In [wheel theory (MathSE)](https://math.stackexchange.com/questions/994508/wheel-theory-extended-reals-limits-and-nullity-can-dne-limits-be-made-to) (see also [Wikipedia here](https://en.wikipedia.org/wiki/Wheel_theory) and [nLab here](https://ncatlab.org/nlab/show/wheel)), $0/0$ is more tractable than usual. There is a unary version of division, $/a$, which is involutive: $//a = a$, and $0/0$ is used as an absorbing element generally. So the theory goes on to allow for working with division by zero to a greater extent than usual.
Now for transfinite cardinals $\aleph_{\alpha}$, division is pretty much not defined. Even self-subtraction is indeterminate in that e.g. $\aleph_0 - \aleph_0$ might cover a situation in which a set of length $\omega$ was subtracted from, say, $\omega + \omega$, leaving countably many elements behind, so that the result of the cardinal subtraction is again $\aleph_0$. Or one might reduce a countable set so as to leave behind 1000 elements, etc. More generally, as $\aleph_0 + n = \aleph_0$, it is possible to have self-subtraction of an aleph leave any *n* behind, etc.
Similarly, then, and akin also to self-division of zero, $\frac{\aleph_0}{\aleph_0} = X$ can take any value in $[1, \aleph_0]$. (But dividing an aleph by itself and getting 0 is not granted.)
So, would it be possible to expand upon wheel theory to introduce $\aleph_0/\aleph_0$ as a sort of "top element" antipodal to $0/0$ as a bottom element? And then to have $/\aleph_0$, etc. as "cardinal infinitesimals"? (I'm trying to see if we can use infinite cardinals instead of infinite surreals or infinite hypernumbers in the Robinsonian sense, as the base for infinitesimals. Having looked over Jech's book on the axiom of choice, I was also minded to consider non-aleph infinite cardinals $A$ such that $1/A$ is more tractable, but getting to use the alephs for this purpose is my dream...)
Ripheus24
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2026-02-25T13:56:58Z (7 months ago)