Infinities being bigger than others, "countable" and "non countable" infinities

[Marshmallow Marshall]

Some context:
https://en.wikipedia.org/wiki/Cantor...gonal_argument
https://www.youtube.com/watch?v=HeQX2HjkcNo

I've been randomly looking into this and... I don't believe Cantor's argument! I'm not great at maths, though, so it's very possible I'm just being dumb, but I think that the argument is flawed (and that the conclusion is too). Here's why: it's impossible to add one to x rank of each number on an infinite list without getting to a number of the list, because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything. Thus, isn't the proof based on a misunderstanding of the concept of infinity?

And yes, it is very ballsy of me to attack something that apparently was proven a century ago, and I dare hope you're gonna prove me wrong or tell me I misunderstood something, O you knowledgeable people :P

[OzyWho]

I remember watching that video. I don't remember anything from it, and this post didn't jog my memory, but it sounds like arguing semantics wrt what is infinity in math?

[DJarJar]

Let's try a proof that some infinite set is countable then. In general, to prove this we need to show that you can map a natural number (these are the numbers 0, 1, 2, 3 4, 5, .... to infinity, sometimes we include 0 sometimes we don't depending on textbook) to every element in the set.

The natural numbers are trivially countable since you can just assign 0 to 0, 1 to 1, 2 to 2, 3 to 3, and so on.

Integers (.... -3, -2, -1, 0, 1, 2, 3 ...) countable proof:
Map 0 to 0, 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3....

Now you may say that "HEY! wait a minute, then you're going to run out of natural numbers long before you finish the integers!" But the natural numbers is an infinite set. Pick any integer and I can give you the natural number that would produce it. We just needed to show that we can assign a unique natural number to every integer. Hence the integers are countable.

Now to show that a set is UNcountable you'd need to prove that there is no formula out there which can map a unique natural number to every single element in the set. Of course trying every possible formula would be impossible, hence we look for a more clever way to prove that any formula will fail.
For cantor's diagonalization theorem, in the video they are proving the set R(real numbers) is uncountable. In the wiki and generally the way the proof is shown, it's used on the set of all infinite length binary sequences. In either case, the idea is we first assume that there DOES exist a formula out there which manages to enumerate all the elements of the target set. Then we show that this enumeration must be missing an element of the set and therefore fails to map a natural number to every element in the set. In both cases, the elements here are all infinite in length. So, while yes we are constructing an infinite length element that is not in the enumeration, that element is still a valid member of the set we are trying to prove is uncountable.

[Renegade]

Fun topic.

[Plotato]

i see that the french have surrendered to an intellectual fight... how shameful...

a countable infinity is defined where you can assign every single element to the set of positive integers (1, 2, 3, ...) which is countable because any baby can count as such. if i wanted to go on an infinitely long but countable tirade on why i hate the french, i can map out every word with a number like so.

- (french -> 1, are -> 2, stupid -> 3, because -> 4... 1000000000 -> retards)...

cantor's argument basically breaks it down to this with real numbers.

real number: real numbers involve rationals or fractions, or decimals... i.e. 0.00000000001231293128093. You can begin to see why it might be difficult to start mapping shit out. Where do you begin?

(for sake of example lets talk numbers between 0 and 1) if we assume, that all these decimals whatever they look like can be counted, we would have a list of them... (1 -> 0.29382, 2 -> 0.849343, etc...) and stacked them on top of each other for reference. what if we say, draw a diagonal down a line of numbers, and change it by adding 1 to each of those numbers... what does that mean? we've made an entirely new number! now you might ask, because you are a simpleton goat, oh! but isn't that number already listed in the infinite set of integers? no. as unintuitive as it sounds, it doesn't. why? because in order to be "countable", you have to have a one-on-one

360px-Bijection.svg.png

connection between 1, 2, infinity (and beyond) with whatever you are mapping to. cantor's diagonalization means we have made an entirely new number that doesn't have a single integer mapped to. as strange as it sounds, we have found an infinite set of decimal numbers of which we suddenly can't map an infinite but countable set of integers to. as such, the set of real numbers is uncountable. now you might ask again, because you are a french simpleton goat, why can't we just add another integer and map it to our new decimal number? because, mr. surrender, to prove that the set of decimals is countable you would have to theoretically show that you can count and match everything once. you don't get to fist another goat in there and call dibs.

i can always find a new decimal number between 0 and 1 that explains the length of your penis, should you deny my claim.

if you are still confused i have a stellar comeback which involves more insulting of the french.

