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.