05609396

j

05609396 Baldwin, John T. Kolesnikov, Alexei Shelah, Saharon The amalgamation spectrum. J. Symb. Log. 74, No. 3, 914-928 (2009). 2009 Association for Symbolic Logic, Poughkeepsie, NY EN disjoint amalgamation property size of models

doi:10.2178/jsl/1245158091

Summary: We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals. Theorem A. For every natural number $k$, there is a class ${\bold K}_k$ defined by a sentence in $L_{\omega_1,\omega}$ that has no models of cardinality greater than $\beth_{k+1}$, but ${\bold K}_k$ has the disjoint amalgamation property on models of cardinality less than or equal to $\aleph_{k-3}$ and has models of cardinality $\aleph_{k-1}$. More strongly, we can have disjoint amalgamation up to $\aleph_\alpha$ for $\alpha<\omega_1$, but have a bound on the size of models. Theorem B. For every countable ordinal $\alpha$, there is a class ${\bold K_\alpha}$ defined by a sentence in $L_{\omega_1,\omega}$ that has no models of cardinality greater than $\beth_{\omega_1}$, but $\bold K$ does have the disjoint amalgamation property on models of cardinality less than or equal to $\aleph_\alpha$. Finally we show that we can extend the $\aleph_\alpha$ to $\beth_\alpha$ in the second theorem consistently with ZFC and while having $\aleph_i\ll\beth_i$ for $0 < i\le \alpha$. Similar results hold for arbitrary ordinals $\alpha$ with $|\alpha|=\kappa$ and $L_{\kappa^+,\omega}$.