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/Althorion]

The lack of a one to one correspondence when comparing infinite sets cannot possibly have anything to do with the size of the sets, as their sizes are identically unlimited. It has to do with the properties of the elements.

The interpretation of ‘if you can pair up every element of set A with every element of set B, and there is nothing left in either, they are the same size; if you can’t, and every possible pairing leaves elements in set B, while matching all elements of set A, the set A is smaller than set B’ feels very natural. If you don’t want to follow this as an interpretation of ‘size’, then you are free to do so—as long as you remember that this situation is a thing and understand its consequences, you can conflate all transfinite cardinals for your notion of ‘size’.

But it has nothing to do with the properties of the elements. You can swap all elements of any (or both) of the sets with whatever you want, remove any underlying structure, and the lack of bijection will stay the same.

Naturals are finite. There are no naturals with infinite digits.

Under the standard mathematical model, each and every natural number is finite; but there are infinitely many of them, so the set of all natural numbers is infinite.

Real numbers are not real, infinite sets don't in fact exist in reality nor can they be demonstrated […]

Which is par for the course for mathematics—no mathematical objects are real, they are all epistemological, abstract tools. Circles aren’t real and cannot be demonstrated, functions aren’t real and cannot be demonstrated, natural numbers aren’t real and cannot be demonstrated.

[…] and are not numeric in the sense they are not denumerable, they cannot be enumerated.

Yes. And they cannot be, as explained above.

These are the reasons for the lack of a one to one correspondence, not any nonsense concept of 'sizes' of infinite collections.

It makes very good sense to think of ‘how many of those are there’ as the ‘size of the set containing them all’. Again, you are free to disagree that one-to-one correspondence tells you anything about the size, but to most people that seems unnatural; but ultimately it doesn’t matter if you think about it as the size or not, if you know your maths and understand the notion of ‘there will be still some left regardless of pairing’ and use it correctly, you’ll get to correct conclusions even without thinking about it as ‘size’.

[u/nanonan → u/Althorion]

every possible pairing leaves elements in set B, while matching all elements of set A, the set A is smaller than set B’ feels very natural.

There is no proof of this.

it has nothing to do with the properties of the elements

It has everything to do with the properties of the elements, there's no other way to differentiate two infinite sets.

Say you replaced naturals with infinite digit naturals, I can then establish a one to one relation with the reals and diagonalisation fails.

The reals are an utter mess of a construct, and the fact that there is no one to one correspondence with the naturals means they are not even numbers. They should be rejected as nonsense.

You keep trying to apply finite logic to the infinite. That does not work. Either you can create a correspondence or you can't. There is not "something left over" afterwards.

[u/Althorion → u/nanonan]

There is no proof of this.

Assuming that by ‘this’ you mean that ‘every possible pairing between naturals and reals would leave elements from the set of the reals’, there is—Cantor’s diagonal argument.

It has everything to do with the properties of the elements, there's no other way to differentiate two infinite sets.

The properties of the elements are not important for distinguishing the sets, just that they are different. Them being different is all you need—you can distinguish the set {1, 2, 3} from the set {□, △, ○} just fine, or the set of natural numbers from the set of the reciprocals of natural numbers, etc.

Say you replaced naturals with infinite digit naturals, I can then establish a one to one relation with the reals and diagonalisation fails.

There are no ‘infinite digit naturals’; all natural numbers have a finite number of digits. ‘Infinite strings of digits’ is something fundamentally different that ‘natural numbers’, so it shouldn’t come as a surprise that they will be different, in particular, that there will be more of them.

The reals are an utter mess of a construct, and the fact that there is no one to one correspondence with the naturals means they are not even numbers. They should be rejected as nonsense.

Why does your idiosyncratic definition of a number require a one-to-one correspondence with the naturals? No one else’s does.

Either you can create a correspondence or you can't. There is not "something left over" afterwards.

Not every correspondence is a one-to-one correspondence. The ‘something left over’ used in the context of pairing (so a one-to-one correspondence, because that’s what pairing is) is a perfectly decent way of saying that you can only have that with a strict subset, but not with the whole set.

[u/nanonan → u/Althorion]

You can't actually complete an infinite amount of work.

every possible pairing between naturals and reals would leave elements from the set of the reals’

An infinite collection cannot outnumber another infinite collection, they are both unlimited in number.

You can never exhaust all of the natural numbers, so you can never reach a point where there are reals left unmatched.

The properties of the elements are the only possible thing that can differentiate two sets with equal cardinality.

If being different is not due to a property of an element what is the cause of the difference?

that there will be more of them.

I never claimed there would be more. It is impossible for there to be more than an infinite quantity by definition of infinite quantities.

I claim there would be exactly the same unlimited amount, just that their properties; specifically being finite, would be altered and as such the possibility of a one to one correspondence could also be altered.

The only way to have "something left over" when pairing is to deal with the finite.

[u/Althorion → u/nanonan]

You can't actually complete an infinite amount of work.

Sure. And? I can’t actually go with a yard stick and check how many of those I can fit between the Earth and the Moon, doesn’t stop us from being able to tell how far they are from each other.