[DJarJar]

The set of rationals is countable though, it’s the irrational that present the problem

Although I guess I’m just responding to a copypasta. I wonder if plotato ever posts his own thoughts

[WrathCyber]

The set of rationals is countable though, it’s the irrational that present the problem

Although I guess I’m just responding to a copypasta. I wonder if plotato ever posts his own thoughts

Off topic but do you have a math-heavy background?

[Lumi]

Let's try a proof that some infinite set is countable then. In general, to prove this we need to show that you can map a natural number (these are the numbers 0, 1, 2, 3 4, 5, .... to infinity, sometimes we include 0 sometimes we don't depending on textbook) to every element in the set.

The natural numbers are trivially countable since you can just assign 0 to 0, 1 to 1, 2 to 2, 3 to 3, and so on.

Integers (.... -3, -2, -1, 0, 1, 2, 3 ...) countable proof:
Map 0 to 0, 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3....

Now you may say that "HEY! wait a minute, then you're going to run out of natural numbers long before you finish the integers!" But the natural numbers is an infinite set. Pick any integer and I can give you the natural number that would produce it. We just needed to show that we can assign a unique natural number to every integer. Hence the integers are countable.

Now to show that a set is UNcountable you'd need to prove that there is no formula out there which can map a unique natural number to every single element in the set. Of course trying every possible formula would be impossible, hence we look for a more clever way to prove that any formula will fail.
For cantor's diagonalization theorem, in the video they are proving the set R(real numbers) is uncountable. In the wiki and generally the way the proof is shown, it's used on the set of all infinite length binary sequences. In either case, the idea is we first assume that there DOES exist a formula out there which manages to enumerate all the elements of the target set. Then we show that this enumeration must be missing an element of the set and therefore fails to map a natural number to every element in the set. In both cases, the elements here are all infinite in length. So, while yes we are constructing an infinite length element that is not in the enumeration, that element is still a valid member of the set we are trying to prove is uncountable.

This is a great explanation

The result obtained through this cannot be unique, as infinity necessarily comprises everything.

The one thing I would add is to address the underlying source of the misunderstanding.

If infinity did comprise everything by definition, then your reasoning would be correct - but something can be infinite, and not contain everything.

For example, there are infinite counting numbers (1, 2, 3, 4, ...)
But the counting numbers will never contain 0.5, no matter how high you count. Nor will they ever contain 1.5, or 2.5, or 3.5. In fact despite being there being infinite counting numbers, there are still infinite numbers that are aren't counting numbers.

The important takeaway stems from a pretty common misunderstanding of what infinity is. Infinity doesn't mean it contains everything, it just means that you can continually list things forever.

===

On a sort of a tangent, what you're describing as infinity (a set that contains everything) can actually be shown to be impossible due to Russell's paradox that they talk about in the Veritasium video you linked.

[yzb25]

Some context:
https://en.wikipedia.org/wiki/Cantor...gonal_argument
https://www.youtube.com/watch?v=HeQX2HjkcNo

I've been randomly looking into this and... I don't believe Cantor's argument! I'm not great at maths, though, so it's very possible I'm just being dumb, but I think that the argument is flawed (and that the conclusion is too). Here's why: it's impossible to add one to x rank of each number on an infinite list without getting to a number of the list, because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything. Thus, isn't the proof based on a misunderstanding of the concept of infinity?

And yes, it is very ballsy of me to attack something that apparently was proven a century ago, and I dare hope you're gonna prove me wrong or tell me I misunderstood something, O you knowledgeable people :P

Well, it might be better for you to flesh out your understanding of this and us to then reply to you. Because this reasoning:

"because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything."

Sounds like an argument for why the statement itself must be wrong rather than any issue with the proof itself. As lag said, you can certainly have infinitely many things without having everything. You just need to have a never-ending amount.

Nonetheless, this discussion is very fun so I'll say a few things --

The proof simply constructs a new real number from a list of other real numbers which is not anywhere on the aforementioned list. Do you have an issue with some aspect of that process? Do you have an issue with the idea of applying an "infinitely long operation" to construct the new real number? Or are you unconvinced that the new real number will necessarily not appear anywhere on the list?

