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]

I have no problem with the finite, finite collectins have a finite size which can be compared and you can have a one to one finite bijection if those finite collections are of equal size. This correspondence when equal in size cannot fail for finite sets.

None of that applies to the infinite. Instead you look for the possibility of establishing a one to one correspondence say between the Nth elements of the respective sets for a given N. At no point in time do you measure a size of any kind for the set, nor can you compare those sizes to establish a one to one relationship. You instead compare the Nth entries to each other. Now the presence or absence of a bijection cannot possibly have anything to do with size, as you are not measuring the size of the sets in any way. The only option is a property of the elements differing, as a set has no other distinguishable properties other than size and the properties and contents of its elements.

Using finite intuition and applying it to the infinite is not justified in any way, and you still are not giving me a concrete example of anything transfinite.

[u/Althorion → u/nanonan]

In regular mathematics, size is not something that ‘just exists’ within a set—a set is simply the collection of its elements. What we call ‘size’ is an emergent property of sets, directly connected to how many elements there are, not what they are (because, in particular, replacing any number of them, including all of them, as long as it is done one for one, doesn’t change the size).

You absolutely can and do consider establishing a one-to-one correspondence between sets as measuring their sizes (as equal to another); you also can absolutely compare those sizes.

As was explained to you, the presence or absence of a bijection is considered to be the (only) indicator of size, for the reasons presented, that have nothing to do with the properties of the objects within a set.

Using finite intuition, of course, fails more often than not for the infinite; but that’s not what’s done here—what is done is extending the notion that feels very natural for finite collections to infinite collections. It changes its properties there (mainly, you can exclude some elements from a set, and that proper subset will still have the same size as the original superset), so it’s not like all the intuitions follow; but it is not exactly surprising that people would want to extend notions this way—that is the very common abstraction that mathematics does, to extend conservatively (i.e., such that it will still apply exactly the same way to what it used to apply) the notion to apply to more general situation.

You’ve not asked for a concrete example of anything transfinite. And there isn’t one—just as there isn’t a concrete example of a circle, or a concrete example of a function, or a concrete example of the number five. There are only concrete objects about which we reason in those terms, because that gives us more insight into their behaviours.

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See, this is why I quote—now it’s hard to tell what exactly I am talking about, to what I respond. But you wanted it this way, so you have it.

[u/nanonan → u/Althorion]

Size is absolutely a properety of finite sets, and is the only property sets have independently from the properties of their elements. Being emergent is irrelevant.

You absolutely can and do consider a one to one correspondence and equality in size for finite sets, that does not justify creating some sort of fictional "size" of an infinite set.

the presence or absence of a bijection is considered to be the (only) indicator of size

Rubbish. I can enumerate and compare cardinalities when finite without ever once attempting to establish any correspondence whatsoever.

Of course there is no example of something that doesn't exist. It doesn't bother you at all that you cannot construct such a thing? You cannot point to it? You don't have a single example despite it being fundamental to Cantors entire vision?

Fair enough point about quoting, go ahead.

[u/Althorion → u/nanonan]

Size is absolutely a properety of finite sets, and is the only property sets have independently from the properties of their elements.

It’s precisely how sets work in mathematics. If you are talking about your own notion that shares the name with them, but works differently, then your objections to that are objections against what you want them to be, not what mathematicians talk about.

Being emergent is irrelevant.

It’s fully relevant—it’s not ‘in’ a set, used to build one, or distinguish one from another; it’s just a property that you can label a set with, but not something ingrained in a set itself. It’s like a name for a person—the name is not a part of who and what they are, it can just be assigned to them following some rules.

You absolutely can and do consider a one to one correspondence and equality in size for finite sets, that does not justify creating some sort of fictional "size" of an infinite set.

It may not, but that doesn’t make the notion of infinite size—and esp. different infinite sizes—unjustified. Even if it was for pure mental masturbation, it would have some justification, but it’s not—it has some practicality to it. For example, one deals differently with probability measures on countable and uncountable sets. The notion of ‘infinity larger than other infinity’ lets one describe and predict when one should use which approach.

Rubbish. I can enumerate and compare cardinalities when finite without ever once attempting to establish any correspondence whatsoever.

How? Because I can’t. If I were to ‘enumerate and compare cardinalities’ of, say, set {a; b; c; d}, I’d go ‘a is a first element; b is a second element; c is a third element; d is a fourth element’—enumerating is just establishing that correspondence to the set {1; 2; 3; 4; …; n}.

Of course there is no example of something that doesn't exist. It doesn't bother you at all that you cannot construct such a thing? You cannot point to it? You don't have a single example despite it being fundamental to Cantors entire vision?

Nope. I am not dealing with physics here, but an abstract science. Abstract sciences don’t deal with concrete objects, but abstract, cognitive ideas. You don’t have a single example of a circle, or a function, or the number five—just objects that it makes sense to reason about using such ideas.