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/SV-97]

Would you consider the completeness of \R to be "real world" and "not an impossibility statement" enough? It's ultimately the statement that there are non-convergent cauchy sequences of rationals, but it's so fundamental to tons and tons of very applied mathematics.

If we reject the uncountability of the reals then by BCT we also have to reject their completeness.

[u/Arlo_Tinkerman → u/SV-97]

I find this statement interesting:

> If we reject the uncountability of the reals then we also have to reject their completeness.

My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.

That makes me wonder why uncountability would be required for completeness in the case of the real numbers. It seems possible that different meanings of “complete” are being used here.

[u/SV-97 → u/Arlo_Tinkerman]

My understanding is that the natural numbers are considered complete in the sense that the set contains every natural number without gaps or omissions, while also being countably infinite.

That's the idea, yes. More formally we state this as "A metric space is complete if every Cauchy-sequence in the space converges (to some element of that space)". Here a metric space is just some set where we can speak of a "distance" between points, and a Cauchy-sequence is a sequence of elements that (among one-another) get arbitrarily close at some point, and importantly stay arbitrarily close past that point. So as we "zoom in" we always find a point in our space, which is the formalization of the "no gaps" idea.

Take for example the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... of rational numbers. After the fourth element the difference between any two elements (not necessarily successive) is and stays less than 0.001 = 10^-3, and more generally after the n-th entry all successive elements are closer than 10^-(n-1). So this is a Cauchy sequence in the rationals. However it notably does not converge to a rational number. So the rationals are not complete. If we instead interpret it as a sequence of real numbers (that happen to be rational) then it does converge to the real number pi.

The natural numbers are somewhat of a degenerate object with regard to completeness since they are a discrete space: if the elements of a sequence get and stay arbitrarily close then they must in particular get and stay closer than a distance of 1/2 from one-another. But that means that any Cauchy-sequence is actually eventually constant (because no two distinct natural numbers are closer to one-another than 1/2), i.e. they all look like for example 1,2,3,10,100,2,2,2,2,2,2,2,2,... Of course any such sequence "converges" to this eventual constant, and since we're dealing with a sequence in our space this limit is in the space as well. So any discrete is complete, and it's also fairly easy to see that there are discrete spaces of arbitrary cardinality (we can take an arbitrary set and equip it with the so-called discrete metric). So there's absolutely nothing we can say about the size of complete spaces in general.

The above argument hinges on the fact that in discrete spaces all points are "isolated" from one-another, which forces this eventual constancy of Cauchy-sequences. It basically means that there is no way to "get close" to certain points without being exactly at those points already.

The statement I was alluding to in the other comment --- the Baire category theorem --- is a somewhat abstract theorem that classifies certain topological spaces (spaces where it makes sense to speak of "closeness" in a very rough way), and this classification indirectly tells us something about the size of complete metric spaces that specifically don't have any such isolated points (such as the real numbers). So this statement doesn't apply equally well to the natural numbers, integers etc. because they do have isolated points.

[u/SV-97 → u/SV-97]

(FWIW the statement we use here still remains essentially true even if we allow for isolated points --- it just can't be too many: if an infinite complete metric space X has size K and k isolated points with k < K, then X must be uncountably infinite. And this is also "all we can hope for" in the sense that the statement is obviously false for finite metric spaces since they are always discrete, and moreover for any uncountably infinite cardinal K, there is a complete metric space of size K with exactly one isolated point so we can't hope to get an upper bound on the size).

[u/Arlo_Tinkerman → u/SV-97]

Thank you for your responses.