I considered deleting the post, but this seems more cowardly than just admitting I was wrong. But TIL something!
this post was submitted on 06 Jan 2024
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That's not really how counting infinite sets works.
Suppose you have the set {1,2,3} and another set {2,4,6}. We say that both sets are of equal cardinality because you can map each element in the first set to a unique element in the second set (the mapping is "one to one"/injective), and every element has something mapped to it (the mapping is onto/surjective).
Compare the number of integers to the number of even integers. While it intuitively seems like there should be more integers than even integers, that's not actually the case. If you map 1 to 2, 2 to 4, 3 to 6, 4 to 8, ..., n to 2n, then you'll see both sets actually have the same number of things in them because that mapping is one to one and onto.
There's similarly the same number of real numbers as numbers between 0 and 1.
But there's more numbers between 0 and 1 than there are integers.