Recursive Functions and Church’s Thesis
1.1. The first part of this book was primarily concerned with mathematical proof ; we showed that the analogous concept in formal languages is that of formal deduction, after which the most interesting results were that certain intuitive mathematical assertions (such as the continuum hypothesis and its negation) are not deducible.
Our primary concern in the second part of the book is the notion of a determinate computational process, that is, the processing of information, or, briefly, the notion of an algorithm. In §2 we give a precise and presumably complete characterization of everything that can be obtained using computational algorithms. Then the most interesting results turn out to be assertions that certain intuitively defined functions cannot be computed by an algorithm (Chapter VI).
Both the theory of proof and the theory of computation can be presented in large part independently of one another. This is the approach we have adopted, even though it does not correspond to the historical development. But when the machinery of both theories has been developed to a certain point, it becomes possible to apply each theory to investigate the other. The third part of the book is devoted to such applications.
KeywordsPartial Function Total Space Recursive Function Computable Function Elementary Operation
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