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We prove that for every Borel equivalence relation $E$, either $E$ is Borel reducible to $mathbb{E}_0$, or the family of Borel equivalence relations incompatible with $E$ has cofinal essential complexity. It follows that if $F$ is a Borel equivalence relation and $cal F$ is a family of Borel equivalence relations of non-cofinal essential complexity which together satisfy the dichotomy that for every Borel equivalence relation $E$, either $Ein {cal F}$ or $F$ is Borel reducible to $E$, then $cal F$ consists solely of smooth equivalence relations, thus the dichotomy is equivalent to a known theorem.
We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consist of so-called emph{Mammen spaces}, where a Mammen space is a triple $(U,mathcal S,mathcal C)$, where $U$ is a non-empty set (the universe), $mathcal S$ is a perfect Hausdorff topology on $U$, and $mathcal Csubseteqmathcal P(U)$ together with $mathcal S$ satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-J{o}rgensen, who conjectured that the existence of a complete Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology. On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants $mathfrak u_M$ and $mathfrak u_T$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. We show $mathfrak u_M=mathfrak u_T=2^{aleph_0}$ follows from Martins Axiom, and, contrastingly, we show that $aleph_1=mathfrak u_M=mathfrak u_T<2^{aleph_0}=aleph_2$ in the Baumgartner-Laver model.
We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorv{c}evi{c} theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todorv{c}evi{c}, Moore, Ngyuen Van Th{e}, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.
The Doob convergence theorem implies that the set of divergence of any martingale has measure zero. We prove that, conversely, any $G_{deltasigma}$ subset of the Cantor space with Lebesgue-measure zero can be represented as the set of divergence of some martingale. In fact, this is effective and uniform. A consequence of this is that the set of everywhere converging martingales is ${bfPi}^1_1$-complete, in a uniform way. We derive from this some universal and complete sets for the whole projective hierarchy, via a general method. We provide some other complete sets for the classes ${bfPi}^1_1$ and ${bfSigma}^1_2$ in the theory of martingales.
The Omitting Types Theorem in model theory and the Baire Category Theorem in topology are known to be closely linked. We examine the precise relation between these two theorems. Working with a general notion of logic we show that the classical Omitting Types Theorem holds for a logic if a certain associated topological space has all closed subspaces Baire. We also consider stronger Baire category conditions, and hence stronger Omitting Types Theorems, including a game version. We use examples of spaces previously studied in set-theoretic topology to produce abstract logics showing that the game Omitting Types statement is consistently not equivalent to the classical one.
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