The following advice may help you to think about it:

Rather than trying to grasp the entire infinite process at once, try to think about what's happening at each entry. You can't possibly grasp the new real number this construction generates, because changing the n'th entry on every n'th row will take infinitely long. However, you can ask me about any of this new real number's digits - ask me what the 50th digit is, ask me what the 100th digit is - and I can run 100 rows down my list and 100 entries along and find the digit and tell you "ah, the new real number's 100th digit is a 1!". This is no different from how you understand any other real number. You do not know every single digit of pi, but given enough time (and enough motivation ^^) you can find the 100th digit or the 200th digit. And that's enough to satisfy you that the number exists and makes sense.

The same is true for the process of checking whether this new real number is different from every entry on the list. You can't possibly check every number on the list to see if it's different from your new real number. Attempting such a thing would be foolish. But, give me any row. Ask for the 1000th row even. I can go to that row, and verify for you that our new real number is indeed different from the number on that row, by plodding along to the 1000th entry and checking that our new real number does indeed have a different digit.

It may sound daft, but maybe try literally carrying out this construction for just a few numbers. Write out 5 or 6 real numbers to 10 digits, then reverse the n'th digit on the n'th row, and stare at the resulting construction until it makes sense how it would work exactly the same in an endless list ^^.

[yzb25]

oh also MM, if you have doubts about the veracity of modern mathematics, you might get a kick out of this --

https://en.wikipedia.org/wiki/Long_line_(topology)

Wikipedia doesn't have a picture for that one lmao ^^

"Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments."

[Plotato]

The set of rationals is countable though, it’s the irrational that present the problem

Although I guess I’m just responding to a copypasta. I wonder if plotato ever posts his own thoughts

set theory union you 600 pound mass

what is a real number to a frenchman with no mathematical background?

[Marshmallow Marshall]

oh also MM, if you have doubts about the veracity of modern mathematics, you might get a kick out of this --

https://en.wikipedia.org/wiki/Long_line_(topology)

Wikipedia doesn't have a picture for that one lmao ^^

"Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments."

To be clear, I have more doubts about my understanding of maths than about maths themselves lol. But anyway, I misspoke and meant infinity comprises everything between 0 and 1 in this case. I thought it was clear at first, but now I see how it isn't. Whoops.
My issue is more that the infinitely long operation is forced to result in a number between 0 and 1 still (else you'd be out of the specific infinity set, which would be "cheating" and wouldn't prove anything). How is a real number between 0 and 1 not comprised in an infinity of real numbers between 0 and 1? Shouldn't everything between 0 and 1 be in there, thus including the result number?

As for your "do it yourself" suggestion, it doesn't solve my issue because I know it's going to give a different number, that's obvious. My issue is with the exact concept of infinity, I guess. It's what aamirus said here:

So, while yes we are constructing an infinite length element that is not in the enumeration, that element is still a valid member of the set we are trying to prove is uncountable.

Also thanks for the walls and fuck you plotato

[Lumi]

oh also MM, if you have doubts about the veracity of modern mathematics, you might get a kick out of this --

https://en.wikipedia.org/wiki/Long_line_(topology)

Wikipedia doesn't have a picture for that one lmao ^^

"Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments."

This is great.

"It's like, the number line... but longer"

Also a perfect example of why I stuck to applied mathematics xP

[yzb25]

To be clear, I have more doubts about my understanding of maths than about maths themselves lol. But anyway, I misspoke and meant infinity comprises everything between 0 and 1 in this case. I thought it was clear at first, but now I see how it isn't. Whoops.
My issue is more that the infinitely long operation is forced to result in a number between 0 and 1 still (else you'd be out of the specific infinity set, which would be "cheating" and wouldn't prove anything). How is a real number between 0 and 1 not comprised in an infinity of real numbers between 0 and 1? Shouldn't everything between 0 and 1 be in there, thus including the result number?

As for your "do it yourself" suggestion, it doesn't solve my issue because I know it's going to give a different number, that's obvious. My issue is with the exact concept of infinity, I guess. It's what aamirus said here:


Also thanks for the walls and fuck you plotato

hmm, maybe there is a misunderstanding about what this infinite list represents exactly. This list is just some arbitrary list of numbers between 0 and 1. The list does not necessarily have the infinity of numbers between 0 and 1. Indeed, we are about to prove it doesn't. The list could be something dumb like this:

1) 0.1
2) 0.01
3) 0.001
4) 0.0001
...

Or it could be a more earnest attempt to hit every number between 0 and 1:

1) 0.110100110001...
2) 0.110001111111...
3) 0.101010110101....
...

It doesn't matter. The point of the proof is that, no matter what infinite list is given, we can apply the construction to get a number between 0 and 1 not on the list.

