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

Question

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

In: 0

Answer 1

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!