Theories of Finite Type Related to Mathematical Practice

Abstract

Publisher Summary This chapter deals with the systems of second order, including a number that was selected for study only on syntactic grounds. The present exposition differs by concentrating on finite-type theories that directly reflect logical features of practice and in which everyday mathematics is readily developed. The chapter also presents closure conditions on universes of sets and functions. The elementary closure conditions lead to finite-type structures. Further, closure conditions include quantification functionals for each domain that is regarded as fixed (or definite) and recursion functionals for natural numbers. A number of finite-type theories based on these closure conditions are also presented in the chapter followed by recursion-theoretic models for finite-type theories.