Is there a stable/consistent extension of wheel theory to alephs?
In wheel theory (MathSE) (see also Wikipedia here and nLab here), $0/0$ 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...)
2 answers
The following users marked this post as Works for me:
| User | Comment | Date |
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| Ripheus24 |
Thread: Works for me This is what I was hoping for, and more. Thank you so much! |
Feb 27, 2026 at 01:50 |
Any commutative semiring with a chosen multiplicative submonoid can be extended to a wheel of fractions. Cardinal addition and multiplication form a commutative semiring (glossing over the set/proper class distinction), so we can easily talk about the wheel of cardinal fractions with respect to the submonoid of multiplication on \(\mathbb{N}_{>0}\). Formally, the elements of this wheel are equivalence classes of pairs of cardinals \(\kappa : \mu\) under this equivalence relation: \[\kappa_1 : \mu_1 \sim_{\mathbb{N}^\times_{>0}} \kappa_2 : \mu_2 \iff \exists n_1, n_2 \in \mathbb{N}_{>0} \text{ such that } n_1\kappa_1 = n_2 \kappa_2 \wedge n_1\mu_1 = n_2\mu_2 .\]
But as usual, we can use the cardinal \(\kappa\) to refer to the class of \(\kappa : 1\) when there is no confusion.
The reciprocal of \(\aleph_0\), then, is the class of \(1 : \aleph_0\), and we can identify \(\aleph_0/\aleph_0\) as the class containing the fraction product \(\aleph_0\cdot 1 : 1 \cdot \aleph_0 = \aleph_0 : \aleph_0\). What else is equivalent to this pair? Exactly the pairs \(\kappa : \mu\) such that there are non-zero naturals \(n_1, n_2\) satisfying \(n_1 \aleph_0 = \aleph_0 = n_2\kappa = n_2\mu\). Clearly, as \(n_1\) and \(n_2\) are finite and non-zero, their specific values don't matter and \(\kappa = \mu = \aleph_0\). So as requested, we have a value \(\aleph_0/\aleph_0\) that stands apart. In fact, in this wheel, every infinite cardinal \(\kappa\) induces a value \(\kappa/\kappa\) that is distinct from \(0/0\), \(1\), and any other \(\kappa'/\kappa'\).
With respect to the suitability of \(/\aleph_0\) as an infinitesimal, we may wish to define a partial order on our wheel of cardinal fractions. Say that \(\kappa_1 : \mu_1 < \kappa_2 : \mu_2\) iff \(\kappa_1\mu_2 < \kappa_2\mu_1\). This partial order on pairs extends consistently to the wheel of fractions, because the equivalence relation \(\sim_{\mathbb{N}^\times_{>0}}\) adds or removes the same positive finite factors to both sides of the inequality when replacing one pair with another in its equivalence class. Then it follows that \(0 < /\aleph_0 < /n\) for every \(n \in \mathbb{N}_{>0}\):
- \(0 < /\aleph_0\) because \(0 \cdot \aleph_0 = 0 < 1 \cdot 1 = 1\)
- \(/\aleph_0 < /n\) because \(1\cdot n = n < 1 \cdot \aleph_0 = \aleph_0\)
With this partial order, we can make some statements about \(\aleph_0/\aleph_0\). While \(0/0\) is incomparable with any other value, we can place \(\aleph_0/\aleph_0\) strictly between 0 and \(\aleph_1\), matching your observation about the range of values \(\aleph_0/\aleph_0\) could take using an intuitive interpretation of cardinal division. Further up the ladder, for any pair of infinite cardinals \(\kappa_1 < \kappa_2\), we have that \(0 < \kappa_1/\kappa_1 < \kappa_2\), but \(\kappa_1/\kappa_1\) is incomparable with any cardinal between (inclusive) 1 and \(\kappa_1\) and with \(\kappa_2/\kappa_2\).
To help demonstrate the structure of this wheel a little bit more fully, I've printed up some operation tables for addition, multiplication, and ordering for some of the simplest cardinal fractions, per the above definitions.
