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Register a new userUniversal and complete sets in martingale theory
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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.
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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.
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-graduate) student in Mathematics should know about foundations of this science.
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 generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.
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 behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of analytic equivalence relations. This is not possible in general. We show that, under some acyclicity (and also topological) assumptions, this is widely possible. In particular, we prove that, for any non-self dual Borel class {Gamma}, there is a concrete finite =< c-antichain basis for the class of Borel relations, whose closure has acyclic symmetrization, and which are not potentially in {Gamma}. Along similar lines, we provide a sufficient condition for =< c-reducing G 0. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.
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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