Hi Georgina,
"The 'division' of reality is not exactly the same as the Liar problem because the Image reality can be part of the object reality while also not being it, so it does not have to be excluded but I think it shows that such a mathematical oddity is not unprecedented."
"The barber paradox" is a variation of the liar's paradox which goes something like -- if the barber shaves all the men who don't shave themselves, who shaves the barber? (If the barber shaves himself, he does *not* shave all the men who don't shave themselves.)
B. Russell denied that any such self-referential statement is more than noise. Let's take it at face value, though, and assume that your image reality (that of the observer, or quantum operator) belongs to the same set as the object reality (the barber who shaves all the men who don't shave themselves) for every act except the self-referential.
This dualism ultimately results in quantum entanglement; i.e., the operator never observes himself, and every individual (every element of the set of those who don't shave themselves) are in a superposition of states (shaved or unshaved) until "shaved." IOW, the operator only belongs to the set when shaving (observing) and at no other time. The entire reality is observer-created.
This raises the question, at what time is the shaver never shaving? -- in actuality, never.
J. A. Wheeler found a way out of this dilemma, by applying information theory to a continuously self-referential universe, objective, unitary and participatory. This restores classical orientation entanglement to quantum mechanics, and obviates quantum entanglement -- because the operator and the object (members of your image and object realities) are correlated at all times. Joy Christian realized that the critical properties of orientability and initial condition are ultimately responsible for all observed quantum correlations, and quantum entanglement is no more than an illusion caused by the assumption of nonlocal events (IOW, the assumption that there are times when the barber isn't shaving).
Point is, one cannot exclude the "barber" from the set of those being shaved without adding a layer of non-physical interpretation to the physics. A local realistic (classical) framework does not need that superfluous interpretation; there is complete 1 to 1 correspondence between elements of the theory and elements of the reality.
Tom