Post History

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 \(1\cdot n = 1 < 1 \cdot \aleph_0 = \aleph_0\)

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.

<details>
<summary>Click to show/hide operation tables</summary>

\[
\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}
\]

</details>

I hope you do something interesting with these!