Relations on computable structures

TL;DR: Gődel's incompleteness theorem from 1931 is an astonishing early result of computable mathematics as discussed by the authors, which established the rigorous mathematical foundations for the computability theory, and showed that there are problems in the theory of ordinary whole numbers which cannot be decided from the axioms.

Abstract: Gődel’s incompleteness theorem from 1931 is an astonishing early result of computable mathematics. Gődel showed that “there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms.” The work of Gődel, Turing, Kleene, Church, Post and others in the mid-1930’s established the rigorous mathematical foundations for the computability theory. However, even in 1930, van der Waerden in his Moderne Algebra I introduced an explicitly given field as one “whose elements are uniquely represented by distinguishable symbols with which addition, subtraction, multiplication and division can be performed in a finite number of operations.” He showed that if a field is given explicitly, then every simple extension of that field is also given explicitly. In his pioneering paper on non-factorability of polynomials from 1930, van der Waerden essentially proved that an explicit field (F,+, ·) does not necessarily have an algorithm for splitting polynomials in F [x] into their irreducible factors. In the 1950’s, Frohlich and Shepherdson used the precise notion of a computable function to obtain a collection of results and examples about explicit rings and fields. Several years later, Rabin and Mal’tsev did an extensive investigation of computable groups and other computable (recursive, constructive) algebras. In the 1970’s, Metakides and Nerode initiated a systematic study of computability in mathematical structures and constructions by using modern computability-theoretic tools, such as the priority method. Computable mathematics explores the algorithmic content (effectiveness) of notions, constructions and theorems in classical mathematics. It starts by defining effective analogues of classical concepts in algebra and model theory. If we begin with structures, and effectivize the notion of a structure, we arrive at the notion of a computable structure. An algebraic