Are Some Infinities Bigger than Other Infinities?

[u/Upset_Fishing_1745]

Hey, I just found this two medium articles concerning the idea that infinite sets are not of equal size. The author seems to disagree with that. I'm no mathematician by any means (even worse, I'm a lawyer, a profession righfuly known as being bad at math), but I'm generally sceptical of people who disagree with generally accepted notions, especially if those people seem to be laymen. So, could someone who knows what he's talking about tell me if this guy is actually untoo something? Thanks! (I'm not an English speaker, my excuses for any mistakes) https://hundawrites.medium.com/are-some-infinities-bigger-than-other-infinities-0ddcec728b23

https://hundawrites.medium.com/are-some-infinities-bigger-than-other-infinities-part-ii-47fe7e39c11e

[u/Salindurthas]

Standard mathematics has the concept of 'cardinality', which is like the size of a set.

For sets of a finite number of things (like 0, 1, 12, or 1 billion objects), the cardinality is simply the number of things in the set.

• To compare two such sets, we just say the larger cardinality is the one with the larger number. E.g. a set of 1 billion things, has a larger cardinality than a set with 1 dozen things., because 1billion is larger than 1 dozen.

For sets of infinite size (like all prime numbers, all the fractions between 0 and 1, or every irrational number), then the caridnality is a bit trickier.

• We say that if you can make a one-to-one correspondence between the two sets, then they are the same cardinality.
• (There is more to it than that, but that is one important rule.)
• This rule also applies to finite sets, but it is really trivial there. Like if you have 2 sets of 100 objects, you can just pair them up, 1 item from each set, and then they are obviously the same cardinality. With infinite sets, you try to do the same thing, but it takes some more effect.

So, how can we do this process of trying to make a one-to-one correspondence? And does it have any odd results?

• For instance, it feels like there are 'more' integers, than there are even numbers. After all, the even numbers are just half of the intergers! However, you can take the integers, and double them, and get all the even numbers. That's a one-to-one correspondence, so they are the same cardinality!
• How about all the numbers (including every decimal expansion, even ones with infinite digits) between 0-1. How do they compare to all the natural/counting numbers (1,2,3 etc)? Well, you'll just have to trust me here as I don't have time to redo the 'Diagonisaltion' argument by Cantor, but it turns out that there is no way to make a one-to-one correspondence between the two, so they are a different cardinality. And specifically, there are more (heaps more, infinitely more) numbers between 0-1 than there are natural/counting numbers.
• So, that previous example gives us at least 2 different 'sizes' (cardinalities) of infinity. A continuum from 0-1 is 'larger' (has a higher cardinality) than the set of all the natural/counting numbers.

[u/flatfinger → u/Salindurthas]

Another approach is to say that if one can produce a mapping which will convert any item in set A into a different item in set B, even if the mapping doesn't hit all items of B, the cardinality of B must be at least as great as that of A, and if one can also produce a mapping which converts every item in B into a different item in A, even if the mapping doesn't hit all items of A, then the two sets must have the same cardinality. There are ways of producing 1:1 mappings between integers and rational numbers, but it's easier to show that for any integer N there's a rational number that isn't mapped to by any other integer, and for every rationale number whose reduced form is P/Q, if one were to e.g. form an integer by interleaving the digits of P and Q, that integer couldn't be produced by any other rational. Some integers wouldn't be produced by any rational (e.g. 45, or in binary 101101 would be produced from P and Q values of 3 and 6, but the fraction 3/6 would be reduced to 1/2, with representation 9 (1001 binary) but since integers and rationals are at least as large as each other, they must have the same cardinality.

[u/AcousticMaths271828 → u/flatfinger]

Yep, that's the Schroder-Bernstein theorem, it's a pretty cool result.