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

Aleph-Null is the answer to the question “How many integers are there greater than zero?” It’s defined as the smallest infinite cardinal number: in other words, anything infinite must be at least as large as Aleph-Null, and the only things smaller than Aleph-Null are all finite. So how does this work? How do you get to be larger than an infinite number? You could instead turn this around and ask something a little different: if Aleph-Null is infinite, then how can it be smaller than other infinite things?

Let’s go back to that original question: how many natural numbers -that is to say, integers greater than zero- are there? Natural numbers are the numbers we use for counting. If I sat down and started to count, I could reach any of the natural numbers, given enough time. It would literally take forever to reach aleph-null, but if I had forever, I could still do it. This is what we call *countably infinite*. If you can assign an integer to every member of an infinite set, then it’s also countably infinite. Let’s say I have a hotel with an infinite number of rooms, and a full bus with an infinite number of seats pulls up: I can put each guest in a room. They are countably-infinite too.

Let’s ask a slightly different question: how many real numbers are there between 1 and 2? Note that I’ve changed the number set here: we’re no longer just dealing with whole numbers, but also fractions and decimals. Now we have to deal with 0.1, but before that you have to get to 0.01, and 0.001, and 0.0001… and all the way down the line. If you sat down to try to count these, you would *never* reach 2, even if you had forever. You’d never even reach 0.1, in fact. This is *uncountably infinite*. And since something you can’t count has to be bigger than something you can, this, then, is larger than Aleph-Null. In fact, it’s Aleph-One.

[Veritasium has a video about this, including the hotel story, and how an infinite hotel can still run out of room](https://youtu.be/OxGsU8oIWjY?feature=shared).