The Gödel Incompleteness Theorem and Decidability over a Ring

TLDR: In this article, the authors give an exposition of Godel's result in an algebraic setting and also a formulation (and essentially an answer) to Penrose's problem and show a converse to this result: any sufficiently infinite ordered field with this latter property is necessarily real closed.

Abstract: Here we give an exposition of Godel’s result in an algebraic setting and also a formulation (and essentially an answer) to Penrose’s problem. The notions of computability and decidability over a ring R underly our point of view. Godel’s Theorem follows from the Main Theorem: There is a definable undecidable set ovis Z. By way of contrast, Tarski’s Theorem asserts that every definable set over the reals or any real closed field R is decidable over R. We show a converse to this result: Any sufficiently infinite ordered field with this latter property is necessarily real closed.