%
\label{prop:set-countable}
The collection of all $\tau$-sets is \textbf{countable}:
\[
    |\mathrm{Set}(\tau)|
    \;=\;
    |\tau\text{-Idx}|
    \;=\;
    \aleph_0.
\]

By definition (Chapter~\ref{ch:membership-divisibility}),
a $\tau$-set is an element of $\tau$-Idx
equipped with the $\in_\tau$ relation.
The underlying objects are the natural numbers
$\underline{1}, \underline{2}, \underline{3}, \ldots$
(after identification with $\mathbb{N}$ via Chapter~\ref{ch:set-operations}).
Since $\tau$-Idx is countably infinite
(it is in bijection with $\mathbb{N}$),
the collection of all $\tau$-sets is countable.