I know basic things about cardinality (I'm only in High School) like that since $\mathbb{Q}$ is countable, its cardinality is $\aleph_0$. Also that the cardinality of $\mathbb{R}$ is $2^{\aleph_0}$.
Are there any direct applications of these numbers outside of theoretical math?
I know this can be convenient for certain proofs and help understanding sets of numbers, but are there any applications of this?
Best Answer
The answer is an obvious no. For two main reasons:
With the exception of occasional naive approach to sets, there is little to no use of set theory outside theoretical mathematics. So any application would be indirect and purely coincidental.
Applied mathematics is not concerned with infinite objects. Let alone "vastly huge beyond any reasonable visualization and imagination of a human being" sizes of infinity.
It is important to understand that mathematics is not "merely a tool for engineers" (or physicists). It is a world filled with magic and mind boggling ideas which have absolutely nothing to do with this physical reality. Infinite sets is one of them. These ideas trickle slowly and some of them eventually get to the point where they have some use, but these uses are far from being "direct" in any sense of the word.
For example, by plain cardinality arguments it is easy to see that almost any function from $\Bbb R$ to itself is not continuous, or even Borel measurable. Almost any continuous function is nowhere differentiable, and almost all the differentiable functions are not continuously differentiable, and so on and so forth (although some of these arguments require more than sheer cardinality).
But have you ever seen someone "applying" everywhere-discontinuous functions to a real world situation? I can't recall anything like that (although it might be in some quantum theory sort of application I am unaware of).
As long as mankind is limited by a finite powers of perception we cannot even distinguish between $100^{100^{100^{100^{100}}}}$ and $\aleph_0$.
Might also be relevant: Can we distinguish $\aleph_0$ from $\aleph_1$ in Nature?