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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Usually because you have a reason to believe they are larger or smaller based on the rules that still apply.
For example, exponential divergent functions (2^x) grow faster than linear divergent functions (2*x). If you chose x to be infinite then whatever undefined thing (2^infinite) would be, it would definitely be bigger than infinite*2 or you break the rules of your basic operants.
You can also take a look at different classes of infinity. For example, there is an infinite amount of integers (1,2,3,4,…)., In theory you could use them to count an infinite amount of objects right?
Now take a look at real numbers (1, – 2.35894, 2/3, pi,…) and try to count them all. 1.0 is the first, 1.1 the second and then.. How to proceed with the third? Is it 1.2? Or 1.11? Or 1.100000001?
It does not matter because even between 1.00000000001 and 1.00000000002 there is still an infinite amount of real numbers. You can just keep padding zeros indefinitely and we haven’t even reached 2 yet. It can’t be done. There are more real numbers than can be counted with an infinite amount of integers.
Clearly there must be more of them by an entire order of definition.
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).
> 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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How can there not be a number smaller than infinity? Literally, all of the real numbers are smaller than infinity.
Aleph-null is the smallest infinity, generally called “countable infinity”
It’s the number of natural numbers that exist (1, 2, 3, 4….)
It’s also the number of whole numbers (0, 1, 2, 3, …)
And the number of integers (…-2, -1, 0, 1, 2…)
And the number of rational numbers (1, 3/2, 1/16, etc)
All of these are for math reasons I won’t go into, but we have proven that all of them are the same total.
The number of real numbers (π, e, 1, sqrt(2), 7/8, etc) is also infinite, but we can prove there are more of those than there are natural numbers, an infinity greater than aleph-null. An uncountable infinity.
There’s a lot to go into with the different types of infinity, but the main difference you should know is countable vs uncountable infinity.
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
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.
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.
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