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Math information representations

  1. Dec 4, 2003 #1
    Any two noticeable R numbers a and b can be an open interval (a,b) of infinitely many R numbers, that cannot be separated form each other by any representation, and each R number can be represented only by aleph0 different representations.

    Is it right ?

    If what i wrote holds, than please tell me how can Cantor use some aleph0 representation to prove that there are more numbers then this aleph0 representation?

    What i clime is: Cantor cannot start by using the assumption saying that he produces its new diagonal number (which is not in the list) from a complete list of R numbers, because a complete list of R numbers has no representation of any kind, therefore no information of any kind, and therefore there is no basis for any conclusion like: R numbers are not countable | R numbers are countable .

    An example:

    0 . ? ? ? ? ? ...
    0 . ? ? ? ? ? ...
    0 . ? ? ? ? ? ...
    0 . ? ? ? ? ? ...
    0 . ? ? ? ? ? ...

    We cannot use the assetion that we have a complete list of R numbers, because
    in this case we cannot find any number in the diagonal, therefore no list, no information, no result (or proof if you like).



    Please show by Math language, how can we come to some result by using no information as a meaningful input?
     
    Last edited: Dec 4, 2003
  2. jcsd
  3. Dec 4, 2003 #2
    Re: Information

    It can be my english, but that doesn't make sense.....
     
  4. Dec 4, 2003 #3
    Thank you Guybrush Threepwood,

    I have changed 'distinguished' to 'noticeable' in my first post.
     
  5. Dec 4, 2003 #4
    well actually my comnfusion is:
    is (a != b) as in "2 distinguished numbers from R"
    or (a = b) as in "cannot be separated form each other by any representation"
     
  6. Dec 4, 2003 #5
    By Modern math a complete list of all R numbers is uncountable, according to Cantor's second diagonal proof: http://home.ican.net/~arandall/abelard/math12/Cantor.html

    I clime that by using the diagonal method we cannot come to this conclusion.
     
    Last edited: Dec 4, 2003
  7. Dec 4, 2003 #6

    Hurkyl

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    Gold Member

    What do you mean by "noticable"?

    Anyways, the real numbers [itex]a[/itex] and [itex]b[/itex] cannt be an open interval; they're real numbers, not intervals. What is correct to state is:

    Given any two real numbers [itex]a[/itex] and [itex]b[/itex], we can form an open interval, which we call [itex](a, b)[/itex], defined by

    [tex]
    (a, b) := \{x | a < x < b\}
    [/tex]

    that is, [itex](a, b)[/itex] is defined to be the set of all real numbers greater than [itex]a[/itex] and less than [itex]b[/itex].


    [itex](a, b)[/itex] contains infinitely many real numbers iff [itex]a<b[/itex]; if [itex]b<=a[/itex], then [itex](a, b)[/itex] is the empty set.


    What do you mean by "representation" and what do you mean by representations seperating real numbers?

    For [itex]a<b[/itex], I can always decompose the open interval [itex](a, b)[/itex] into, for instance, three disjoint pieces, though I'm not sure if this has any bearing on your statement because I don't know what you mean (iow I'm purely guessing at what you could mean):

    [tex]
    (a, b) = (a, \frac{a+b}{2}) \, \dot{\cup} \, \{\frac{a+b}{2}\} \, \dot{\cup} \, (\frac{a+b}{2}, b)
    [/tex]


    Again, what is a representation?


    For reference:

    General form of a proof by contradiction:

    P is the statement in which we are interested.

    e.g. P := "There is a 'list' of all real numbers"

    Assume P is true, and use this assumption to prove P is false.

    Therefore, we can logically conclude that P is false. (Because if P is true, we can derive a contradiction)

    In symbolic logic:

    [tex](P \Rightarrow \neg P) \Rightarrow \neg P[/tex]
     
    Last edited: Dec 4, 2003
  8. Dec 4, 2003 #7
    Hi Hurkyl,

    Thank you very much for your reply.


    By "noticable" i mean that a < b.

    'representation' for some R number can be by base^power method with finite or infinitely many digits like 0.46385... or by using equivalence classes of increasing sequences of rational numbers like 1/2 - 3/45 + 7/88 ... and so on.

    You wrote:
    What i clime is: Cantor cannot start by using the P statement saying that he produces its new diagonal number (which is not in the list) from a complete list of R numbers, because a complete list of R numbers has no representation of any kind, therefore no information of any kind, and therefore there is no basis for any conclusion like: R numbers are not countable | R numbers are countable .

    An example:

    ? . ? ? ? ? ? ...
    ? . ? ? ? ? ? ...
    ? . ? ? ? ? ? ...
    ? . ? ? ? ? ? ...
    ? . ? ? ? ? ? ...

    We cannot use the assumption that we have a complete list of R numbers, because in this case we cannot find any number in the diagonal, therefore no list, no information, no result (or proof if you like).


