Introduction to the handbook of Recursive Mathematics

Abstract

This chapter describes the historical roots of recursive mathematics and discusses some of the major themes that have arisen in its study. The development of recursion theory in the 1940s and early 1950s by Kleene, Post, Peter, Myhill, Rice, and many others provided the basic tools for the first results in recursive mathematics, which appeared in the 1950s and early 1960s. In particular, Frolich, Shepherdson, and Rabin provided explicit counterexamples of recursive fields that failed to have factorization algorithms that justified Van der Waerden's claim. A number of important common mathematical themes have emerged from past work on recursive mathematics. These include fixing a given recursive structure—ideally the one that is universal for a large class of recursive structures—and studying the complexity of various model-theoretical and algebraic constructions on that structure; finding necessary and sufficient conditions for the existence of recursive or constructive models of a theory with given properties; and determining whether the class of all recursive models of a given structure is computable.