Who's wasting whose time??

It was you who asked me a question: "In what sense is the set {0,1,2,3,4,...} twice as large as the set {0,2,4,6,8,...}?"

I gave you a layperson's answer: "The intuition would be something like this: half the members of the former are missing from the latter, so the former must be "twice as large" as the latter. Or perhaps it's something like this: the set of odd positive integers {1,3,5,7,9,...} is the same size as the set of even numbers {0,2,4,6,8,...}. Throw the two sets together, and you get the set {0,1,2,3,4,...}, which must "therefore" be twice as big as either of the other two."

Then you say that if I don't produce for you an example of something entirely irrelevant to the issue you asked about, I wasted your time by answering your question.

I picture two inquisitive people. One wonders how old our universe is, whether there are other universes, and if so how many. The other wonders how many sets of natural numbers there are, how many finite sets of natural numbers there are, and whether those two sets of sets of natural numbers are the same size. By careful reasoning, the second demonstrates the correct answers to her questions, which answers always have been, are now, and forever will be correct.

By observation and careful reasoning, the first comes up with a rough answer to his first question, and wild speculation about the second. Both are trying to satisfy their curiosity about the true states of affairs. One conspicuous difference is that the first relies on (notoriously unreliable but corrigible) observation as well as reasoning. Another is that the second knows she is right, whereas the first is entitled to no such confidence.