[1] https://inference-review.com/article/loebs-theorem-and-curry...
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.
Or is this too reductive?
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.
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.
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.
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.
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.
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.
https://www.youtube.com/watch?v=PpSxqde0af4
This is another good exposition.
Löb gets you to the main idea faster, but Gödel numbering is the part that makes it feel like the system is actually doing it itself.
Without that step, it can start to feel a bit too close to the liar paradox.