Dear Olaf,
once more an essay with Gödel's results involved. And I like it. But I also have some suspicions.
If I take your analogy serious, I conclude that you have proven consciousness to be an emergent phenomenon. Since consciousness is needed to at all be aware of something like 'emergence' and consciousness in your framework is an emergent property of reality, it follows that consciousness has proven to itself to be an emergent property.
But now 'emergence' is 'provable' and not anymore a true, but unprovable statement.
So, once more, what is 'provable' seems to depend heavily on what one puts into the equation in the first place as an axiom. Since your axiom is 'emergence', your output will be 'emergence'.
I really don't know how such logical figures should or can say anything true and reliable about *physical* reality and the underlying origins (if there are some) of for example consciousness.
For me, it seems to only say that we don't know how consciousness comes about from dumb matter, because we cannot prove this to be factually the case. But this does not mean that consciousness must be 'emergent', in the sense that a clump of matter awakes to some awareness.
I think what you did is simply state that the concept of emergence cannot be proven to be evidently valid for 'things' like consciousness, quantum behaviour, gravity and cosmology.
Isn't there a big danger that we label things we cannot formalize in the traditional Newtonian manner as 'emergent' - and by the very premise that consciousness should itself be 'emergent' it then seems that the very concept of 'emergence' naturally emerges not only in the fact that consciousness exists, but then also 'necessarily' in the fact that this consciousness can facilitate at all the idea of 'emergence'?
The big question for me is whether or not the concept of emergence is merely an - unprovable - idea, or rather a physical necessity.
Another question for me is what or who adds new 'axioms' that can built the upwards ladder of emergence? Surely, all we can observe in physical reality should be consistent with other facts. Therefore the hierarchy of axiomatic systems should be more or less well-ordered. But is there an *end* to this hierarchy - and if yes, why and where does it stop and if no, what does the latter then mean at all for physics?