TITLE: Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?
QUESTION [9 upvotes]: Can we even find examples of infinity in nature?

REPLY [2 votes]: 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 $\aleph_1$.