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

Imagine you have an infinitely long line of some object, say apples for the sake of example. The number of apples in this line is infinite, more specifically it is equal to aleph null.

However, there are “larger” infinities. Take that same line of apples and in between each one place another apple. You now have twice as many apples, even though you already had infinitely many.

Infinite numbers are a much more abstract concept than just “the biggest number”, so you can do a lot of things with them that from afar seem counterintuitive