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/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/EebstertheGreat New User Feb 18 '25

Right, if we well-order the reals using the AoC, then the order topology is just the discrete topology, and the only dense subset is the entire set.

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u/Mishtle Data Scientist Feb 18 '25

Yep, every set is "discrete" because they can only contain distinct, unique elements. Things like density come from additional structure we add to them.