"Wheel Theory", Extended Reals, Limits, and "Nullity": Can DNE limits be made to equal the element "$0/0$"?

First of all I'd like to say that I'm very amused by the fact that you and I independently decided to call wheel theory's $0/0$ 'nullity' after James Anderson's 'transreal arithmetic'.

I'm fairly sure that if you take the real projective line topology on $\mathbb{R}\cup\{\infty\}$ and append $\Phi$ as an open extension topology (i.e. the open sets are precisely the pre-existing open sets in $\mathbb{R}\cup \{\infty\}$ and the entire space $\mathbb{R}\cup\{\infty,\Phi\}=\odot_\mathbb{R}$) then you get a topological wheel. Furthermore I think this may be the only way to extend the ordinary real projective line to get a topological wheel (largely because of reasoning similar to that in your edit), but I haven't proved that.

This topology is somewhat reminiscent of generic points in the Zariski topology on the spectrum of a ring in that nullity is 'next to' every other number, but it's not exactly the same. Also it's somewhat natural in that it's the quotient topology of $\mathbb{R}^2$ under the equivalence relation $(a,b)\sim(c,d)$ iff $(a,b)$ and $(c,d)$ are not $(0,0)$ and $(a,b)=(e\cdot c,e \cdot d)$ for some nonzero $e$, which is just the construction of the real projective line without deleting $(0,0)$.

As far as the limits are concerned you almost get what you want. Every sequence converges to $\Phi$ and at most one other point. The non-$\Phi$ limit point exists iff the sequence converges in the real projective line topology and is equal to that limit.

Furthermore I think that this may be the best that you can do. If you consider any series $a_n\in\mathbb{R}$ that normally does not converge, but in your topological wheel converges to $\Phi$, then the series $(a_n, -a_n)$ converges in the product topology $\odot_\mathbb{R} \times \odot_\mathbb{R}$ to $(\Phi,\Phi)$, addition is a continuous map $+ : \odot_\mathbb{R} \times \odot_\mathbb{R} \rightarrow \odot_\mathbb{R}$ therefore the sequence $a_n - a_n=0$ must converge to $\Phi+\Phi=\Phi$ as well as the obvious limit of $0$.

Personally, I tend to think that “nullity” is is exactly the wrong name for 0/0, as “null” means “nothing” and 0/0 is anything but. Rather, I'd call it “omnity” after the fact that 0/0 is usually left undefined because it could literally be anything. My personal inclination would be to also use ⊙ to denote 0/0 precisely because that's also the preferred symbol of the wheel, thus reinforcing the notion that the element is a stand-in for “could be anything”; but I can understand the problems of conflation they would cause. Alternately, the symbol could be an underscore “_”, denoting the “fill in the blank” nature of the element.

My only substantive difference with Wheel Theory has to do with 0^0: as I understand it, the limit as you approach 0^0 is the same as 0/0; but the value of 0^0 itself should be 1, for much the same reason why 0! is 1: you're dealing with an empty product, which is 1.

As I see it, the most important contribution of Wheel Theory is an explicit unary operator for multiplicative inversion, debited by a prefixed “/”. Prior to Wheel Theory, I couldn't find a notation for inversion that wasn't some sort of binary operator, whether it be “1/x” or “x^{-1}”. It wasn't really all that important until you got to things like the Riemann Sphere, where we started getting examples of “inverses” that didn't give a product of 1. But at that point, it becomes quite important.