Jenann Ismael at the IAI:
Any attempt to describe the universe as a totality inevitably involves self-reference. This isn’t something that one often confronts in physics. Most day-to-day physics is modelling other systems: cells, gases, planets. We maintain a separation of subject and object, or of investigator and system being investigated. And even though cosmology is explicitly devoted to the study of the universe as a whole, it is customary in cosmology to maintain the imaginative fiction that we – the people modelling the universe – are looking at it from the outside. We adopt, that is to say, the God’s Eye View.
Ultimately, though, we are part of the universe. And that means that however we regiment the universe, whatever regime we work in, if we aim for a theory that describes all of existence, self-reference is unavoidable. Any system that is modelling the universe as a whole – aiming for full coverage of all of existence – is going to encounter self-reference. This is something that we can ignore in some contexts. It matters in others.
The people that have unavoidably encountered it are people who are trying to program an artificial general intelligence (an AGI).
More here.