Theories fully involving Time are Necessarily inequivalent

Let T be any well-defined theory that involves time. T could be a temporal logic, a theory of temporal moments, or a theory of things that evolve in time, etc. Say

(1) T is full iff it accounts for everything I know about time

The idea is a "full" theory that involves time in some way can account for the Moving Point argument, i.e. it can account in all respects for the "meta-time" behavior of a point moving along (in "real time", so to speak) of the theory's own model of time. You could interpret a set of axioms (relevant to time) {A_i} in a model M, both given by a Kripke structure K. But you don't have to, so I'll just call any theory involving time T. The point (more or less) is that one can always form the singleton set s = {T}. Having done this, the Moving Point argument applies again, and T necessarily doesn't account for all temporal behavior. Thus T is not "full" because it cannot account for everything we know about time. Here are two responses to this (not necessarily the only two):

(2) There is no full theory T. Boring.

(3) I am in possession of a full theory T_1, but I can't communicate all of it to you, and you are in possession of a full theory T₂, but can't communicate all of it to me.

If we demand of a theory that it *realistically* incorporates time (3) is the avenue to explore. The idea is there is a full theory T₁, and a full theory T₂, and they are necessarily ontologically distinct. They are necessarily distinct in the ontology of the theories themselves. So (3) is a kind of ontological consistency condition on any full theory involving time. (It doesn't matter if your universe has only physical instantiations of what people think are full theories, or if your universe includes the existence of mental objects that are full theories: any representation of the of theory in the universe (of the theory) will do).

Presumably, there are isomorphic subtheories of T₁ and T₂ that allow us to compare notes on what we know about time (though T₁ and T₂ themselves need not be isomorphic). But they are necessarily not equivalent. Therefore they do not have the properties we usually ascribe to numbers (or to the set {0, {0}} as given by ZFC). We assume when we teach someone 2 + 3 = 5, this fact is true independent of who considers it (or knows it), and (usually) it's independent of whether or not anyone is around to consider the fact. These are true regardless of the formal definition of these numbers and operations, and regardless of whether numbers are supposed to "exist" in some sense or not. That's fine. But if T is full the ontological situation is necessarily different (assuming (3)).

The idea that 2 + 3 = 5, however it's ontologically instantiated, is an equivalence class of it's instantiations (regarding it as a potentially full theory). Happily, can be sloppy about this equivalence class: we can think of it as having a representative element that's true, or of all of them being true, or we can assume it's a true assertion about the equivalence class independently of whether there is anything in the universe, etc. ("true" means all past current and future behaviors and objects that exist in any way and their relations are invariant with respect to any ontological instantiation of 2 + 3 = 5). The point is for *numbers* all of these are valid. But a (necessarily) full theory of time requires more careful ontological distinctions.

I'll argue this has critical consequences for any theory of physics that wants to be realistic with regard to time (and which have not been incorporated yet). And I'll argue this idea of a full theory of time extends to the notion of a full theory of qualia and to the notion of a full theory of existence.