Note that the title is misleading. This is really countable dimension factored spaces, which is much better, since it allows for the possibility of something kind of like continuous time, where between any two points in time, you can specify a time strictly between them.
Yeah, also note that the history of X given Y is not actually a well defined concept. There is only the history of X given y for y∈Y. You could define it to be the union of all of those, but that would not actually be used in the definition of orthogonality. In this case hF(X|y), hF(V|y), and hF(Z|y) are all independent of choice of y∈Y, but in general, you should be careful about that.
I think that works, I didn't look very hard. Yore histories of X given Y and V given Y are wrong, but it doesn't change the conclusion.
I could do that. I think it wouldn't be useful, and wouldn't generalize to sub partitions.
I don't know, the negation of the first thing? A system that can freely model humans, or at least perform computation indistinguishable from modeling humans.
If by "pure" you mean "not related to history/orthogonality/time," then no, the structure is simple, and I don't have much to ask about it.
Yeah, this is the point that orthogonality is a stronger notion than just all values being mutually compatible. Any x1 and x2 values are mutually compatible, but we don't call them orthogonal. This is similar to how we don't want to say that x1 and (the level sets of) x1+x2 are compatible.
The coordinate system has a collection of surgeries, you can take a point and change the x1 value without changing the other values. When you condition on E, that surgery is no longer well defined. However the surgery of only changing the x4 value is still well defined, and the surgery of changing x1 x2 and x3 simultaneously is still well defined (provided you change them to something compatible with E).
We could define a surgery that says that when you increase x1, you decrease x2 by the same amount, but that is a new surgery that we invented, not one that comes from the original coordinate system.
Thanks for writing this.
On the finiteness point, I conjecture that "finite dimensional" (|B| is finite) is sufficient for all of my results so far, although some of my proofs actually use "finite" (|S| is finite). The example with real numbers is still finite dimensional, so I don't expect any problems.