[yzb25]

Maybe the problem is how you're thinking about infinity, as you say. You have an infinitely long list, therefore you can include everything between 0 and 1. Because you have "endless ammunition" anything not on the list can be added to the list. Is that what you're thinking?

[Plotato]

My issue is more that the infinitely long operation is forced to result in a number between 0 and 1 still (else you'd be out of the specific infinity set, which would be "cheating" and wouldn't prove anything). How is a real number between 0 and 1 not comprised in an infinity of real numbers between 0 and 1? Shouldn't everything between 0 and 1 be in there, thus including the result number?

i knew you would still be confused about this, oh, how the germans have stumped the french once again

mathematically, countability is defined of whether you can assign 1, 2, etc. to whatever you want to map to on a one-to-one basis, no element left unjerked. if the list is infinitely long, then, as unintuitive as it sounds, then a countably infinite list is full... matched with infinite elements, on a one-to-one basis.

Cantor's argument comes with the above presupposition that you can build such a list where you can map every integer (1 -> 0.1232, etc) with a real number once (therefore making it countable). the order of the list doesn't matter. now why wouldn't such a list already have the whatever number produced by diagonalization? no, i can, through diagonalization, come up with a new number that won't have an sole integer assigned to. i know it sounds unintuitive, but i will make bold the affirmative. the case is: i have produced a new number of which there is not an assignment of an integer, therefore proving my idea that such a list is countable wrong. guess its not countable. this idea also implies the varying sizes to infinity, as we can't match one-to-one every element from one set of numbers to another without leaving something behind...

the thing with math is that you have to explicitly define, prove and state everything in a sheer logical manner. the definition of "countable" is mentioned earlier, and if i, for some reason, can't make that one-to-one complete map from the integers to whatever, the definition is now uncountable by contradiction. intuition doesn't matter. somethings in math are just unintuitively the case. yes, every number that would exist exists between 0 and 1... can you "count" them all, given the definition of "countable"? if you're still thinking that you could always generate a new integer for a new real number produced, well that would just break the definition of "countable" that we have, which is a pretty good one, based off of mathematical reasoning. if we hold the definition of "countable" to be the case, well, you can't have your intuition go your way.

please understand you escargot

[Marshmallow Marshall]

Maybe the problem is how you're thinking about infinity, as you say. You have an infinitely long list, therefore you can include everything between 0 and 1. Because you have "endless ammunition" anything not on the list can be added to the list. Is that what you're thinking?

Anything between 0 and 1 not only can be added to the list, but is on it. That's the assumption I'm making. If it's just an arbitrary list of some stuff between 0 and 1, then sure, it works, but isn't that... not infinite?

[Plotato]

Anything between 0 and 1 not only can be added to the list, but is on it. That's the assumption I'm making.

yes but is it countable?

[yzb25]

Anything between 0 and 1 not only can be added to the list, but is on it. That's the assumption I'm making. If it's just an arbitrary list of some stuff between 0 and 1, then sure, it works, but isn't that... not infinite?

Hm... Maybe it would benefit both of us for you to be more cautious with how you phrase these things. An arbitrary list of some stuff may very well be unending (which is exactly what the word infinite means). What I assume you mean is, "clearly that list isn't complete".

What I am asserting is there is no complete list of all the numbers between 0 and 1, whether it's unending or not. By virtue of merely demanding that you have to write each number down, one by one, you have already imposed a subtle restriction on your ability to account for every number. Even if you write down one entry per second from now until the end of time, you will still be missing a number. That is what the proof illustrates, and that is what makes it profound.

[yzb25]

If you do not necessitate that the numbers are being written down, one at a time, once per second, forever, then perverse things can happen. For example, if you allow god to take the pen, maybe god is capable of instantaneously producing a vast list that includes every such number. Maybe he uses his long line for space to do it (lol) but that is neither here nor there.

As Plotato is repeating, the list needs to be countably infinite, which I suppose for our purposes is the mathematically formal way of ruling out the possibility god took the pen. It restricts you to either using a numbered well-defined list, or some kind of scenario where you only get to write down one entry per second forever.

