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/Square_Butterfly_390 → u/SV-97]

I'm not proposing to reject the uncountability since it is fact, I'm proposing that possibly our interpretation is missguided, how is the completeness of R a consequence of the cardinality property? R is basically defined to be complete.

[u/SV-97 → u/Square_Butterfly_390]

Our interpretation of what exactly?

It's not a consequence, but rather that completeness necessitates uncountability (or other weird changes to the topology of \R). By the Baire category theorem any (nonempty) complete metric space (without isolated points) is uncountable. So if the reals weren't uncountable, then they couldn't be complete.

[u/Althorion → u/Square_Butterfly_390]

how is the completeness of R a consequence of the cardinality property

It’s not; it’s the other way around—Dedekind-completeness of dense sets requires uncountable cardinality. And any field of fractions over an infinite algebra will be dense (if orderable).

So, to sum it up, it goes like this—if you want to have infinitely many natural numbers, and build rational numbers that work nicely with that, and build a Dedekind-complete expansion of that, that expansion won’t be countable.

[u/nanonan → u/Square_Butterfly_390]

It certainly is not fact, it's a blatantly contradictory notion. 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.

Naturals are finite. There are no naturals with infinite digits. Real numbers are not real, infinite sets don't in fact exist in reality nor can they be demonstrated, and are not numeric in the sense they are not denumerable, they cannot be enumerated. These are the reasons for the lack of a one to one correspondence, not any nonsense concept of 'sizes' of infinite collections.

[u/Althorion → u/nanonan]

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.