We use iterations of elementary embeddings derived from countably complete ideals on ω1 to provide a uniform proof of some classical results connecting the number of models of cardinality א1 in various infinitary logics to the number of syntactic types over the empty set. We introduce the notion of an analytically presented abstract elementary class (AEC), which allows the formulation and proof...

Elementary Classes Motivations and Directions John T. Baldwin Why AEC? Categoricity and Complex Exponentiation Excellence– Generalized Amalgamation Eventual Categoricity Core Mathematics again Abstract Elementary Classes Motivations and Directions

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism X × <mml:mrow class="MJ...