[Plotato]

What I am asserting is there is no complete list of all the numbers between 0 and 1, whether it's unending or not. By virtue of merely demanding that you have to write each number down, one by one, you have already imposed a subtle restriction on your ability to account for every number. Even if you write down one entry per second from now until the end of time, you will still be missing a number. That is what the proof illustrates, and that is what makes it profound.

this is misleading

the set of integers will always have a bigger number, this is an approach but never reach argument which implies the set of integers is uncountable, but it is. cantor's diagonalization demonstrates by contradiction that you can generate more real numbers than a prescribed infinite number of integers, therefore making the set of real numbers, not countable by mathematical definition.

personally, analogies that try to "list" or "count" infinities will always end up being confusing because the verb implicitly ascribes a mathematical countability (i can count +1 everytime) or uncountability (i can go on forever, can't count forever) to it, depending on whoever's interpretation. some infinities will be smaller or larger in size, or the number of elements contained in them, but the only thing defining countability infinite is whether there is mathematical indication that you can map one-to-one from the infinite set of integers to whatever.

[OzyWho]

Guys, let's talk about something less smarty stuff.
Like, what’s your favorite sleeping position?

[Plotato]

Guys, let's talk about something less smarty stuff.
Like, what’s your favorite sleeping position?

your mother likes to be spooned so there

[OzyWho]

your mother likes to be spooned so there

Are you the yesterday's fat guy or that old hairy dude from the day before?

[yzb25]

this is misleading

the set of integers will always have a bigger number, this is an approach but never reach argument which implies the set of integers is uncountable, but it is. cantor's diagonalization demonstrates by contradiction that you can generate more real numbers than a prescribed infinite number of integers, therefore making the set of real numbers, not countable by mathematical definition.

personally, analogies that try to "list" or "count" infinities will always end up being confusing because the verb implicitly ascribes a mathematical countability (i can count +1 everytime) or uncountability (i can go on forever, can't count forever) to it, depending on whoever's interpretation. some infinities will be smaller or larger in size, or the number of elements contained in them, but the only thing defining countability infinite is whether there is mathematical indication that you can map one-to-one from the infinite set of integers to whatever.

I didn't mean to suggest in the forever-list you're missing a number because you haven't reached it yet. I meant even "after" completing the forever-list, you still wouldn't have every number. You could list every natural number given "forever", by writing the number n at second n, or every integer by alternating between writing a negative and a positive each second. Understand? This is a legitimate way to think about it. If the thought is clearly conveyed, the forever-list is no different from a function from the naturals.

I agree it might be better to let go of the forever-list stuff if it's derailing things and just work with a function from the naturals. I just like it because you can explain it without giving explicit formulae. >.>

[Plotato]

Are you the yesterday's fat guy or that old hairy dude from the day before?

take a look at yourself in the mirror and figure out which of the genes passed

[DJarJar]

I’m glad he at least stopped talking in pure copypastas

[Marshmallow Marshall]

yes but is it countable?

Uncountable, but any real number between 0 and 1 is comprised anyway, so the real number you would "create" technically already is part of the infinity. What you're proving is that uncountable and countable infinities are different (and that the integers infinity is countable, while the [0, 1[ infinity isn't), but not that one is bigger than the other, since both can go on forever. There aren't "more numbers" in one set than in the other, even though it's impossible to pinpoint a rank in the real numbers list unlike in the integers list. You seemed to be debating the fact that the real numbers infinite set is uncountable... which nobody disputed, as far as I know.

I'm totally making you all waste your time on explaining this to me XD

[Plotato]

clearly god doesnt exist because in no way this frenchman was the image of man he ever wanted to design

you wanna know something cool? the smallest mathematical infinity is the set of positive integers (1, 2, 3...) . there is no infinity of a smaller size. anything smaller would have a provable way of showing that it is finite and bounded.

in order for an "infinity" to be the same size you would have to provably show that you can do a one-to-one map of every integer in the infinite set to every element of another set. this is where your holier-than-thou-but-stupid-as-fuck showerthought comes in:

"There aren't "more numbers" in one set than in the other, even though it's impossible to pinpoint a rank in the real numbers list unlike in the integers list. You seemed to be debating the fact that the real numbers infinite set is uncountable... which nobody disputed, as far as I know."


thank god for adding "as far as I know" because it just means youre an idiot for unknowingly disputing the math

this is true for countable infinities, because all countable infinities are the same size, as ascribed to by some mapping of the positive integers. now, the important thing to realize with this statement is that the list is full.

full.

or that every thing is mapped. whatever. if for some reason i have something unique and spare and wanted to assign an integer to it, nope. all engaged here. here comes the reason.

