Is it possible to label particles of a continuum body?

Question

In basic continuum mechanics (e.g. fluid dynamics), we label particles of the continuum, i.e., each particle can be identified by a label, e.g., $p$. Then other quantities are defined accordingly, e.g., velocity: $V=\frac{\partial r(p,t)}{\partial t}|_p$ where $r$ is the location vector of $p$ and $t$ is time.

Now, I was wondering what kind of number $p$ is? It cannot be a natural number e.g. $1,2,3,...$, because the set of particles of continuum is uncountably infinite, while the set of natural numbers is countably infinite, so there cannot be a one-to-one correspondence. So, is it true to say $p\in\mathbb{R}$ (real numbers)?

Answer 1

Is it true to say $p \in R$ (real numbers) ?

Yes, or more generally $p \in R^n$ i.e. $p$ is a tuple of real numbers. For example, one way to label infinitesimal fluid parcels in a continuum model is to label them by their location at some fixed time $t_0$, so that

$p = (x,y,z) \iff \vec r(p, t_0) = (x,y,z)$