I’m a mathematician who spent many hours thinking about Goedel theorem and adjacent topics. The author in a self-deprecating way says that he’s not a mathematician, but just a programmer. Bear no mind to that: this is one of the best popular expositions of Goedel theorem I’ve seen. Everything is very accurately explained, there are no silly mistakes and untruths one often sees mentioned in context of Goedel theorem, and even the original proof is sketched in a very approachable manner, striking a very good balance between getting across all crucial ideas, and skipping most of the distracting technical aspects (which are important and interesting, but I think only to mathematicians, haha). Great job all around.
Unfortunately, you missed that the author's proof does not actually prove inferential undecidability (sometimes called "inferential incompleteness") of Russell's Principia Mathematica for the following reason:
The [Gödel 1931] proposition *I'mUnprovable* does **not** exist in Principia Mathematica because it violates restrictions on orders of propositions that are necessary to avoid paradoxes (such as Russell's Paradox). Gödel numbers (and in the author's case Lisp expressions) leave out the order of a proposition with the consequence that the Diagonal Lemma **cannot** be used to construct the proposition *I'mUnprovable*.
Furthermore, existence of the proposition I'mUnprovable contradicts the following fundamental theorem of provability that goes all the way back to Euclid:
I'm going to need someone to come along and write up a layman's summary of this article. :) Does it change the end result of Gödel? Or is it basically the equivalent of saying what some restricted programming languages are doing, that we can get "good enough" results and not have these problems by restricting what we can do in the language?
The article linked below explains why [Gödel 1931] did not prove inferential undecidability of Russell's Principia Mathematica and likewise why the formalization of [Gödel 1931] in Lisp proof being discussed is also invalid:
It absolutely is jargon. The fact that you explained it by saying it's "mathematics" instead of "English" suggests as much. Jargon means that it's technical language specific to a field and not used in everyday vernacular. What you wrote can't even be typed on a standard keyboard. It's jargon.
I'm pointing it out because its use is not so innocent. Jargon is often used to obfuscate meaning, especially when it means something that would otherwise be easy to say plainly.
I think I maybe noticed another typo... The last two axioms don't have matching parentheses? It looks like one more is opened than closed if I counted right.
