Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
Can we even find examples of infinity in nature?
If by nature mean physical universe, note that there is no proof that an infinite set exists. While some of us would like to believe that universe is infinite, this cannot be proved. You also have to remember that while we wish to perceive reality as a continuous process, we cannot measure it in a precise accuracy. Given two pieces of metal of length 1m (by the same ruler) they might differ by a few atoms, which means an actual increase in length since while very very small atoms still have a physical size. So we cannot really point out an infinite object in the universe, but we also fail pointing out something which is "exactly half" of another thing, etc.
If, on the other hand, you mean "in mathematics outside of set theory", then $\aleph_1$ is somewhat hidden. The reason is that in non-set theoretic constructions there is often little importance to cardinalities but rather countability; the cardinality of the continuum; and perhaps the power set of that.
However there are still mild appearances of $\aleph_1$. For example Shelah proved in 1980 that there is a group of size $\aleph_1$ such that every proper subgroup of it is countable. Another example (although slightly more set theoretic in nature) is the solution for Whitehead problem's for $\aleph_1$. If we assume in addition to ZFC a certain set theoretical assumption then every $W$-group of size $\aleph_1$ is free; if we prove a different assumption (again, with ZFC) then there is a $W$-group of size $\aleph_1$ which is not free.
Somewhat related:
- Proofs given in undergrad degree that need Continuum hypothesis?
- Examples of sets whose cardinalities are $\aleph_{n}$, or any large cardinal. (not assuming GCH)
- Does infinity and zero really exist?
How about this : the set of the possible repartition of probability of presence of, say, an electron is a continuum (well at least in most models!), while the set of its possible energy levels is countable.
The question as asked is somewhat ill-defined. I'm taking the word "nature" to mean "the thing described by physics", but what does it mean to say that some mathematical object is found "in nature"?
Such a phrase can only have meaning relative to some scheme of associating mathematical objects to physical "things". It's easy to construct such a scheme: e.g. if I consider an interpretation where I associate "$\aleph_1$" to the apple sitting on the table and don't assign meaning to any mathematical objects, then I have found $\aleph_1$ in nature: it's sitting on the table.
Maybe more interesting is a more 'standard' interpretation associated with a physical theory. The interpretations typically seen in a physical theory don't tend to associate cardinal numbers to objects. In particular, they don't assign $\aleph_1$ to any sort of object, and this fact is a banality.
But, in this case, we can turn to higher-order concepts. e.g. a version of Newtonian physics based upon ZFC+CH says that the number of points in any region of space is $\aleph_1$.