Chains of Structures and of Theories
A seminar talk at Sabancı University, March 20, 2013.
The union of a chain of fields is a field. The union of a chain of vector-spaces with their scalar-fields is still a vector-space, but it may have strictly lower dimension than the spaces in the chain. A model-theoretic result of the 1950s called the Chang--Los-Suszko Theorem relates these observations to the logical form of the theories of the structures in the chains.
Instead of looking at chains of models of a fixed theory, one may fruitfully look at chains of theories themselves. Such a chain might consist of the theories of fields equipped with finite numbers of commuting derivations; or of the theories of vector-spaces with predicates for linear dependence of finite numbers of vectors. I shall discuss some results concerning these and other examples.
My notes (12 pp., A5 paper, revised April 4, 2013):