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Register a new userPseudocompactness and the Uniform Metastability Principle in Model Theory
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We prove that uniform metastability is equivalent to all closed subspaces being pseudocompact and use this to provide a topological proof of the metatheorem introduced by Caicedo, Duenez and Iovino on uniform metastability and countable compactness for logics.
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This is a largely expository paper about how groups arise or are of interest in model theory. Included are the following topics: classifying groups definable in specific structures or theories and the relation to algebraic groups, groups definable in stable, simple and NIP theories, definable compactifications of groups, definable Galois theory (including differential Galois theory), connections with topological dynamics, model theory of the free group.
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.
Let ${mathfrak C}$ be a monster model of an arbitrary theory $T$, $bar alpha$ any tuple of bounded length of elements of ${mathfrak C}$, and $bar c$ an enumeration of all elements of ${mathfrak C}$. By $S_{bar alpha}({mathfrak C})$ denote the compact space of all complete types over ${mathfrak C}$ extending $tp(bar alpha/emptyset)$, and $S_{bar c}({mathfrak C})$ is defined analogously. Then $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$ are naturally $Aut({mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{bar c}({mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{bar c}({mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{bar alpha}({mathfrak C})$ in place of $S_{bar c}({mathfrak C})$.
We survey discrete and continuous model-theoretic notions which have important connections to general topology. We present a self-contained exposition of several interactions between continuous logic and $C_p$-theory which have applications to a classification problem involving Banach spaces not including $c_0$ or $l^p$, following recent results obtained by P. Casazza and J. Iovino for compact continuous logics. Using $C_p$-theoretic results involving Grothendieck spaces and double limit conditions, we extend their results to a broader family of logics, namely those with a first countable weakly Grothendieck space of types. We pose $C_p$-theoretic problems which have model-theoretic implications.
Given a cardinal $kappa$ and a sequence $left(alpha_iright)_{iinkappa}$ of ordinals, we determine the least ordinal $beta$ (when one exists) such that the topological partition relation [betarightarrowleft(top,alpha_iright)^1_{iinkappa}] holds, including an independence result for one class of cases. Here the prefix $top$ means that the homogeneous set must have the correct topology rather than the correct order type. The answer is linked to the non-topological pigeonhole principle of Milner and Rado.
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