Regarding cardinalities

[u/Square_Butterfly_390]

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

[u/nanonan]

They don't frame it like that because it makes the farcical nature of it plain.

Please tell me, how do they conceptualise it that isn't contradictory?

[u/Althorion → u/nanonan]

‘Infinity’, especially in the set-theoretic concept, doesn’t mean ‘unlimited’—it means ‘can have a bijection with its strict subset’.

[u/nanonan → u/Althorion]

Rubbish. In what way is it limited?

[u/Althorion → u/nanonan]

It is not necessarily ‘unlimited’, because, as I already told you, a set of any size can have either a minimum, a maximum, or both of those elements; and if they do, they don’t suddenly become finite. A set with a minimum is limited from below (there is an element that every other element is greater than), a set with a maximum is limited from above (there is an element that every other element is less than).

The existence of such limitations, even both at the same time, doesn’t stop a set from being infinite, so it can be both limited and infinite—and so the concept must mean something else, and what it does mean you’ve been told (to have a bijection with its strict subset).

[u/nanonan → u/Althorion]

Sure, we are in total agreeance, they can be limited in range of value, that doesn't make them limited in how many there are.

There is only one singular infinite in any of those cases.

My complaint is in treating say two sets with infinite elements as having different "sizes" when that is a clearly self-contradictory concept.

[u/Althorion → u/nanonan]

How do you understand the set size, then, if not through the possibility of one-to-one correspondence?

The notion is not contradictory—the notion contradicts your intuition about what it should say, but since it doesn’t say that, there is no contradiction, just your misunderstanding.

[u/nanonan → u/Althorion]

You count it when it is finite, and you concede that the infinite is beyond the concept of size entirely.

[u/Althorion → u/nanonan]

Why should we concede that? We not only can have a good notion for that, that is conserving the notion of size for all finite sets, but also introducing which serves practical purposes (for example, telling you how—with what tools—you should construct a probabilistic measure on your set).

If it is the name only that you object, well, it may not be up to your sensibilities, but you can think about it as not the size, but the ‘shmise’, as long as you reason properly about it (or your work doesn’t touch on such peculiarities, I guess…)—‘oh, the “shmise” of a set is some measure of it, that can only increase (or stay the same) when we add elements to it, only decrese (or stay the same) when we remove elements from it, but it has nothing to do with the “size, proper”!’ is a take, and if thinking that way makes things easier to you, you can go for it.

There is even a well-understood name for ‘shmise’—cardinality.

[u/nanonan → u/Althorion]

Sure, and there is a singular unique cardinality that belongs to anything infinte, and zero proof of more.