Hi Alyssa,
and yes I agree, lots of moving parts!
"I can't help but think a mathematical model that captures the subjectivity of an observer could be represented with some kind of set theory."
To paraphrase Husserl's notion that 'consciousness is consciousness of something' one might say that 'observation is observation of sets of something'? Would an intuitionistic set theory then be a good place to start?
"I think this 'cut' between an observer and the world should have a big impact on the dynamics of both the world and an observer, especially if the observer's dynamics are not fixed in time."
What if the observer simply IS their observable world which includes observations of their own and others bodies, and the unobserved remainder is the potentially observable world? The cut would then be between the model's observable world (as a dynamic flux of observations) and its potentially observable world; or between the phenomenal actuality of - and non-phenomenal potentiality for - observational experience. Something like an 'objective world' could then emerge from the observational flux as an abstraction of that flux.
I think the 'cut' between observer and world, or subject and object, is just such a philosophical abstraction, and an unhelpful one at that.
So in terms of modelling an observable universe, such a universe would by simple definition have an observer at its centre, and there would be no need to separate the two. Much like it's difficult to physically separate a photon emitted from a distant star and its incidence on a retina with the subsequent neuronal stimulations and behavioural outputs ... where's the non-arbitrary 'cut'? The totality of the potentially observable universe is, in principle at least, described by the wave function for that universe, with the flux of actual observations being the ongoing, finite and actual 'measurement' outcome. Is this a quantum computing problem? And does the fact that the observational sub-system can also feed back on the whole system complicate things?
"I think what makes humans so interesting is our ability to extend our computation power beyond the brain, which I personally think why computers and machines are so important to collective human tasks (this is the extended model of cognition in psychology)."
Embodied cognition and Clark and Chalmers' extended mind theory are all part of what appears to me to be an increasingly commonly held notion that the 'individuated, isolated subject/observer' is something of an abstract concept and possibly a dead end path for thinking. Even Everett conceived of his observer sub-system as a physical automaton 'in' its branch world 'in' the universe. But I can't see how this third-person godlike analytic perspective can logically be part of a perspectival model for an observer dependent reality. Cos if we're all Wigner, and God is dead, then who's observing us?