Definitions of Recursive Functions

This chapter shows that certain kind of definitions of functions and relations always lead to recursive functions and relations. The simplest kind of definition of a function has the form. The chapter also highlights that certain kind of definitions of functions and relations always lead to recursive functions and relations. The intention is to expand the class of symbols which may be used on the right side of explicit definitions of recursive functions and relations. For simplicity, such symbols are called recursive symbols. The chapter presents several Propositions. It explains that if a function or a relation has an explicit definition or an inductive definition or a definition by cases in terms of recursive symbols, then it is recursive. Recursive symbols include variables, names of recursive functions and relations, propositional connectives, and bounded quantifiers.