 1 moves. The possible moves are described by the edge relation E ⊆ V × V . We call w a successor of v if (v, w) ∈ E, and we denote the set of all successors of v by vE. To d...

On the surface, the emergent descriptive theory of represented spaces is the extension of both classical [36] and effective [52] descriptive set theory from its usual setting of Polish spaces to the much larger class of represented spaces. There is far more to it, though: By extending the setting, suddenly connections to areas such as category theory (with a certain topos-theoretic flavour) and...

Metric spaces, Borel and analytic sets, Baire property and measurability, dichotomies, equivalence relations. Possible topics: determinacy, group actions, rigidity, turbulence.

We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from P. Our main focus lies on questions of derandomisation, and we prove that there is ...

Throughout the development of finite model theory, the fragments of first-order logic with only finitely many variables have played a central role. This survey gives an introduction to the theory of finite variable logics and reports on recent progress in the area. For each k 1 we let Lk be the fragment of first-order logic consisting of all formulas with at most k (free or bound) variables. Th...