Abstract
Publisher Summary This chapter presents the first draft of the axioms for computation theories. A definition of computation theories is proposed and these structures are studied to get the first step in classifying the known theories. One of the difficulties in trying to compare and classify these theories has been the lack of a definition of recursion theory. The chapter outlines the basic properties of computation theories. This includes proving some of the basic results of ordinary recursion theory, and identifying some of the known theories as computation theories. The important concept of finiteness relative to a theory is introduced and studied. The approach in the chapter differs from the recursive and bounded definition of metarecursion theory. A set is finite relative to a theory if the functional representing quantification over that set is computable in the theory. This is a natural approach and allows for direct generalization of the fundamental properties of finite sets.
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