Click to show/hide operation tables
\[ \begin{array}{c|c*{15}c} + & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & 1 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \hline \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ 0 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & 1 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ /\aleph_1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & /\aleph_1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ /\aleph_0 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & /\aleph_0 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ 1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & 1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & 2 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \aleph_0 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \aleph_0 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \aleph_0 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \aleph_1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \aleph_1 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ /0 & \tfrac{\aleph_1}{0} & \tfrac{\aleph_0}{0} & /0 & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & /0 & /0 & \tfrac{\aleph_0}{0} & /0 & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ \end{array} \]\[ \begin{array}{c|c*{15}c} \times & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & 1 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \hline \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ 0 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & 0 & 0 & \tfrac{0}{\aleph_0} & 0 & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ /\aleph_1 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ /\aleph_0 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_0}{\aleph_0} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ 1 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & 1 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \aleph_0 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_0}{\aleph_0} & \tfrac{\aleph_0}{\aleph_0} & \aleph_0 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & \tfrac{0}{\aleph_0} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \aleph_1 & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & \tfrac{\aleph_1}{\aleph_0} & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \aleph_1 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0}\\ /0 & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & /0 & \tfrac{\aleph_0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \tfrac{\aleph_1}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{0}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0} & \tfrac{\aleph_1}{0}\\ \end{array} \]\[ \begin{array}{c|c*{15}c} \overset{^?}{\smash{<}} & \tfrac{0}{\aleph_1} & \tfrac{0}{\aleph_0} & 0 & /\aleph_1 & \tfrac{\aleph_0}{\aleph_1} & \tfrac{\aleph_1}{\aleph_1} & /\aleph_0 & \tfrac{\aleph_0}{\aleph_0} & 1 & \aleph_0 & \tfrac{\aleph_1}{\aleph_0} & \aleph_1 & \tfrac{0}{0} & /0 & \tfrac{\aleph_0}{0} & \tfrac{\aleph_1}{0}\\ \hline \tfrac{0}{\aleph_1} & = & \parallel & \parallel & < & < & < & < & < & < & < & < & < & \parallel & < & < & <\\ \tfrac{0}{\aleph_0} & \parallel & = & \parallel & < & < & < & < & < & < & < & < & < & \parallel & < & < & <\\ 0 & \parallel & \parallel & = & < & < & < & < & < & < & < & < & < & \parallel & < & < & <\\ /\aleph_1 & > & > & > & = & \parallel & \parallel & < & < & < & < & < & < & \parallel & < & < & <\\ \tfrac{\aleph_0}{\aleph_1} & > & > & > & \parallel & = & \parallel & < & < & < & < & < & < & \parallel & < & < & <\\ \tfrac{\aleph_1}{\aleph_1} & > & > & > & \parallel & \parallel & = & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & < & < & <\\ /\aleph_0 & > & > & > & > & > & \parallel & = & \parallel & < & < & < & < & \parallel & < & < & <\\ \tfrac{\aleph_0}{\aleph_0} & > & > & > & > & > & \parallel & \parallel & = & \parallel & \parallel & < & < & \parallel & < & < & <\\ 1 & > & > & > & > & > & \parallel & > & \parallel & = & < & < & < & \parallel & < & < & <\\ \aleph_0 & > & > & > & > & > & \parallel & > & \parallel & > & = & < & < & \parallel & < & < & <\\ \tfrac{\aleph_1}{\aleph_0} & > & > & > & > & > & \parallel & > & > & > & > & = & \parallel & \parallel & < & < & <\\ \aleph_1 & > & > & > & > & > & \parallel & > & > & > & > & \parallel & = & \parallel & < & < & <\\ \tfrac{0}{0} & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & \parallel & = & \parallel & \parallel & \parallel\\ /0 & > & > & > & > & > & > & > & > & > & > & > & > & \parallel & = & \parallel & \parallel\\ \tfrac{\aleph_0}{0} & > & > & > & > & > & > & > & > & > & > & > & > & \parallel & \parallel & = & \parallel\\ \tfrac{\aleph_1}{0} & > & > & > & > & > & > & > & > & > & > & > & > & \parallel & \parallel & \parallel & =\\ \end{array} \]I hope you do something interesting with these!
(Update: this answer is not complete; I made a misstep, as you can see in footnote 2. Sorry!)
I shall use a slightly different formalization of wheel theory I read from Sobociński's excellent Graphical Linear Algebra1, where the numbers on the wheel are considered as linear relations between input and output.
Thus, for example, ordinary numbers $n$ are considered as relations ${(x,y): y=nx}$ where the input is $n$ times the output. The inverses of such numbers, $\frac1n$ are considered as relations ${(x,y): x=ny}$ where the output is $n$ times the input.
The only difference between this theory and ordinary wheel theory is that there are two different instances of $\frac00$: one is the trivial relation $\{(x,y): 0x=0y\}$ that allows any input and output (Sobociński calls this $\bot$) and the other is the maximally restrictive relation $\{(x,y): x=0∧y=0\}$ that constrains both input and output to be 0 (Sobociński calls this $\top$).
It is not difficult to extend this to any abelian group. The only snag is that cardinals, as observed in the OP, constitute an abelian monoid and not an abelian group, so we get addition, multiplication, and division but not subtraction.2
However, as we still have subtraction of finite integers, the situation is not unlike that which we already have with wheels and $\frac00$. More precisely, adjoining alephs to Sobociński's system gives us a proper class of new operations $\aleph_\alpha$ each of which are equal to their antipodes and dominate any operators less than or equal to themselves (i.e. $\aleph_\alpha + x = \aleph_\alpha$ whenever $x ≤ \aleph_\alpha$).
But this answer is incomplete, as it is not fully clear how equational rules that allow addition would fit elegantly in this system (e.g. what is $\frac{\aleph_0}{\aleph_0} + \frac00$?).
1(See chapter 26 of his online book, "Keep calm and divide by zero".)
2 The usual way to make an abelian monoid into an abelian group is to form its [Grothendieck group](https://en.wikipedia.org/wiki/Grothendieck group), whose elements are differences $a - b$ where $a - a$ is set to 0 for all $a$. But this does not work properly, because e.g. $1 = (\aleph_0 - \aleph_0) + 1 = \aleph_0 + (-\aleph_0 + 1) = \aleph_0 + (-\aleph_0) = 0$.
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