Not a mathematician and it's been a number of years since I read GEB, but my rough takeaway of the incompleteness is self-referential systems are magical because they can create statements that can't be proven in that system. "This sentence is false" or "Can God create a burrito so hot He cannot eat it" and all that.

So if Wheeler is saying the universe comes out of quantum observation, then the connection seems to be a self-referential Strange Loop of consciousness/observation/participation along the lines of "we're just the universe observing itself"

Wouldn't that be a limitation of language rather than the ability to form concepts?

Again, not a mathematician so I'm likely butchering all this, but my layperson's understanding is Godel showed there's a kind of equivalence mapping between english language and mathematical symbolic language. It's all just information juggling (and you can use the equivalence to translate things you can prove with mathematics into the other trickier languages).

So if sub-matter quantum-woo is just information juggling, then it's the ability to have self-referentiality that makes for some interesting properties.

I don't know the context, but Claus Kiefer was on the Physics Frontiers podcast recently talking about this paper:

https://arxiv.org/abs/2305.07331

Gödel's undecidability theorems and the search for a theory of everything

"I investigate the question whether Gödel's undecidability theorems play a crucial role in the search for a unified theory of physics. I conclude that unless the structure of space-time is fundamentally discrete we can never decide whether a given theory is the final one or not. This is relevant for both canonical quantum gravity and string theory."

> His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring[0].

So why leave it solely to the mathematicians.

[0] https://www.quantamagazine.org/how-godels-proof-works-202007...

On these, you can define some paradoxes in compiler (lisp specially) by redefining predicates (functions that return either true or false). On most cases, it either returns an error, or spawns a debugger.

I think it's simply that Godel's incompleteness theorem had strong, foundation-shaking implications outside the context of mathematics alone. Philosophy and science were greatly affected.