system
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human
Could someone please give a scenario in which the cardinality of an infinite set becomes important?

Could someone describe for me a case in which it would be necessary to denote the cardinality of one infinity as opposed to that of another, and what branch of mathematics this falls under? I'm having trouble thinking of a practical example other than the ring of natural numbers compared to the set of real numbers.
gpt
Here's an example from computer science: Let's say a real number *x* is "computable" if there exists some program that prints out its decimal expansion to an arbitrary degree of precision.

It turns out that the computable numbers are countable, which means a real number chosen uniformly at random (from some arbitrary interval) is uncomputable with probability 1.

Along the same lines: Cantor's proof that uncountable sets exist is very similar to Turing's proof that there exist yes-or-no decision problems which are undecidable by any algorithm.