Continuous Model Theory

Publisher Summary This chapter discusses continuous model theory. The sets of formulas and sentences are defined in the usual manner. Connectives are interpreted as functions mapping the cartesian powers. While it is relatively easy to find examples of continuous logics, it is not so certain that many adequate logics can be found. This is because the point 1 ∈ X has to have certain special properties which would preclude X from being an arbitrary compact Hausdorff space. A large body of results in 2-valued model theory is concerned with the problem of providing syntactic characterizations for classes of models which are closed under certain algebraic operations. A class K M is said to be an existential, universal, universal-existential, or positive class if it is an arbitrary intersection of finite unions of classes. A theorem analogous to the above four theorems and which does not appear to carry over to the general case is the previously mentioned result of Keisler on Horn classes and reduced products.