G\"odel for Goldilocks: A Rigorous, Streamlined Proof of (a variant of) G\"odel's First Incompleteness Theorem

Gödel’s famous incompleteness theorems (there are two of them) concern the ability of a formal system to state and derive all true statements, and only true statements, in some fixed domain; and concern the ability of logic to determine if a formal system has that property. They were developed in the early 1930s. Very loosely, the first theorem says that in any “sufficiently rich” formal proof system where it is not possible to prove a false statement about arithmetic, there will also be true statements about arithmetic that cannot be proved. Most discussions of Gödel’s theorems fall into one of two types: either they emphasize perceived cultural and philosophical meanings of the theorems, and perhaps sketch some of the ideas of the proofs, usually relating Gödel’s proofs to riddles and paradoxes, but do not attempt rigorous, complete proofs; or they do present rigorous proofs, but in the traditional style of mathematical logic, with all of its heavy notation, difficult definitions, technical issues in Gödel’s original approach, and connections to broader logical theory before and after Gödel. Many people are frustrated by these two extreme types of expositions and want a short, straight-forward, rigorous proof that they can understand. Over time, various people have realized that somewhat weaker, but still meaningful, variants of Gödel’s first incompleteness theorem can be rigorously proved by simpler arguments based on notions of computability. This approach avoids the heavy machinery of mathematical logic at one extreme; and does not rely on analogies, paradoxes, philosophical discussions or hand-waiving, at the other extreme. This is the just-right Goldilocks approach. However, the available expositions of this middle approach have