Classification theory for non-elementary classes. I: The number of uncountable models of \(\psi \in L_{\omega _ 1,\omega}\)
The paper is related to the following problem due to Baldwin: Can a sentence of the language L(Q) have exactly one uncountable model? The author continues his investigation from one of his previous papers [Isr. J. Math. 20, 127-148 (1975; Zbl 0324.02038)] in the context of \(L_{\omega_ 1,\omega}\). Assuming that \(2^{\aleph_ n}<2^{\aleph_{n+1}}\) holds for every \(n<\omega\), the main theorem states that every sentence in \(L_{\omega_ 1,\omega}\) has many non- isomorphic models of size \(\aleph_ n\) for some \(n<\omega\) or has models in all infinite cardinalities. It follows from the theorem that every sentence in \(L_{\omega_ 1,\omega}\) has at least \(2^{\aleph_ 1}\) non-isomorphic uncountable models. The central idea of the paper is to transfer properties of models of cardinality \(\aleph_ n\) to countable models and then back from countable models to models of higher cardinalities. To this end, the author introduces the concepts of n-dimensional diagrams, good sets and generalized amalgamation which culminate in the concept of excellent classes - the cutting edge for classification of atomic models of some countable theories. Excellent classes are parallel to totally transcendental theories, they have models in all infinite cardinalities, satisfy the amalgamation property and Łos conjecture. A non-structure theorem is proved for non-excellent classes stating that they have many non-isomorphic models in some \(\aleph_ m\) if they have an uncountable model and if \(2^{\aleph_ n}<2^{\aleph_{n+1}}\) holds for an appropriate n.
scientific article; zbMATH DE number 3878903