---

you are guessing (because clearly there is no thinking here) that such infinite whatevers exist on a whim that contain every whatever that exists within those infinities, and as such all infinities are by nature of "completeness" or whatever are equal in size.

this is kind of the hypothesis that cantor thought (because he is a mathematician and more importantly, not french) to contradict with his diagonalization theory, which introduces the idea of uncountable infinities, or infinities that are larger than countable ones.

this is true: all uncountable infinties are larger than countable infinities. to reiterate: all countable infinities are the same size, and are the smallest of the infinities. i shall simplify why.

let us assume again that your line of thinking is true (its not) that the integer and real number infinity is the same size. firstly , you are already contradicting yourself by assuming the truth that the integers are countable and real numbers aren't, yet they are the same size ("go on forever"). what is the distinction here? i can come up with more integers just as i can decimal numbers. if for some reason i cant do a one-to-one map of a forever list between these two sets, what does that imply about the size of one set to another?

but if we assume that the integers and real numbers are the same size (in your words, "but not that one is bigger than the other, since both can go on forever") we can do a one-to-one map for each individual integer to a real number in their respective infinite sets. i can assure you, however, that i can generate an entirely new number, unique in property to every existing real number infinitely listed from nitpicking one number in every real number in this infinite list. this implies that i have a new number not already mapped. completely new, never before seen. because there is an entirely new unmapped real number, this makes this new list uncountable, and contradicting your hypothesis. i already know at this point you are still thinking that such a number is already listed somewhere in the infinity but if you are still thinking this then clearly i have proof that robespierre's reign of terror has lead to lineages of families born without their heads.

putting """""philosophy"""""" in a now considerably mathematically formalized idea of "infinity" is just going to screw with your line of thought and i suggest you drop it

[Plotato]

and as far as the rules go i see nothing against posting porn in horrendous topics so i demand this slight be righted by the other powers that be

[yzb25]

Uncountable, but any real number between 0 and 1 is comprised anyway, so the real number you would "create" technically already is part of the infinity. What you're proving is that uncountable and countable infinities are different (and that the integers infinity is countable, while the [0, 1[ infinity isn't), but not that one is bigger than the other, since both can go on forever. There aren't "more numbers" in one set than in the other, even though it's impossible to pinpoint a rank in the real numbers list unlike in the integers list. You seemed to be debating the fact that the real numbers infinite set is uncountable... which nobody disputed, as far as I know.

I'm totally making you all waste your time on explaining this to me XD

As a matter of fact, you have given our hollow lives purpose!

[yzb25]

It might help to bare in mind when we talk about the "size" of these infinite sets, that is a very informal way of referring to something called their "cardinality". In the popular culture, we've gotten very used to talking about "cardinality" as a measure of "size", but it may be slightly more accurate to think about cardinality in terms of "information".

For example, if you consider the set of positive whole numbers (1,2,3,4,5...) vs. the set of even numbers (2,4,6,8,10...) the first set seems strictly larger than the second set (in some sense, it has literally double the stuff). However, from the point of view of "cardinality", they both have the same amount of information. I can label every positive whole number with a unique even number like so, in a well-defined manner:

2->1
4->2
6->3
...

And when we say the real numbers have a higher cardinality, we are somehow making a statement that the real numbers are simply too complicated to be encoded in terms of positive whole numbers. There is no way of labelling every real number with a unique positive whole number.

If we could label every real number with a unique positive whole number, that would be kind of revolutionary for our notation. We use these garish "infinite decimals" to encode real numbers... but no matter how many decimal places you write down, there's still so many possible numbers you could be referring to when you write the next digits! If we could encode every real with a natural, we'd have a way of finitely expressing every real number at once. Can you imagine?! Well, we literally can't, but still!

[oops_ur_dead]

You see, mathematicians needed to come up with numbers past infinity so that they could calculate the amount of time it took them before they would get laid.

[Marshmallow Marshall]

It might help to bare in mind when we talk about the "size" of these infinite sets, that is a very informal way of referring to something called their "cardinality". In the popular culture, we've gotten very used to talking about "cardinality" as a measure of "size", but it may be slightly more accurate to think about cardinality in terms of "information".

For example, if you consider the set of positive whole numbers (1,2,3,4,5...) vs. the set of even numbers (2,4,6,8,10...) the first set seems strictly larger than the second set (in some sense, it has literally double the stuff). However, from the point of view of "cardinality", they both have the same amount of information. I can label every positive whole number with a unique even number like so, in a well-defined manner:

2->1
4->2
6->3
...

And when we say the real numbers have a higher cardinality, we are somehow making a statement that the real numbers are simply too complicated to be encoded in terms of positive whole numbers. There is no way of labelling every real number with a unique positive whole number.

