Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
If there’s any way that you can invent a “pairing rule” between two sets A and B, such that every element in set A has exactly 1 unique partner in set B, and vice versa, then you have proven that the sets are the same size.
If you can’t(*) do that, but you *can* invent a “pairing rule” such that every element in A has a partner in B, *but* not every element in B has a partner in A, then you can say that set B is strictly larger than set A.
So here’s an example of two sets:
Set A is all the real numbers from 0 to 1. Set B, is all the real numbers from 0 to 2.
You might be tempted to think that set B is bigger, since it’s a wider interval, right?
But here’s a “pairing rule.” For every number in A, you multiply it by 2 to find its partner in B. And for every number in B, you divide by 2 to find its partner in A. This means the sets are the same size!
edit:
*: where I said “if you can’t,” that should probably actually say: “if you can prove it’s impossible”. Because it’s conceivable that there *is* a possible pairing-rule which assigns unique partners to both sets, but you just couldn’t figure out what that rule would be.
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