> While with differently-sized infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrela

> While with differently-sized infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else. The distinction between countable and uncountable is very practically useful. You can always encode the elements of a countable set such that every element is represented in the computer by a finite-length bit string, but you can never design an encoding that does this for every member of an uncountable set. No exceptions. In addition, this notion of distinguishing countability from uncountability can be applied to show that the set of Turing machines is countable (up to isomorphism). This immediately informs one of the limits of computation, and why, e.g. trying to design a computer or program that performs exact arithmetic operations on real numbers in general, or say finding the limit of a series in real numbers in general, is as achievable as squaring the circle. And make no mistake, computational modeling that today already permeates every aspect of society would look very different if exact operations on real numbers were computable and efficient. (and for the general programmer, no need to worry about the imprecision of floats and especially with financial stuff)