If we could label every real number with a unique positive whole number, that would be kind of revolutionary for our notation. We use these garish "infinite decimals" to encode real numbers... but no matter how many decimal places you write down, there's still so many possible numbers you could be referring to when you write the next digits! If we could encode every real with a natural, we'd have a way of finitely expressing every real number at once. Can you imagine?! Well, we literally can't, but still!

Lol

I think I understand a little better now (it basically comes down to plotato's thing about mapping integers and real numbers one to one, right?), and well... I guess that's why I didn't go too far into maths xD.

[Oberon]

@Marshmallow Marshall are you still confused about this, Frenchman? I can offer a more intuitive explanation:

The set of integers are numbers with a distance equal to 1 between them. If you draw a tick at every single natural number along a line, you will see a distance of 1 between each. However, this cannot be done with irrational numbers because every irrational number consists of a countable infinity of digits after the decimal point. This means that in order to reach the immediate neighbour of a given irrational number, you have to add or subtract an amount of the form 0.00000.....01 = 1 / infinity = 0 from each. You cannot do this.

Basically the irrational numbers are an infinity of infinities. In fact, if you were to say that the rational/integral numbers have a cardinality of, say, N, then the irrational numbers have a cardinality that is "conceptually" equal to 2 ^ N: if you were to take the set of all possible subsets (including the whole set) of integral numbers, this set of sets would have the same cardinality as the set of irrationals: basically, there are as many subsets of the integral numbers as there are rational numbers.

This number is bewilderingly large because this infinity is the BINARY POWER ​of another infinity.

[Oberon]

Fun fact: you can extend this indefinitely. There are also "numbers" that have an uncountably infinite number of digits after the decimal point, in which case this set of numbers has a cardinality that is equal to (N!)!, aka equal to the cardinality of the set of all the subsets of irrational numbers.

[Marshmallow Marshall]

I am fine, lest Plotato makes his stellar comeback involving more insulting of the French. Thank you :P

[DJarJar]

I am fine, lest Plotato makes his stellar comeback involving more insulting of the French. Thank you :P

mm choosing to not be nerdy to avoid getting picked on by a bully? was sc2mafia grade school all along?

[Marshmallow Marshall]

mm choosing to not be nerdy to avoid getting picked on by a bully? was sc2mafia grade school all along?

Lmao

Don't worry, it's just that I had private classes on the topic, and Mag laid it out right here lol

[Oberon]

i knew you would still be confused about this, oh, how the germans have stumped the french once again

mathematically, countability is defined of whether you can assign 1, 2, etc. to whatever you want to map to on a one-to-one basis, no element left unjerked. if the list is infinitely long, then, as unintuitive as it sounds, then a countably infinite list is full... matched with infinite elements, on a one-to-one basis.

Cantor's argument comes with the above presupposition that you can build such a list where you can map every integer (1 -> 0.1232, etc) with a real number once (therefore making it countable). the order of the list doesn't matter. now why wouldn't such a list already have the whatever number produced by diagonalization? no, i can, through diagonalization, come up with a new number that won't have an sole integer assigned to. i know it sounds unintuitive, but i will make bold the affirmative. the case is: i have produced a new number of which there is not an assignment of an integer, therefore proving my idea that such a list is countable wrong. guess its not countable. this idea also implies the varying sizes to infinity, as we can't match one-to-one every element from one set of numbers to another without leaving something behind...

the thing with math is that you have to explicitly define, prove and state everything in a sheer logical manner. the definition of "countable" is mentioned earlier, and if i, for some reason, can't make that one-to-one complete map from the integers to whatever, the definition is now uncountable by contradiction. intuition doesn't matter. somethings in math are just unintuitively the case. yes, every number that would exist exists between 0 and 1... can you "count" them all, given the definition of "countable"? if you're still thinking that you could always generate a new integer for a new real number produced, well that would just break the definition of "countable" that we have, which is a pretty good one, based off of mathematical reasoning. if we hold the definition of "countable" to be the case, well, you can't have your intuition go your way.

please understand you escargot

I have a much simpler explanation.
What is a natural number? It's a finite sequence of digits. What is the totality of natural numbers? An infinite sequence of digits, ordered in a certain way.
What is a irrational number? An infinite sequence of digits. Every irrational number corresponds to a certain permutation of the set of natural numbers. If you take all these permutations together, it is obvious there are more irrationals than naturals. You effectively run into an infinity of infinities

[Plotato]

I have a much simpler explanation.
What is a natural number? It's a finite sequence of digits. What is the totality of natural numbers? An infinite sequence of digits, ordered in a certain way.
What is a irrational number? An infinite sequence of digits. Every irrational number corresponds to a certain permutation of the set of natural numbers. If you take all these permutations together, it is obvious there are more irrationals than naturals. You effectively run into an infinity of infinities

Your definitions are awful, inaccurate and unhelpful.

