If a number is infinite (like “Aleph-Null”) then how can there be numbers larger or smaller than it? Shouldn’t that be impossible?

Question

If a number is infinite (like “Aleph-Null”) then how can there be numbers larger or smaller than it? Shouldn’t that be impossible?

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Answer 1

If you have two sets of items, and you can draw a line from each item of one set to each item of another set with no leftovers, then the two sets are the same size. If you do have leftovers in one set but none in the other, then the first set is bigger than the second set. Cantor’s diagonalization proof is a proof that the set of all integers is smaller than the set of real numbers, despite both being infinite.