Generalizing classical and effective model theory in theories of operations and classes (Q810008)

The author proposes a family of theories of operations and classes with an aim of developing abstract versions of model-theoretic results. The theories in question are two-sorted, with one kind of variable for individuals and the other for classes. All the theories used assume a common group of axioms. The author works mainly with three theories, \(FMT_ 0\), FMT, \(FMT_{\Omega}\). The aim of the author's work is to give an abstract development of model theory in these theories and in a way to look as much as possible like classical mathematics. \(FMT_ 0\) generalizes portions of countable and recursive model theory; FMT generalizes portions of countable and hyperarithmetical model theory and \(FMT_{\Omega}\) provides a generalization of the classical L(Q)- completeness theorem and of an admissible version of L(Q)-completeness due to Bruce and Keisler. The paper continues the program originated by Feferman.