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We introduce the notion of first order amenability, as a property of a first order theory $T$: every complete type over $emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure in the same variables. Amenability of $T$ follows from amenability of the (topological) group $Aut(M)$ for all sufficiently large $aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [Amenability, connected components, and definable actions; E. Hrushovski, K. Krupi{n}ski, A. Pillay], we prove that if $T$ is amenable, then $T$ is G-compact, namely Lascar strong types and Kim-Pillay strong types over $emptyset$ coincide. This extends and essentially generalizes a similar result proved via different methods for $omega$-categorical theories in [Amenability, definable groups, and automorphism groups; K. Krupi{n}ski, A. Pillay] . In the special case when amenability is witnessed by $emptyset$-definable global Keisler measures (which is for example the case for amenable $omega$-categorical theories), we also give a different proof, based on stability in continuous logic. Parallel (but easier) results hold for the notion of extreme amenability.
Let $alphageq 2$ be any ordinal. We consider the class $mathsf{Drs}_{alpha}$ of relativized diagonal free set algebras of dimension $alpha$. With same technique, we prove several important results concerning this class. Among these results, we prove
We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version
We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For exam
We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.
In this paper the 3-valued paraconsistent first-order logic QCiore is studied from the point of view of Model Theory. The semantics for QCiore is given by partial structures, which are first-order structures in which each n-ary predicate R is interpr