Abstract
This chapter discusses ordinal definitions and their closure ordinals. Inductive definitions are basic to the development of recursion theory. The chapter also defines ordinal recursion and recursion in a partial functional by means of inductive definitions. A brief summary of results on inductive definitions over the natural numbers is presented. The chapter explores the relation between non-monotone inductive definitions and ordinal recursion. Ordinal recursion can be defined by an inductive operator, and in return the theory of inductive definitions can be developed within the framework of ordinal recursion. The chapter provides an overview of ordinal recursion. The inductive operator (actually a class of operators) which defines ordinal recursion is discussed in the chapter. The chapter discusses two related operators with reference to this definition. The two propositions, which demonstrate the desired closure, can be proven by means of the recursion theorem.
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