# DISJOINT AMALGAMATION IN LOCALLY FINITE AEC

@article{Baldwin2017DISJOINTAI, title={DISJOINT AMALGAMATION IN LOCALLY FINITE AEC}, author={John T. Baldwin and Martin Koerwien and Michael C. Laskowski}, journal={The Journal of Symbolic Logic}, year={2017}, volume={82}, pages={98 - 119} }

Abstract We introduce the concept of a locally finite abstract elementary class and develop the theory of disjoint $\left( { \le \lambda ,k} \right)$ -amalgamation) for such classes. From this we find a family of complete ${L_{{\omega _1},\omega }}$ sentences ${\phi _r}$ that a) homogeneously characterizes ${\aleph _r}$ (improving results of Hjorth [11] and Laskowski–Shelah [13] and answering a question of [21]), while b) the ${\phi _r}$ provide the first examples of a class of models of a…

## 15 Citations

Disjoint n-Amalgamation and Pseudofinite Countably Categorical Theories

- MathematicsNotre Dame J. Formal Log.
- 2019

This paper shows that if a countably categorical theory T admits an expansion with disjoint $n-amalgamation for all $n$ then T is pseudofinite, and two generic theories of equivalence relations are examined, and it is shown that both are pseud ofinite.

Shelah’s eventual categoricity conjecture in universal classes: part II

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- 2017

AbstractWe prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the…

Shelah's eventual categoricity conjecture in universal classes: Part I

- MathematicsAnn. Pure Appl. Log.
- 2017

A lower bound for the Hanf number for joint embedding

- MathematicsFundamenta Mathematicae
- 2022

In [13] the authors show that if $\mu$ is a strongly compact cardinal, $K$ is an Abstract Elementary Class (AEC) with $LS(K)<\mu$, and $K$ satisfies joint embedding (amalgamation) cofinally below…

Higher-dimensional obstructions for star reductions

- MathematicsFundamenta Mathematicae
- 2021

A $*$-reduction between two equivalence relations is a Baire measurable reduction which preserves generic notions, i.e., preimages of meager sets are meager. We show that a $*$-reduction between…

The first omega alephs: From simplices to trees of trees to higher walks

- MathematicsAdvances in Mathematics
- 2021

Amalgamating many overlapping Boolean algebras

- Mathematics
- 2016

In general, two overlapping Boolean algebras always extend to a common Boolean algebra, but three may not. We prove a new sufficient condition for $n$ overlapping Boolean algebras to have a common…

Non-absoluteness of Hjorth's Cardinal Characterization

- Mathematics
- 2021

In [5], Hjorth proved that for every countable ordinal α, there exists a complete Lω1,ω-sentence φα that has models of all cardinalities less than or equal to אα, but no models of cardinality אα+1.…

Characterizing the existence of a Borel complete expansion

- Mathematics
- 2021

We develop general machinery to cast the class of potential canonical Scott sentences of an infinitary sentence Φ as a class of structures in a related language. From this, we show that Φ has a Borel…

A survey on tame abstract elementary classes

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- 2015

Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification…

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