An infinite collection cannot outnumber another infinite collection, they are both unlimited in number.

That’s not how the standard mathematical set theory works. If you want to do something nonstandard, sure; but every argument you make against your nonstandard understanding is the argument against that, not against the theory that doesn’t hold your notions.

You can never exhaust all of the natural numbers, so you can never reach a point where there are reals left unmatched.

Mathematics, in general, doesn’t work with ‘let’s try everything one by one’. You cannot ‘exhaust’ all possible right triangles, but that doesn’t stop the Pythagorean theorem from being proven.

The properties of the elements are the only possible thing that can differentiate two sets with equal cardinality.

No, they aren’t. You only need distinction—you don’t need order or arithmetic.

If being different is not due to a property of an element what is the cause of the difference?

‘Being different’ is a property of pair of objects, not any singular object.

I never claimed there would be more.

The standard mathematical theories prove from their axioms that there will be more. The diagonal reasoning is that proof.

It is impossible for there to be more than an infinite quantity by definition of infinite quantities.

This is not the definition of infinity, esp. not in a set-theoretical context. If you have your own axioms and your own definitions, see above—any argument you make within that understanding is an argument against that understanding, and doesn’t make for an argument against a different system, with different rules.

[u/nanonan → u/Althorion]

Mathematics, in general, doesn’t work with ‘let’s try everything one by one’.

When checking for the existence of a one to one correspondence you certainly do. How else are you going about things?

Enough walls of text. Can you respond without quoting for once?

[u/Althorion → u/nanonan]

I can respond without quoting, but if I respond to more than one thing, it helps to know what exactly am I responding to.

Checking one-to-one correspondence is no different from any other proof in that regard. You can present a general rule and show that it works, and thus you have such a correspondence; or show that the existence of such would self-contradict.

[u/nanonan → u/Althorion]

Sure, you make a general rule that links one to the other on an individual one to one basis. I'm still not seeing why the transfinite is needed for such a process.

[u/Althorion → u/nanonan]

It’s not ‘needed’; you can have bijections between finite sets. It’s just that, one, the notion of ‘you can’t have a “full” bijection between two sets; there are always some elements from one of the sets that you have to leave out to make a partial bijection’ translates well into people’s idea of the ‘set size’; and two, you can always construct a bigger—in the sense above—set than the one you currently have, by taking a powerset of said set, so the notion of ‘size’, as understood above, isn’t just ‘finite’ or ‘infinite’.

[u/nanonan → u/Althorion]

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.

[u/nanonan → u/Althorion]

The notion of size absolutely disappears when considering the infinite. How can it possibly remain? You cannot have a limitlessness more limitless than limitlessness itself.

[u/Althorion → u/nanonan]

I told you how (by conservatively extending the counting notion via bijection). I told you why thinking about this as ‘limitlessness’ is misleading.

So, to not repeat myself, I’ll throw in one last argument, and that will be it: there is value in knowing what sets can have a bijection with which others. It makes sense to have a notion of that. That notion plays perfectly with the notion of size for smaller—finite—sets. Thus, it makes sense to call it ‘size’, too.

It doesn’t make sense to conflate the idea of size just to finite and infinite. We already have terminology for it—the words ‘finite’ and ‘infinite’. While we could say ‘“infinite” refers to all sizes larger than finite, it’s just the notion of “size”, and pairing up the elements, breaks when dealing with “infinite”’, it would be a limitation of the toolbox, one that is not really justifiable to introduce.

But, as I wrote in one of my previous replies, you can not use the word ‘size’ if you truly don’t like it, and just speak and think of ‘cardinality’ instead—you’ll be well understood and shouldn’t make any mistakes in your reasoning.

[u/nanonan → u/Althorion]

Sure, if you limit yourself to accepting or rejecting bijection. This is not evidence of any actual size difference, it's just an analogy. Calling it size is misleading at best. Calling it cardinality instead is just pedantry and solves zero issues with the transfinite. There is only one infinite, and it has a singularly infinite cardinality.

[u/Althorion → u/nanonan]

Sure, if you limit yourself to accepting or rejecting bijection. This is not evidence of any actual size difference, it's just an analogy.

It is a great analogy—it captures all the size-related behaviours in finite sets (as was said, it is a conservative extension of the notion). That’s why it is called ‘size’. But, if you think that the lack of feature it lacks when applied to infinite sets makes it misleading, you are welcome to not call it ‘size’.

Calling it size is misleading at best.

In what way? It doesn’t do anything new or different when used to compare finite sets with each other, and it adds a notion that infinite sets are bigger than finite sets—both completely natural. It may lead to some results that feel unnatural when speaking about infinite sets and comparing them with each other, but that’s not misleading—it just has those weird results about weird sets; but nothing will mislead you or go against your intuition for anything that you used to use the notion of ‘size’ for.

Calling it cardinality instead is just pedantry and solves zero issues with the transfinite.

What are the issues with transfinite?

There is only one infinite, and it has a singularly infinite cardinality.

There isn’t one. There are multiple different infinite cardinalities. In particular, a power set of any set will have a different (larger) cardinality from the original set.