The amalgamation spectrum

by Baldwin and Kolesnikov and Shelah. [BKSh:927]
J Symbolic Logic, 2009
For every natural number k^*, there is a class boldmath {K}_* defined by a sentence in L_{omega_1, omega} that has no models of cardinality > beth_{k^*+1}, but mbox {boldmath {K}}_* has the d isjoint amalgamation property on models of cardinality <= aleph_{{k^*}-3} and has models of cardinality aleph_{{k^*}-1} . More strongly, For every countable ordinal alpha^*, there is a class mbox {boldmath {K}}_* defined by a sentence in L_{omega_1, omega} that has no models of cardinality > beth_{alpha}, but mbox {boldmath {K}}_* has the disjoint amalgamation property on models of cardinality <= aleph_{alpha} . Similar results hold for arbitrary kappa and L_{kappa^+, omega} .


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