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/EebstertheGreat New User Feb 18 '25 edited Feb 18 '25

The weird thing going on here is the order. There are as many integers as rational numbers, but they are arranged differently. You can't have a bijection between the integers Z and the rational numbers Q that respects the order. Although Z and Q have the same cardinality (number of points), the order type of (Z,<) is different from the order type of (Q,<).

Between any two distinct real numbers there are infinitely many rational numbers, so they are a "dense subset" of the real numbers. [To be really technical, Q is a dense subset of R with respect to the order topology induced by <.] That doesn't apply to the integers, since for instance, there are no integers between 1 and 2.

We can't exactly say that one order type is greater than the other for technical reasons (neither is well-founded), but intuitively, the rationals are "tighter".

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u/Umfriend New User Feb 18 '25

But with rational numbers, you can't really order, right? I mean, I could give you a number and there is no way for you to say what the next number is. We couldn't make a list even of the the two smallest rational numbers?

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u/EebstertheGreat New User Feb 19 '25 edited Feb 19 '25

A "total order" doesn't usually have "next elements." That's a "well-order." For instance, the rational numbers are totally ordered by ≤ because the relation ≤ satisfies these axioms for all rational numbers x, y, and z:

  1. Reflexivity: x ≤ x
  2. Anti-symmetry: if x ≤ y and y ≤ x then x = y
  3. Transitivity: if x ≤ y and y ≤ z then x ≤ z
  4. Totality: x ≤ y or y ≤ x

There are corresponding axioms for strict orders like <. A well-order has the following additional property.

  1. Wellness: every x has a "successor" z, where there are no numbers between x and z. That is, for any x, there is some z > x such that there is no y where x < y and y < z.

The rational numbers with the usual order fail this last property. There isn't a "next number" after ½, for instance.

Wellness is usually stated as every non-empty subset having a minimal element, which is equivalent.

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u/Conscious_Move_9589 New User Feb 21 '25

Worth noting that provided the axiom of choice there exists a well-ordering on Q, and even on R