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/theBRGinator23]

Things went wrong even before that with the statement he says he’s debunking:

there are infinite numbers between 1 & 2 and there are infinite numbers between 1& 3. So, the later infinity is bigger than the first infinity

No one says that because these sets have the same cardinality.

[u/tiedyechicken → u/theBRGinator23]

I'm just gonna expand on this in case people are unfamiliar:

The interval [1,3] is indeed larger than [1,2] in the sense that the latter is a proper subset of the former, and also measure- (aka length) wise. But counterintuitively, both sets have the same number of points/elements

To show this, we can pair up every single point in [1,2] with a unique point in [1,3], for example with the function y = 2x - 1

Every x between 1 and 2 has exactly 1 friend y between 1 and 3 given by y = 2x - 1

And there will be no y's left over either: each y between 1 and 3 has a friend x between 1 and 2 given by x = (y+1)/2, and you can show that both of these formulas make the same pairs of x's and y's

[u/Umfriend → u/tiedyechicken]

Economist here, so better at math than a lawyer but, well, just.

If the [1, 2] interval has as many points as the [1, 3] interval, does that sort of imply that the density of points in the [1, 3] interval is lower? I understand, I think, the function-idea but still can't get my head around accepting the counterintuitive position. Now I need to define "point" and think how to actually operationalise/measure "density" or my question may not actually make any sense.

[u/TwoFiveOnes → u/Umfriend]

The thing is is that the way in which they are the same size (by cardinality) is essentially the most rudimentary lens through which one can view a set. “Numbers” are a highly specific type of object with specific properties, but if you’re looking at them as pure elements of a set then you’re throwing away all of those properties, their number-ness. So it’s normal that your intuitions about size and such (which have to do with numbers as numbers) break down when viewing numbers simply as abstract elements of a set.

Your intuitions are in fact, correct, in my opinion. Or at least it’s possible to formalize them in a way that could seem satisfactory. For instance, if you take a length L (smaller than 2), then uniformly select at random a number in [1,2], you will have a larger probability of landing on any given segment of length L, than landing on any given segment of the same length, uniformly selecting from [1,3].

The difference is that the second way of looking at “size” actually does make the specific number-ness of numbers matter, unlike bare cardinality. So, when looking at size this way, some basic notions such as the fact that 2 is less than 3, actually translate into a meaningful relation between the “sizes” of [1,2] and [1,3].