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

Here is how I understood it.

Imagine you have a water molecule, H2O. 2 hydrogen atoms, one oxygen atom. In other words, you have two times more H than O.

Now imagine you have an infinite amount of water. That means that you have infinite amount of H atoms and O atoms. But H infinity is still 2 * O infinity. In this case, ∞ =/= ∞.

Consider infinity as a placeholder for numbers so big that they cannot be calculated. Notice the plural, numberS. There is not one infinity, but many (infinity of infinities, hehe). Every single infinity is a placeholder for different number. If we know relations between those numbers (e.g. number of H atoms = 2 * number of O atoms), the same relations will apply to their placeholders.