View Poll Results: Is 0.999... exactly equal to one?

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  • Yes, they are equal.

    40 54.79%
  • No, they are not equal.

    33 45.21%
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Thread: Is 0.999... equal to one?

  1. #5041
    Senior Member john_gabriel's Avatar
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    Default How did Cantor arrive at the idea of a countable set?

    So how did Cantor get the idea that the set is countable? And what exactly does "countable" mean?

    Test your knowledge before we investigate what countable means, by choosing the one correct answer:

    1. is countable because it contains natural numbers.

    2. is countable because you can count the elements.

    3. is countable because each element has a unique name.

    4. is countable because every element has an ordinal value.

    So, what do you think is the ONLY correct answer?

    I will reveal the answer only if at least one person attempts the question.
    You will be surprised at what it means to be countable. I can bet a few dollars that many topologists never fully understood this concept.
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  2. #5042
    Senior Member john_gabriel's Avatar
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    Default The end of Cantor's flawed Diagonal Argument.

    The end of Cantor's flawed Diagonal Argument.

    There is a lot of confusion among many set theorists and topologists regarding what it means exactly for a set to be countable.

    I will explain in detail how our delusional Cantor came upon the idea and then show you a very simple way to determine whether a set is countable, by defining it in one simple sentence.

    While there are those who try to distinguish between finite and infinite sets, there really is ZERO difference in the definition of countable. For example, Countable set tries to distinguish between countable finite and infinite sets.

    As you shall shortly see, our young Cantor, being the abstract learner that he was (I can’t think of many Jews who are constructive learners! I myself am an abstract learner too), loved to imagine. I love to imagine also, except what I imagine is real, but the same can’t always be said for Cantor.

    Cantor knew that he could name every element of a finite set, but he fantasized about being able to name every element of an infinite set. The need to name each element, was of paramount importance in his mind, for without this ability, he could not well define the concept of set. After all, what use is a set, if its contents cannot be identified? Of course this is perfectly valid reasoning, since Cantor had not yet lost his marbles.

    In due time, he experienced an epiphany concerning certain sets. Cantor soon realized that every member of the set of natural numbers already had a name! How is this possible? No, he was not thinking of 1, 2, 3, etc. You can only count so far before you run out of names, e.g. what do you call 234, 475, 573, 488, 111, 352 followed by a Googleplex of zeroes?

    Well, you call it by the digits that are used in the given radix system, which is always a UNIQUE identifier! Voila! Cantor thought in terms of binary at first, but there is no difference between any radix system that is used, i.e. the representation is always UNIQUE. It is this representation that he adopted as the name of each element. (*)

    Using this scheme, he could write down as many elements as he desired, by means of their names (not an index). The process of writing down elements or listing them came to be known under many different terms: representation, enumeration, denumeration, etc. However, all he meant was that the elements could be listed!

    So now, Cantor had the means to list as many elements of the infinite set of natural numbers as he desired. The realization that he could list the set of integers and rational numbers soon followed. The inevitable happened next: what about real numbers? Could he name every real number? Well, had he known about Gabriel’s decimal tree, he would have soon realised that provided one assumes real numbers can all be represented as decimals, then it too would be possible to list these also. Unfortunately, he resorted to the use of sophistry, which is clearly evident in his flawed Diagonal Argument.

    You see, representation is nine tenths of enumeration (naming the elements). How can anyone name objects that can’t be represented? Ludicrous of course! So in our young handsome Jew’s mind, the real numbers could not be listed. In fact, the real numbers can't actually be listed for two reasons: (a) they don't exist. (b) not all magnitudes and all incommensurable magnitudes can be represented in decimal.

    How did the idea of a one-to-one correspondence come about? A little reasoning reveals immediately that the idea was created solely for the purpose of teaching these somewhat complex concepts to constructive learners. See, there is really no need for any set to be in one-to-one correspondence with the natural numbers, in order to be considered countable.

    The only criteria, is that a set must contain elements that can be listed, given the importance of using unique names. The word bijection did not even exist until 1963. However, using the idea of bijection makes it easier to determine whether a given set is countable or not. In other words, making the concepts easier to learn for constructive learners.

    To be sure, if one can name all the elements of an infinite set, then the set is countable. Another way of saying this, is that if the elements can be listed or written down, the set is countable. In fact, this is the ONLY criteria Cantor uses in his flawed Diagonal Argument. What this means is that his argument is clearly defeated by Gabriel’s tree without a further enumeration process as I demonstrated by creating an index set, which I then showed to be of the same cardinality as the set of natural numbers.


    And now let’s examine the question posed in comment 5041.

    The first option is FALSE. A set is not countable because it contains natural numbers. It is countable because its members can be listed. For example, the set {knife, fork, spoon} is countable but none of its members are natural numbers.

    The second option is also FALSE because counting elements has NOTHING to do with a set being countable.

    The last option is FALSE because sets can contain elements that do not have ordinal values. Again, {knife, fork, spoon} is one such set. The example set can be assigned ordinal values, but that does not affect the outcome.

    The correct answer is OPTION 3, that is, a set is countable if, and only if, each element has a unique name, that is, each element can be systematically listed or written down.

    In earlier comments I showed that the so-called set of reals is denumerable by including a secondary enumeration in the form of an index set. Most of you are absolute imbeciles and would not have seen the result if I had done it any other way. Mark Chu Carroll is a fine example.

    I tried to use Socratic reasoning but encountered the same ignorance and obstinacy as I have with Grapes and other members of this forum. If you look at previous responses, you will see that Grapes has difficulty just answering simple questions.

    Carroll knew that he was being reeled in as a fish from the water. Without realizing it, Carroll had already agreed that all the real numbers were in fact in my tree, and consequently they had ALREADY been LISTED!

    Carroll began to flounder, and suggested that 1/3 was not in my tree. Well, as you can see, this is obviously untrue.

    You have been greatly privileged to read the end of this matter here on STATU. I hope that you will study my comments with lots of diligence!

    Finally, I hope that you will take a couple of weeks to learn my New Calculus! All you need is a basic high school knowledge of mathematics. In fact, it is better if you have not studied any college math at all.


    (*) It is this same realisation that kills his flawed Diagonal Argument. If all the "real" numbers can be represented uniquely in decimal, and a scheme exists to list them, then they must be countable! Tsk, tsk. Poor delusional Cantor!

    For a proof that real numbers do not exist:

    Is 0.999... equal to one?

    Very late EDIT:

    I posted this Quiz on Quora, and only one graduate student from MIT got it right - Anders Kaesorg.
    Last edited by john_gabriel; 11-01-2014 at 07:12 PM.
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  3. #5043
    Senior Member john_gabriel's Avatar
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    Default Re: The end of Cantor's flawed Diagonal Argument.

    More thoughts on the infinite sum concept.

    Given that and , it is very tempting to reason as follows:

    Well, the limit of all the partial sums is , so why not just define it to be as Euler did?

    1. Because the sum that matters is not a partial sum. It is an ill-defined sum without a last term.

    2. Such definitions lead to nonsense such as 1/3 = 0.333... and 1 = 0.999...
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  4. #5044
    Senior Member caper's Avatar
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    Default Euler's Mistake in the Digit Proof

    Euler's "blunder" revealed in rigorous detail:

    Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"











    First off, since when did become a variable that we needed to solve for ?? It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".

    The huge blunder here is the combination of multiplying by 10 and subtracting a series from the resulting series, namely, 10 x 0.999... - 0.999...
    There are other underlying assumptions too, but I addressed those earlier in the thread (specifically, multiplying an infinite series by anything which causes a carry or decimal shift)

    Let's do 10n - n using the definition of n







    Now, what Euler does, is subtract the 2 series,

    ...Except, what he failed to show everyone, or assumed was ok, was that he shifted the second series to the right by one decimal place:





    How he does this is by doing to following:

    I believe Colin pointed this out at one point and said this was "ok" to do. At first glance, the fractions are all equivalent... except...
    ...the SERIES has changes slightly:

    [ Notice the starting index value. ]

    This allows the subtraction to occur as :

    instead of (The correct way)

    is really equal to

    So, by the Property of Addition and Subtraction of Summations:



    we can see that we shouldn't shift the series, because that uses the starting index: instead of

    Let's use the proposed theorem 17, to subtract the 2 series, with equal starting indices:

    Unaltered series:

    or, using the

    Altered Series with using Theorem 17 properly:

    Let's do a few terms, shall we?:





    I believe you get the point now, that by changing the series, you change the results.

    Therefore Euler mistakenly assumed that he could slide the series over to the right, and "cancel off" all the decimal digits, incorrectly so, in order to achieve his desired result of 9n = 9.

    We are not allowed, during this proof, to use the definition of 0.(9) = 1 because the point of the proof was to prove that the 2 numbers are equal "only by doing some basic algebra".

    It was a nifty trick that he used, to manipulate the series in order to "cancel off" all the decimals.

    We know what results we should get at EVERY point in every equation, if we define x = a, then x always equals a, and I showed that by NOT assuming things as Euler did, we do end up with the proper result and not some illusion where (a) becomes (b) with some hocus pocus. If we started with x = 1, and ended up with x = 2, would THAT be ok too? I think not...

    The conclusion is that 0.999... does not = 1 in the end. It equals 0.999... so he proved NOTHING, or, maybe the fact that extra care is due when performing "basic number manipulation" on an infinite series.

    Another source of the Converging Series Identities. The difference of converging series is the same logic as summation of series:

    Last edited by caper; 02-24-2014 at 07:18 PM.
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  5. #5045
    Senior Member grapes's Avatar
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    Default Re: Euler's Mistake in the Digit Proof

    Quote Originally Posted by caper View Post
    Euler's "blunder" revealed in rigorous detail:

    Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"











    First off, since when did become a variable that we needed to solve for ?? It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".

    The huge blunder here is the combination of multiplying by 10 and subtracting a series from the resulting series, namely, 10 x 0.999... - 0.999...
    There are other underlying assumptions too, but I addressed those earlier in the thread (specifically, multiplying an infinite series by anything which causes a carry or decimal shift)

    Let's do 10n - n using the definition of n







    Now, what Euler does, is subtract the 2 series,

    ...Except, what he failed to show everyone, or assumed was ok, was that he shifted the second series to the right by one decimal place:





    How he does this is by doing to following:

    I believe Colin pointed this out at one point and said this was "ok" to do. At first glance, the fractions are all equivalent... except...
    ...the SERIES has changes slightly:

    [ Notice the starting index value. ]

    This allows the subtraction to occur as :

    instead of (The correct way)
    So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

    Weirdly enough I agree with that equation.

    Probably, next post, you'll try to argue against it.

    Or are you going to argue against 8.999... = 8 + 0.999...

    Because if you use those two "correct" equations in your other proof, you still get

    0.999... = 1
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  6. #5046
    Senior Member astromark's Avatar
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    Default Re: Cantor's Diagonal Argument a fraud.

    Quote Originally Posted by Yuspoov View Post
    No one can. Good thing it's defined.



    Ah well, if someone says it's common sense, how could anyone argue with that?
    ~ and if I maintain a honesty I must fess up to wanting some argument for the joy of a conversation is not wasted on me.. The contributor 'Yuspoov' has not understood the question as I do. That this is a assumption that my understanding is better, or different than his. and my reasoning is as this... Those three digits or points ... are a clear indication of intent to convey a term undefined to infinite. If I add them at the end of a numeric expression like pi as 3.142... It is a display of understanding that the term is ill defined. Yes that term can and is worked with to a very accurate end result. That the addition of the three dots just shows I understand the reality of that term. In the case of 0.999... it shows that at no point can the term be equal to another number expressed as 1. That 'Yuspoov' sees this as something else just indicates to me he does not UNDERSTAND as I do. As for the quip regarding my common sense. You can not find argument with what remains mine.
    Last edited by astromark; 01-30-2014 at 09:02 PM.
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  7. #5047
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    Default Re: Cantor's Diagonal Argument a fraud.

    Quote Originally Posted by caper View Post
    Euler's "blunder" revealed in rigorous detail:

    Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"











    First off, since when did become a variable that we needed to solve for ??
    "n" here represents a value we need to solve for, if we want to answer to the question which is the topic of this thread. It does not represent a variable.

    It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".
    Just so. And Euler's argument is not intended to change the value of "n", it is to express the unchanged value of a recurring decimal in a different form as a ratio of integers (or a single integer).

    Quote Originally Posted by grapes View Post
    So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

    Weirdly enough I agree with that equation.

    Probably, next post, you'll try to argue against it.

    Or are you going to argue against 8.999... = 8 + 0.999...

    Because if you use those two "correct" equations in your other proof, you still get

    0.999... = 1
    Yes.

    If it is correct that

    9n = 8.999...

    then

    8n = 8.999... - 0.999... = 8

    therefore

    n = 8/8 = 1

  8. #5048
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    Default Re: Cantor's Diagonal Argument a fraud.




    So far so good. Just added equal numbers to both sides

    wrong. You can't subtract n from one side and .999... from the other. n has to be subtracted from both sides. So

    Then you have to convert n into numbers that can be treated properly.


    not



    That's math trickery.
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  9. #5049
    Senior Member caper's Avatar
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    Default Re: Euler's Mistake in the Digit Proof

    Quote Originally Posted by grapes View Post
    So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

    Weirdly enough I agree with that equation.

    Probably, next post, you'll try to argue against it.

    Or are you going to argue against 8.999... = 8 + 0.999...

    Because if you use those two "correct" equations in your other proof, you still get

    0.999... = 1
    I'm sorry? Did you have a counter argument? Are there any mistakes ?
    Didn't think so.
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  10. #5050
    Senior Member caper's Avatar
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    Default Re: Cantor's Diagonal Argument a fraud.

    Quote Originally Posted by Colin View Post
    "n" here represents a value we need to solve for,
    NO we don't ! It is not a variable!
    n = 0.999... !! Right from the start. That is really bad understanding of variables and constants.

    Yes.
    If it is correct that
    9n = 8.999...
    then
    8n = 8.999... - 0.999... = 8
    therefore
    n = 8/8 = 1
    More fail for you and grapes both. Sorry. Didn't you learn anything from the comment I posted?? Would you wager your house on that ?









    Cross check our work:





    As it should. The more mistakes you make, the more I will point them out. Haven't you realized yet, that this proof is WRONG? If it is mathematically invalid, there is nothing you can do get the result you want when it is done properly.

    Quote Originally Posted by grapes View Post
    Because if you use those two "correct" equations in your other proof, you still get 0.999... = 1
    No. No you don't. Well, maybe you do, but it is wrong.

    Any object assigned to a letter is always that object. I can repeat that if you like. It doesn't get any more basic than that.

    Quote Originally Posted by astrotech View Post



    That's math trickery.
    Except that 9.999... - 0.999... = 8.999...
    Therefore:
    8.999... = 8.999...
    Last edited by caper; 01-30-2014 at 09:11 PM.
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