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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.
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,math
This is a short introductory course to Set Theory, based on axioms of von Neumann--Bernays--Godel (briefly NBG). The text can be used as a base for a lecture course in Foundations of Mathematics, and contains a reasonable minimum which a good (post-g
We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generat
The literature provides dichotomies involving homomorphisms (like the G 0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation b
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 Omitti