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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> If a number is infinite (like “Aleph-Null”) then how can there be numbers larger or smaller than it?
There are lots of different notions of numbers and sizes in mathematics. One of the most basic and intuitive concepts of “size” is cardinality. If two sets can be matched up one-to-one, then they have the same cardinality. For example, we can match up {1, 2, 3} with {4, 6, 7}, so they have the same cardinality of 3. We can’t match up {1, 2} with {4, 6, 7}: there is always one left over from the second set, so it has a larger cardinality. If you allow for infinite sets, then we can play the same game there. {1, 2, 3, …} (the positive integers) can be matched up with {2, 4, 6, …} (the positive even numbers) because we can, for example, define a rule in which each number from the first set is matched up with the number that is twice as large from the second set.
Aleph null is defined as the cardinality of the set of integers. Clearly, we can’t match up any finite set with an infinite set, so it’s trivial to see that there are cardinalities smaller than aleph null. It takes a bit more work to show that there are larger cardinalities
> Shouldn’t that be impossible?
Why?
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