Introduction to the handbook of recursive mathematics

Abstract

The chapter presents an introduction to the handbook of Recursive mathematics. Recursive or computable mathematics is the study of the effective or computable content of the techniques and theorems of mathematics. This introductory describes the historical roots of recursive mathematics, including a discussion on some of the major themes that have arisen in the study of recursive mathematics. The development of recursion theory in the 1940's and early 50's by Kleene, Post, Peter, Myhill, Rice, and many others, provided the basic tools for the first results in Recursive Mathematics, which appeared in the 1950's and early 60's. In particular, FrÖlich, Shepherdson, and Rabin provided explicit counterexamples of recursive fields, which failed to have factorization algorithms that justified Van der Waerden's claim. A number of important common mathematical themes emerged from the past work on recursive mathematics. These include: fixing a given recursive structure, ideally one that is universal for a large class of recursive structures, and study the complexity of various model-theoretical and algebraic constructions on that structure; find 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.