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Math Help - Questions About Cantor's Diagonal

  1. #1
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    Questions About Cantor's Diagonal

    I struggle with whether I believe that real numbers with infinite decimal expansions which human beings haven't identified exist or not and, in contemplating whether actual infinities exist or not, I wonder if the difference between arbitrarily large and infinite has been properly recognized, but is that irrelevant? Is it just assumed that they exist for the sake of the proof?

    Why can't cardinality of "infinite sets" be defined in terms of limits as infinity is "approached," so that, say, the cardinality of the set of "all" integers is twice the cardinality of the set of "all" even integers, since that's true of every finite number of even integers? In comparing the cardinality of the set of "all" integers to the cardinality of the set of "all" even integers, is it not at least arbitrary to map integers to even integers, one to one, rather than map each even integer to the identical member in the other set, leaving odd members in the set of "all" integers unmapped?

    Doesn't Cantor's proof just prove that there are more real numbers than each real number has decimal places, as is true for any set of all numbers between 0 and 1 with a finite number of decimal places? Is the debate over whether any purported list of real numbers is "square" or not moot, since, if it isn't, the greater cardinality of the set of "all" real numbers over the set of "all" integers must be true, anyway?

    What practical applications does proving that the cardinality of the set of "all" real numbers is greater than the cardinality of the set of "all" integers have?

    I just posted a message a few days ago in a thread that was active a few years ago, but I can't find it. Was it deleted? Here is the link:

    http://mathhelpforum.com/peer-math-r...t-wrong-3.html
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  2. #2
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    Re: Questions About Cantor's Diagonal

    Quote Originally Posted by CaptiveSlave View Post
    I struggle with whether I believe that real numbers with infinite decimal expansions which human beings haven't identified exist or not
    What do you mean by "identified" here?

    and, in contemplating whether actual infinities exist or not
    What do you mean by "actual infinities"?

    , I wonder if the difference between arbitrarily large and infinite has been properly recognized, but is that irrelevant? Is it just assumed that they exist for the sake of the proof?
    There is a very sharp and well defined difference between "arbitrarily large" and "infinite".

    Why can't cardinality of "infinite sets" be defined in terms of limits as infinity is "approached," so that, say, the cardinality of the set of "all" integers is twice the cardinality of the set of "all" even integers, since that's true of every finite number of even integers? In comparing the cardinality of the set of "all" integers to the cardinality of the set of "all" even integers, is it not at least arbitrary to map integers to even integers, one to one, rather than map each even integer to the identical member in the other set, leaving odd members in the set of "all" integers unmapped?
    Working with finite sets, we define cardinality by saying that two sets have the same cardinality if and only if there exist a "one to one" function mapping one set "onto" the other. Here, "one to one" means that no member of the range set has two or more members of the domain set mapped to it and "onto" means every member of the range set has at least one member of the domain set mapped to it. So "one to one" and "onto" means every member of the range set has exactly one member of the domain set mapped to it. That, in turn, means that the mapping is invertible and that it defines an "equivalence relation". In particular, a finite set has "n" members if and only if there exist a one to one mapping from the set onto {1, 2, 3, ..., n}.

    We extend that to infinite sets precisely by saying that two sets have the same cardinality if and only if there exist a "one to one function" from one "onto" the other. The set of all integers has the same cardinality as the set of all even integers because the function n-> 2n is both "one to one" and "onto". But the point is that there exist such a function. The fact that there also exist non-one to one functions is irrelevant.

    Doesn't Cantor's proof just prove that there are more real numbers than each real number has decimal places, as is true for any set of all numbers between 0 and 1 with a finite number of decimal places?
    ??? That's NOT true of "any set of all numbers between 0 and 1 with a finite number of decimal places"

    Is the debate over whether any purported list of real numbers is "square" or not
    What "debate" are you talking about?

    moot, since, if it isn't, the greater cardinality of the set of "all" real numbers over the set of "all" integers must be true, anyway?
    Why do you think that "must" be true?

    What practical applications does proving that the cardinality of the set of "all" real numbers is greater than the cardinality of the set of "all" integers have?
    Well, for one it is important in showing that we can assign a real number to every point on a line.

    I just posted a message a few days ago in a thread that was active a few years ago, but I can't find it. Was it deleted? Here is the link:

    http://mathhelpforum.com/peer-math-r...t-wrong-3.html
    Last edited by HallsofIvy; September 3rd 2014 at 03:53 PM.
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    Re: Questions About Cantor's Diagonal

    Quote Originally Posted by HallsofIvy View Post
    What do you mean by "identified" here?
    By "identified," I mean described in some other way than an arbitrary, infinite decimal expansion. I'm not very confident that I'm going about imagining what a real number is efficiently. I see decimal places as far as the eye can see, but that stops at a point that's arbitrarily large, not infinite. I'm more comfortable with a number such as pi than I am with one that has not been so identified. No matter how many decimal places out we go with a number such as pi, we can tell which number belongs there. I'm fairly clear about the idea that all real numbers form a continuum, but I struggle to decide if any of them, individually, exist. I'm skeptical that even a number as simple as 1/3 can be expressed in decimal form. I was taught that the square root of a negative number is an "imaginary number." That's what a number with an infinite decimal expansion, at least one that hasn't been otherwise defined, seems to me.

    What do you mean by "actual infinities"?
    I don't know. I was thinking of the kind that Cantor's proof assumes. That I don't believe such things exist may constitute my primary difficulty with the proof, but I'm willing to listen to what definitions have been given.

    ??? That's NOT true of "any set of all numbers between 0 and 1 with a finite number of decimal places"
    Maybe I wasn't clear. I was referring to how, with all finite number of decimal places, the list of members with that many decimal places is longer than each member has decimal places. Let Nx be the set of all numbers between 0 and 1 which have a finite number of decimal places, x being the number of decimal places. If x = 1, the cardinality of Nx is 9:

    .1
    .2
    .3
    ...
    .9

    If x = 2, the cardinality of Nx is 99:

    .01
    .02
    ...
    .99

    As x increases, so does the ratio between the cardinality of Nx and x: 999/3, 9999/4, etc. Let R be the set of all real numbers between 0 and 1. If all members of R can be expressed with infinite decimal expansions and if a diagonal can be constructed from a list of all of them, then there can be no more members than each member has digits in its decimal expansion. I believe that Cantor's proof assumes that it starts out with a complete list of real numbers, but I don't see why the number that is constructed from the altered diagonal not being on the list can't invalidate the assumption that there can be no more members than each member has digits in its decimal expansion rather than the assumption that the initial list was a list of all real numbers. And if this is the proper conclusion, doesn't it, also, prove that the cardinality of the set of all real numbers is greater than the cardinality of the set of all natural numbers?
    Last edited by CaptiveSlave; September 3rd 2014 at 09:04 PM.
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