Omitting types for infinitary [0, 1]-valued logic

We describe an infinitary logic for metric structures which is analogous to $L_{\omega_1, \omega}$. We show that this logic is capable of expressing several concepts from analysis that cannot be expressed in finitary continuous logic. Using topological methods, we prove an omitting types theorem for countable fragments of our infinitary logic. We use omitting types to prove a two-cardinal theorem, which yields a strengthening of a result of Ben Yaacov and Iovino concerning separable quotients of Banach spaces.Comment: 22 page

Omitting uncountable types, and the strength of [0,1][0,1]-valued logics

We study $[0,1]$-valued logics that are closed under the {\L}ukasiewicz-Pavelka connectives; our primary examples are the the continuous logic framework of Ben Yaacov and Usvyatsov \cite{Ben-Yaacov-Usvyatsov:2010} and the {\L}ukasziewicz-Pavelka logic itself. The main result of the paper is a characterization of these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages

Omitting Types and the Baire Category Theorem

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.Comment: 17 page

Model-theoretic Characterizations of Large Cardinals

We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In \bL_{\kappa, \kappa}, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vop\v{e}nka's Principle

Erd\H{o}s-Rado Classes

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erd\H{o}s-Rado Theorem--into the notion of a combinatorial Erd\H{o}s-Rado class. These classes are closely related to Erd\H{o}s-Rado classes, which are those from which we can build generalized indiscernibles and blueprints in nonelementary classes, especially Abstract Elementary Classes. We give several examples and some applications

On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

We study finitely generated models of countable theories, having at most countably many nonisomorphic finitely generated models. We intro- duce a notion of rank of finitely generated models and we prove, when T has at most countably many nonisomorphic finitely generated models, that every finitely generated model has an ordinal rank. This rank is used to give a prop- erty of finitely generated models analogue to the Hopf property of groups and also to give a necessary and sufficient condition for a finitely generated model to be prime of its complete theory. We investigate some properties of limit groups of equationally noetherian groups, in respect to their ranks

Omitting types in operator systems

We show that the class of 1-exact operator systems is not uniformly definable by a sequence of types. We use this fact to show that there is no finitary version of Arveson's extension theorem. Next, we show that WEP is equivalent to a certain notion of existential closedness for C$^*$ algebras and use this equivalence to give a simpler proof of Kavruk's result that WEP is equivalent to the complete tight Riesz interpolation property. We then introduce a variant of the space of n-dimensional operator systems and connect this new space to the Kirchberg Embedding Problem, which asks whether every C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. We end with some results concerning the question of whether or not the local lifting property (in the sense of Kirchberg) is uniformly definable by a sequence of types in the language of C$^*$ algebras.Comment: 25 pages; final version to appear in Indiana University Mathematics Journal; significant clarification of the exposition and a couple new results, including the fact that LLP is equivalent to the local matrix ultraproduct lifting propert

The \Gamma-Ultraproduct and Averageable Classes

We consider an ultraproduct that is designed to omit a fixed set of unary types $\Gamma$, called the $\Gamma$-ultraproduct. The $\Gamma$-ultraproduct is not always well-behaved, but we discuss several general conditions under which it is and several examples

Free algebras, amalgamation, and a theorem of Vaught for many valued logics

We investigate atomicity of free algebras and various forms of amalgamation for BL and MV algebras, and also Heyting algebras, though the latter algebras may not be linearly ordered, so strictly speaking their corresponding intuitionistic logic does not belong to many valued logic. Generalizing results of Comer proved in the classical first order case; working out a sheaf duality on the Zarski topology defined on the prime spectrum of such algebras, we infer several definability theorems, and obtain a representation theorem for theories as continuous sections of Sheaves. We also prove an omitting types theorem for fuzzy logic, and formulate and prove several of its consequences (in classical model theory) adapted to our case; that has to do with existence and uniqueness of prime and atomic models.Comment: arXiv admin note: substantial text overlap with arXiv:1304.0707, arXiv:1304.0612, arXiv:1304.0760, arXiv:1302.304

Spaces of Types in Positive Model Theory

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory