Infinities being bigger than others, "countable" and "non countable" infinities
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  1. #1

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

    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
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  2. #2

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

    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?
    Quote Originally Posted by Oberon View Post
    from time to time, I act very much like a normal person.

  3. #3

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

    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.
    Last edited by aamirus; August 18th, 2021 at 05:55 PM.
    Have you ever heard the tragedy of Darth Jar Jar the wise?

  4. #4

  5. #5

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

    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.

  6. #6

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

    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
    Last edited by aamirus; August 18th, 2021 at 08:43 PM.
    Have you ever heard the tragedy of Darth Jar Jar the wise?

  7. #7

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

    Quote Originally Posted by aamirus View Post
    The set of rationals is countable though, itís the irrational that present the problem

    Although I guess Iím just responding to a copypasta. I wonder if plotato ever posts his own thoughts
    Off topic but do you have a math-heavy background?

  8. #8

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

    Quote Originally Posted by aamirus View Post
    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

    Quote Originally Posted by Marshmallow Marshall View Post
    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.
    Quote Originally Posted by SuperJack View Post
    We can just ban people randomly, forcing them to visit our website.

  9. #9

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

    Quote Originally Posted by Marshmallow Marshall View Post
    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 ^^.
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  10. #10

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

    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."
    Last edited by yzb25; August 19th, 2021 at 04:02 AM. Reason: added the quote lul
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  11. #11

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

    Quote Originally Posted by aamirus View Post
    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?
    Last edited by Plotato; August 19th, 2021 at 05:14 AM.

  12. #12

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

    Quote Originally Posted by yzb25 View Post
    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
    Spoiler : Quotes :
    Quote Originally Posted by S-FM Hey peter View Post
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    Mallow are you really an anti vaxxer
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  13. #13

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

    Quote Originally Posted by yzb25 View Post
    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
    Quote Originally Posted by SuperJack View Post
    We can just ban people randomly, forcing them to visit our website.

  14. #14

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

    Quote Originally Posted by Marshmallow Marshall View Post
    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.
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  15. #15

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

    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?
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  16. #16

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

    Quote Originally Posted by Marshmallow Marshall View Post
    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

  17. #17

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

    Quote Originally Posted by yzb25 View Post
    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?
    Spoiler : Quotes :
    Quote Originally Posted by S-FM Hey peter View Post
    There are two wolves inside you. One is addicted to crack. The other one is also addicted to crack. You are addicted to crack.
    Quote Originally Posted by Stealthbomber16 View Post
    MM IS AN ANTI-VAXXER
    Quote Originally Posted by BananaCucho View Post
    Mallow are you really an anti vaxxer
    Quote Originally Posted by The Lawyer View Post
    Besides your lamp and your refridgerators, do you find anyone else suspicious?
    Quote Originally Posted by Renegade View Post
    God is a goofy loser.

  18. #18

  19. #19

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

    Quote Originally Posted by Marshmallow Marshall View Post
    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.
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  20. #20

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

    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.
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  21. #21

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

    Quote Originally Posted by yzb25 View Post
    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.

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  24. #24

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

    Quote Originally Posted by Plotato View Post
    your mother likes to be spooned so there
    Are you the yesterday's fat guy or that old hairy dude from the day before?
    Quote Originally Posted by Oberon View Post
    from time to time, I act very much like a normal person.

  25. #25

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

    Quote Originally Posted by Plotato View Post
    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. >.>
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

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  28. #28

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

    Quote Originally Posted by Plotato View Post
    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
    Spoiler : Quotes :
    Quote Originally Posted by S-FM Hey peter View Post
    There are two wolves inside you. One is addicted to crack. The other one is also addicted to crack. You are addicted to crack.
    Quote Originally Posted by Stealthbomber16 View Post
    MM IS AN ANTI-VAXXER
    Quote Originally Posted by BananaCucho View Post
    Mallow are you really an anti vaxxer
    Quote Originally Posted by The Lawyer View Post
    Besides your lamp and your refridgerators, do you find anyone else suspicious?
    Quote Originally Posted by Renegade View Post
    God is a goofy loser.

  29. #29

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

    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
    Last edited by Plotato; August 19th, 2021 at 07:25 PM. Reason: because i know this frenchman will go out of his way to read this shit and nitpick something to reaffirm his idiocy so i made

  30. #30

  31. #31

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

    Quote Originally Posted by Marshmallow Marshall View Post
    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!
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  32. #32

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

    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!
    Quote Originally Posted by Blinkstorteddd02 View Post
    naz, he's claiming to have been at your house last night and infected you. I know u were drunk but PLEASE try as hard as you can to remember... That burning you felt the next morning when you went pee was from me, not him.

  33. #33

  34. #34

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

    Quote Originally Posted by yzb25 View Post
    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.
    Spoiler : Quotes :
    Quote Originally Posted by S-FM Hey peter View Post
    There are two wolves inside you. One is addicted to crack. The other one is also addicted to crack. You are addicted to crack.
    Quote Originally Posted by Stealthbomber16 View Post
    MM IS AN ANTI-VAXXER
    Quote Originally Posted by BananaCucho View Post
    Mallow are you really an anti vaxxer
    Quote Originally Posted by The Lawyer View Post
    Besides your lamp and your refridgerators, do you find anyone else suspicious?
    Quote Originally Posted by Renegade View Post
    God is a goofy loser.

  35. #35

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

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