What are the possible applications of Countable Infinite Sets and Power Sets in areas that are not strictly mathematical?
Also I want to know the significance they carry. What was not possible before the concepts of countability and power sets were introduced and what became possible afterwards? How did the introduction of these concepts change our thinking and outlook and the areas they affect?
Please explain in plain language, possibly with examples, as I am from a non-mathematical background. Thank you in advance.
PS: The applications need not be in strictly practical fields either (as Asaf Karagila keeps pointing there aren't any). Please help!
The notion of cardinality as formalized and explored by Cantor introduced a whole new set of questions about the reals that wouldn't have occurred to mathematicians before Cantor. Before Cantor, it was well-known that the reals and the rationals were very different infinite sets, but the difference was generally described by the fact that the reals have a completeness property (loosely, convergent sequences of reals converge to another real -- not so in the rationals, it isn't too difficult to construct a sequence of rationals that converges to $\sqrt{2}$). Once you can distinguish countable and uncountable sets (so after Cantor), the whole picture changes.
For example, in terms of the first-order theory of linear order types (in particular, ignoring algebraic properties), the reals can be completely characterized by the following properties:
1) uncountable
2) no endpoints
3) dense in itself (between any two distinct reals there is another real)
4) no uncountable strictly monotonic sequences
5) contains a countable dense subset (the rationals -- i.e. between any two distinct reals there is a rational)
(If you're wondering why completeness isn't on this list, completeness is a second-order property, quantifying over sets of reals -- "all sets which have an upper bound have a least upper bound.")
Note, though, how many of these properties rely on cardinality. The next interesting question is whether all of these properties are necessary. In particular, consider dropping the last property. Is there a linear order type which satisfies the first 4 properties but contains no subset order-isomorphic to the reals? The answer turns out to be yes -- first construct an uncountable tree with no uncountable branches or levels (actually a counterexample to what would be a natural generalization of Konig's lemma), then look at the lexicographic order on nodes of that tree. The result is a linear order as described above, known as an Aronszajn line (or Specker type). This whole construction can be done in ZFC.
There are other interesting questions here. Consider whether there are alternatives to the last property above. Perhaps it is enough to consider that the linear order has a countable chain condition -- i.e. there are no uncountable collections of pairwise disjoint open intervals. Does this give order types which contain a suborder isomorphic to the reals? This question turns out to be independent of ZFC -- the counterexample is called a Souslin line, and whether such a thing exists depends on the model of ZFC you are using (some have them, some don't).
Again, the point is that these questions about the reals, and possible alternatives to them, would not have occurred to anyone before the development of a formal definition of cardinality for infinite sets, in terms of bijections. Cantor's ideas brought on something of a crisis in our understanding of the reals, and the results (mostly independence results lately) continue to flood in.