A natural number to put it simply is a number represented without fraction or complex numbers. The set of all natural numbers (1, 2, 3, 4,...) is our definitional basis for the countability of infinite sets.

Your third line misrepresents the whole argument because the set of rational numbers are countable and are a certain permutation of the natural numbers. You haven't defined what "certain permutation" means and therefore is moot and your definition of irrational number is also completely unhelpful. Permutation implies a certain order and as far as it goes, the decimal construction of irrational numbers have no order to them (and therefore can't be represented as a fraction).

You can't say it is obvious that there is "more of something". There are no more even numbers than there are natural numbers. There are no more numbers of the Fibonacci sequence than there are natural numbers. Alluded to earlier there is no more fractional numbers than there are natural numbers. Cantor's diagonalization states specifically that such a construction exists where you can generate an entirely new number from an infinite list of numbers that was not previously recorded in an infinite mapping with the natural numbers. This isn't an "inbetween argument" like the way you put it. Infinity does not work the way you describe it like it has a certain scale or something.

[Oberon]

Your definitions are awful, inaccurate and unhelpful.

You can't say it is obvious that there is "more of something". There are no more even numbers than there are natural numbers. There are no more numbers of the Fibonacci sequence than there are natural numbers. Alluded to earlier there is no more fractional numbers than there are natural numbers. Cantor's diagonalization states specifically that such a construction exists where you can generate an entirely new number from an infinite list of numbers that was not previously recorded in an infinite mapping with the natural numbers.

Your third line misrepresents the whole argument because the set of rational numbers are countable and are a certain permutation of the natural numbers.

The set of rational numbers is a permutation of the naturals. Yes, hence why the set of rationals is countable. But for irrational numbers, it is every irrational number, itself, that is a permutation of the entire set of natural numbers and is, hence, countably infinite.

You haven't defined what "certain permutation" means and therefore is moot and your definition of irrational number is also completely unhelpful.

A permutation is a certain way of ordering some set.

Permutation implies a certain order and as far as it goes, the decimal construction of irrational numbers have no order to them (and therefore can't be represented as a fraction).

Yes, indeed. However, I'm talking about the permutations of the set of natural numbers, not reals. Whether or not the real numbers have a well-ordering is not relevant to my point.

This isn't an "inbetween argument" like the way you put it. Infinity does not work the way you describe it like it has a certain scale or something.

It's not an in-between argument. You misunderstood my point about the "infinity of infinities"; I'm not talking about all the segments of [0, 1) that you can copy paste onto each other over and over, but I'm talking about the numbers themselves being infinite; their representation is infinite. If their representation is infinite, you clearly need a "higher" ordinal to index into the list of infinite things: a number that can index into an infinite list of finite things has less information than it, because... you can change a finite number number of digits in a finite number, and get a fully unique, distinct number. For infinite numbers, if you change only a finite number of the first few digits you've written, there are still many numbers you could be referring to.

So perhaps it is unsurprising that if you have an infinite list A of infinite things, that you have more information in A than in the infinite list B of finite things.

[Oberon]

A natural number can be represented in any number encoding; for example, in binary, every natural number is some sequence of 0s and 1s. In decimal, it's a sequence of the decimal digits (0, 1, ... 9). Depending on which encoding you pick, if you try to enumerate every element in the sequence (0, 1, 2...), you get a sequence that looks slightly different, and you'll notice the length of some strings changes depending on the number of digits you use. However, it cannot be disputed that the encoding does not have any impact on the number of elements in the sequence.

You can think of the "rational" numbers as being the natural numbers, but encoded in a number system wherein there are 11 digits: the 10 digits from (0, ... 9) and an 11th digit for representing fractions, '/'. Seeing as we've stated that it does not matter how many digits you use to represent the natural numbers, it is perhaps unsurprising that there are as many rational numbers as natural numbers. You of course have to add the constraint that '/' can never precede a sequence of all 0s, but that's a relatively minor thing. Even without that constraint they're still the same set.

This is not the case for the irrational numbers, because every irrational number is a countably infinite sequence of digits.

[Oberon]

I actually have no idea how you could misunderstand my point so hard lmao, unless you just didn't read or something