By way of background for the uninitiated, "countability" is a concept related to set theory. At first blush it may seem that some sets are finite and some are infinite, and all infinite sets are the same size. But it can be shown that that last assertion isn't true. Some infinite sets are bigger than others. An infinite set is "countable" if it is the same size as the set of counting numbers (1,2,3, etc). Another way of looking at it is that a countable set can be lined up and counted off -- "The first one, the second one," etc. You'll never finish counting them off because the set is infinite. But no matter what element I think of, you will get to it eventually. Relating that last description to the counting numbers, we can line up the counting numbers in order (first 1, then 2, then 3...) and count them off. We'll never finish. But you can pick any number (say, 153,472) and we will get to it eventually. The counting numbers are countable. The rational numbers (numbers that can be expressed as one integer divided by another) are countable. The real numbers aren't countable.
Anyway, the person who was explaining this was correct in what he said. But the other one was just not getting it. He kept saying things along the lines of "but you'll never get done" and "But how can it be countable if you can't finish counting it?" I realized what his hangup was. He was focused on what he felt the definition of "countable" should be, based on his real world understanding of the word. But in math, the definitions are precise. They are often motivated be real world perceptions, but that desn't mean that they fully align. And this is an example. You may not be able to finish counting the rationals (or the counting numbers for that matter), but they are countable.
I wanted to interrupt them to say, "don't think about what counting means to you. Think about what the definition says."
But I didn't. I just shook my head and kept on playing Trivia Crack.
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