Why the hell are there so many “infinities”, in the field of mathematics specifically, and what do they mean and their differences?

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I saw a yt vid on largest numbers etc, and extremely baffled by how many versions of infinities are. Didnt know it has premium version like absolute and the regular infinity. lol

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Ok so the first infinity, what most people think of when they think infinity is countable infinity. That’s the infinity of the whole numbers, you just keep adding one.

beyond that there are two kinds of infinity cardinal infinity and ordinal infinity.

Ordinal infinity omega is just the regular counting number that come after countable infinity. from there you can have omega + 1, omega +2 etc. when you get to the number that is after all the omega + somethings you get to 2 x omega. You can use this scheme to get to really big levels of ordinal inifinty. We use ordinal infinity when the order of the numbers matter.

Cardinal infinity is often used when the order of the numbers doesn’t matter just the size of the infinity. We say that countable infinity, the smallest infinity, is aleph 0. aleph 1 is the next biggest infinity, Omega, beyond this my math isn’t strong enough to really talk about higher alephs.

There are many concepts unrelated concepts that are covered by “infinity”.

There is the concept of “growing without bounds”, attached to limits and usually representes by ∞.

There is the concept of “How many things do I have?” which is attached to cardinals and represented by Aleph and Beth numbers (among others).

There is the concept of “How do I arrange/order infinitely many things?”, which builds up on cardinals (infinitely many things), and is attached to ordinals and represented by omega and epsilon numbers (among others).

Then, as we explored these, we realized wouldn’t it be nice if we could deal with these infinite numbers just like we do with normal ones, or at least as much as possible. At the same time, we were still trying to find a way to make formal Newton’s calculus and the problem of infinitely small values (ghosts of departed quantities) turns out, it can all be made quite formal and attached to the hyperreals.

Each of these kinds of infinities represent a different concepts and need to be manipulated using different rules. Because infinity is weird.

Growing without bounds doesn’t really have any subtleties. If it grows without ever stoping then that’s that. Not much else to do.

Counting (Cardinals) is different though. Basically to say we have the de number of two kinds of things, what we do, is make sure we can match them one-by-one without repeat or forgetting anyone. Sure there are infinitely many natural numbers, and adding one or even doubling the amount of numbers we have doesn’t really change anything. But there are some operations that will change that.

For example, we can’t match naturals and real numbers one-by-one. This proven by showing that if we had such a matching, we could find a real number that hasn’t been matched yet, but no natural number that hasn’t been matched, even though there are infinitely many of them.

Ordering (ordinals) is a bit more subtle. We need the one-to-one correspondence, but we also need to do it so the order is preserved. So adding an element at the end is not the same as adding at the beginning once you have infinitely many of them.

All of these are only worried about operations about the natural numbers (addition and multiplication basically) because it’s all about counting.

The hyperreals are what happens when we try to smooth out these differences but add division as well. You end up with not only infinitely big (called transfinite) numbers, but also infinitely small ones (infinitesimal).

In any case, things get weird fast when you try dealing with infinity.

Great answers here. The best ELI5 version I can think of is “Why? Well, we tried to connect each kind of infinity to each other by mapping each value between them and found some were larger than others”

This is an other example of languages like English (or Latin this case infinite meaning “without end”) created to barter goats and tell stories around the dinner table … are inadequate to explain the full breadth of logic … or in physics, the true nature of the fabric of reality.

It’s because of the definition of infinity and how we group numbers together. Infinity just means something that you could count to for forever without finding a “biggest number.” We can count in any way that we choose though. So I can say that there are infinite numbers of whole numbers. I could also say that there is an infinite number of whole numbers divisible by 5. The infinities differ because of the way we count them, so in the earlier example there are more whole numbers that can exist than the number of whole numbers that are divisible by 5. So in essence those two infinities are mathematically different.

Say you have two groups of things, group A and group B, and you want to see if both of the groups have the same number of things. One good way to do this would be to start pairing them up. you take one thing from group A and you match it with one thing from group B. If you can keep doing this until you run out of things and everything from group A is matched with something from group B and there are no things left not matched up in either group. Now you know that group A and group B both have the same number of things.

Great, but what if the groups are infinitely big? you’d never get to the end. but this doesn’t matter. As long as you have a definite rule for the matching up and you know for sure that every item in group A is going to match to exactly one item in group B, even if both sets are infinite you know for sure they are the same size.

So if we take the group of normal counting numbers: 1,2,3,4… we know this group is infinite and any other group of things that we can have some rule for matching up perfectly to the counting numbers in some way is the same size. We call this size of infinite groups ‘countably infinite’.

weirdly the set of all fractions: 1/2, 3/5, 7/8, 22/9 etc even though there are an infinite number of them between any two counting numbers, is the same size. We can prove this using a clever rule of matching them up:
https://www.homeschoolmath.net/teaching/rational-numbers-countable.php

but some sets are bigger than this ‘countably infinite’. There is an interesting proof of this called Cantor’s diagonal argument:

‘https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Basically, if we start by assuming the set of real numbers is countable. Then we could make a list of them all in order. now you make a new number by taking the first digit of the first number in the list and changing it to something else, you take the 2nd digit from the 2nd number in the list and again change it. you keep going this way taking each new digit from the next number in the list but changing it. Now your new number is definitely not anywhere in that list since each digit is different in at least one place to all the numbers in the list. Therefore the set of real numbers must not be countable!