Computable model theory

TL;DR: In the last few decades there has been increasing interest in computable model theory and the work of Turing, Gödel, Kleene, Church, Post, and others in the mid-1930s established the rigorous mathematical foundations for the computability theory.

Abstract: In the last few decades there has been increasing interest in computable model theory. Computable model theory uses the tools of computability theory to explores algorithmic content (e¤ectiveness) of notions, theorems, and constructions in various areas of ordinary mathematics. In algebra this investigation dates back to van der Waerden who in his 1930 book Modern Algebra de…ned an explicitly given …eld as one the elements of which are uniquely represented by distinguishable symbols with which we can perform the …eld operations algorithmically. In his pioneering paper on non-factorability of polynomials from 1930, van der Waerden essentially proved that an explicit …eld (F;+; ) does not necessarily have an algorithm for splitting polynomials in F [x] into their irreducible factors. Gödel’s incompleteness theorem from 1931 is an astonishing early result of computable model theory. Gödel 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 Turing, Gödel, Kleene, Church, Post, and others in the mid-1930’s established the rigorous mathematical foundations for the computability theory. In the 1950’s, Fröhlich and Shepherdson used the precise notion of a computable function to obtain a collection of results and examples about explicit rings and …elds. For example, Fröhlich and