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DISJOINT AMALGAMATION IN LOCALLY FINITE AEC
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 1 / March 2017
- Published online by Cambridge University Press:
- 21 March 2017, pp. 98-119
- Print publication:
- March 2017
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- Article
- Export citation
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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 complete sentence in 
${L_{{\omega _1},\omega }}$ where the spectrum of cardinals in which amalgamation holds is other that none or all.
