From discrete to continuous arguments in logic. What is needed and why
In recent years there has been considerable activity in generalizing to continuous settings techniques from logic (set theory and model theory) that initially were devised for discrete settings. Several model-theoretic frameworks have been proposed as formalisms for the continuous setting (Chang-Keisler, Henson, Ben-Yaacov); however, they have turned out to be equivalent. I will state a maximality theorem that, among other things, characterizes such frameworks. The characterization is in terms of omitting types, and applies not only to the already proposed formalisms, but to a wide range of logics, namely, logics for which certain topologies are uniformizable.