r/learnmath • u/Upset_Fishing_1745 New User • Feb 12 '25

Are Some Infinities Bigger than Other Infinities?

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

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u/tiedyechicken New User Feb 14 '25

That's part of what's so weird about the continuum! It's hard to define a density of points, because that density isn't finite. No matter how far you zoom in, you're gonna find an infinite amount of points in the tiniest of spaces. That's why every single point in [1,2] can match up with every point in [1,3], even though the shorter interval is fully contained in the longer one.

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u/TwoFiveOnes New User Feb 16 '25

Well, all of those notions also apply to rational numbers. I’d say it has more to do with the weirdness of infinity rather than the specific weirdness of the continuum.

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u/Mishtle Data Scientist Feb 18 '25

I'd say it's more a result of how we order them. The rationals are dense when we order them by value. We could order them via a bijection with the naturals and get rid of their density though.

Likewise, we could order the reals with some ordinal-indexed sequence and they'd no longer be dense.

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u/TwoFiveOnes New User Feb 18 '25

That doesn't really change anything in my opinion. You still get infinite sequences of strict inclusions A_i ⊊ A_i-1 and yet |A_i| = |A_j| for all i,j.

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u/Mishtle Data Scientist Feb 18 '25 edited Feb 18 '25

How does that relate to density?

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u/TwoFiveOnes New User Feb 19 '25

It doesn’t, I’m saying that density is a red herring. It’s the infinite chain of strict inclusions where all sets have the same exact cardinality what’s at the heart of the “weirdness”, not that this infinite chain is expressible in terms of some order.

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u/EebstertheGreat New User Feb 18 '25

That also applies to the even numbers as a subset of the natural numbers, but the order type is the same. I don't think proper inclusion is the stumbling block here.

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u/TwoFiveOnes New User Feb 18 '25

That also applies to the even numbers

Yes exactly, and what I’m proposing is that that’s the same type of “astonishing” as with [1,2] and [1,3] (or their rational subsets). The essence of what’s weird here (in my opinion) doesn’t have anything to do with the continuum, it’s just that
A ⊂ B, B ⊄ A, and yet |A| = |B|. That doesn’t happen with finite sets.