This blog post gets way too caught up in Gödel numbers, which are merely a technical detail (specifically how the encoding is done is irrelevant). A clever detail, but a detail nonetheless. Author gets lost in the sauce and kind of misses the forest for the trees. In class, we used Löb's Theorem[1] to prove Gödel, which is much more grokkable (and arguably even more clever). If you truly get Löb, it'll kind of blow your mind.
As a non-mathematician I always wondered about one thing. Because the way I interpret the Incompleteness Theory is that "you cannot have a universal system of infinite expressiveness, because you will need a more expressive one to prove it".
In other words, you can't have a top-down universal system. But you very well can have well described ones perfectly describe observable behaviour without defects.
We have these things called systems: a "system" is anything that follows rules: a board game, traffic, the English language, math, C++, etc. Some systems are smart and they can talk about themselves, but others can't. For example, Tic-Tac-Toe can't talk about Tic-Tac-Toe, but English can talk about English.
Gödel is interested in smart systems because dumb systems are boring.
Some systems are useful: they are "useful" if they always say true things. So math is more useful than English. I can lie in English, but I can't lie in math. (Formally, this is what we call consistency).
So here's a problem for you: suppose we have a smart-useful-system, call it SUS. SUS should be able to say "SUS is useful." It can talk about itself and it can't lie, so we should have no problem, right?
Gödel showed that if our system can actually say that about itself, it wasn't useful to begin with. For a few centuries, philosophers and mathematicians were trying to come up with the "one perfect system": useful, smart, and also complete (it can say all true things), and a few more properties. Turns out such a system is impossible.
NB: I use the words "say" or "talk about" in a very hand-wavy fashion, sometimes I mean Prove(), sometimes I mean Entail(). The details are very nuanced, and this isn't meant to be a deep dive.
Other names for Gödel encoding: Digital. Binary. Zorros and Unos.
Today Gödel encoding is so pervasive, it’s easy to miss that everything is trivially Gödel encladed. Because like most everything invisible, it’s right in front of us.
We Gödel our memes and gift cards, and (pick your poison) pr0ns. Colors and AI’s, lax ASMR’s and our (sneaky don’t read me) terms of service. Even this very small humble .
Gödel isn’t eating the world. Gödel already pööped it.
That period was encoded in a symbol string, i.e. it is a bit string.
Today we encode everything in bi-symbol strings.
This was not common when Gödel crafted his incompleteness theorem. And at the time it was a novel approach for setting up a context for testing the limits of computing.
Some people can still be struck by it as novel when reading the proof, because in context it was, and still feels that way. But today "symbol string" representation is ordinary and pervasive.
Morse didn't conceptually extend encoding to self-referential symbolic systems. Morse's insight was pure communication of symbols devoid of meaning.
Important but nowhere near the same.
Today, general symbolic encoding is viewed as trivial. Every symbol we have is pervasively encoded as bits, so of course entire expressions are. So Morse's code might seem comparable.
But what Gödel invented went well beyond Morse. We are just jaded with regard to his insight now.
Of course you can encode self-references in morse code, how could morse prevent that? Just use the same lisp syntax as in the article and then encode using morse code instead of Gödel numbering.
The purpose of Gödel numbering is to represent an arbitrary-length string of symbols as a single integer which allows you to manipulate it using Peano arithmetic.
But it is not like Gödel invented binary as you seem to suggest. Baudot code (a 5-bit character encoding) was in use in 1870’s.
In any case, Gödel-numbering is the least interesting part of the the theorem. The groundbreaking idea is creating statements about theorems.
Yeah, I think it would be better to first explain the liar's paradox to give the broad brush strokes, and then go into the details of Gôdel numbering.
It seems like most expositions of Gödel's incompleteness theorem go into a surprising amount of detail about Gödel numbering. In a way it's nice though, because you see that the proof is actually pretty elementary and doesn't require fancy math as a prerequisite.
[1] https://inference-review.com/article/loebs-theorem-and-curry...