    Please show by Math language, how can we come to some result by using no information as a meaningful input?
     
    Last edited: Dec 4, 2003
  9. Dec 4, 2003 #8
    is there some other posibility?


    also i think you don't understand the meaning of a proof by contradiction. What Cantor did was the following:

    1) assume that the statement "real numbers are countable" is true. Thus there exists a list of all the real numbers.

    2) Show that the list is not complete (i.e there eis at least one real number that is not in the list)

    3) Since the list is not complete, it means it does not contain all the real numbers.

    see, just like Hurkyl said....
     
  10. Dec 4, 2003 #9
    Hi Guybrush Threepwood,

    Please read what i write very carefully, because i am talking about a very fine idea that it is hard to understand from first look.

    The main idea is this:

    For any mathematical conclusion you need some input (some information).

    When Cantor finds some irrational number, which is not in the list, all he can say that the list is not complete, and if he add this number to the list then there is another number which is not in the list, and so on.

    If the list of R numbers was really a complete list, then the diagonal number must be in the list.

    But then we have no list at all and we can't conclude that R is uncountable or R is countable, because we have no digits (the input information) for Cantor's function that define the new number, which is not the list.

    So the basic law here is: No (input) information, no proof.

    All we can get is a closed loop, based on base^power representation method, which gives us a list of 2^aleph0-1 numbers, where the missing one is some diagonal number, which is not in the list, and this is because we technically still did not find the way to represent 2^aleph0 numbers.
     
    Last edited: Dec 4, 2003
  11. Dec 4, 2003 #10
    and I ask again, is there another posibility for representing numbers?

    the Cantor's proof doesn't care in which base you choose to represent the real numbers. It can be base 2, base 10, or base 16. He does however use the proof by contradiction.
     
  12. Dec 4, 2003 #11
    Yes you can use instead of base^power representation method (and by writing this i mean any base value) the equivalence classes of increasing sequences of rational numbers.

    For example:

    1 <--> 1/2 3/45 7/88 ...
    2 <--> 3/4 2/67 9/44 ...
    3 <--> 1/9 8/17 3/37 ...
    ...

    Then if our rule is to add 1 to the numerator then in this case Cantor's function result is: 2/2 3/67 4/37 ... which is a new sequence not in the list.

    But it really doesn't matter, because still we can clime that we have some sequence as Cantor's function result, only because we deal with 2^aleph0-1 sequences, where the missing one is Cantor's function result, and this missing sequence depends on some abitrary order and the rule which we use to define Cantor's function result.

    When we have 2^aleph0 sequences, Cantor's function result is unknown because we have no input at all (no rational form, no base^power form, no what so ever).

    Because we are "runnig after our own tail" in a closed system, we cannot find a proof by contradiction.

    But again these is because we did not find yet a way to represent 2^aleph0 numbers.

    For more detailed information, please read this:

    http://www.geocities.com/complementarytheory/NewDiagonalView.pdf
     
    Last edited: Dec 4, 2003
  13. Dec 4, 2003 #12
    how can you say that??? You wrote the input yourself........
    I can't quite understand you.
    Are you saying that you can't write real numbers in Cantor's proof??
    Also the equivalence classes (whatever...) you used are equivalent with the usual representation of real numbers (waht you call base^power)

    how did you come up with that number? you're using base 2 or something? and 2aleph0 is still aleph0...
     
  14. Dec 4, 2003 #13
    Guybrush Threepwood,

    Cantor's diagonal proof says that aleph0 < 2^aleph0 = |R|.

    Please tell me if you know this.
     
  15. Dec 5, 2003 #14
  16. Dec 5, 2003 #15
    Last edited: Dec 5, 2003
  17. Dec 5, 2003 #16
    And my argument is:

    Cantor cannot prove that 2^aleph0 > aleph0, by using its
    diagonal method, because of a very simple reason, which is:

    When we have the complete list of 2^aleph0 numbers, then we dont have any known method that can represent its numbers, therefore without any input (which is some R number for Cantor's function, which its output is some R number not in the list) we HAVE NO WAY to conclude one of the two options, which are:

    1) All R numbers are uncountable.

    2) ALL R numbers are countable.

    We can have some R number as an output of Cantor's function (which is some R number not in the list) if and only if we have some R number as its input.

    When we have 2^aleph0 R numbers, then we dont know how to represent the input, therefore NO INFORMATION can only lead to NO CONCLUSION.
     
    Last edited: Dec 5, 2003
  18. Dec 5, 2003 #17
    I don't understand what you mean here

    Cantor said "let's assume that all real numbers are countable". That means we will be able to make a list of all the real numbers. The list has [itex]\aleph_0[/itex] entries (because we assumed the real numbers are countable). I dont' know what list of [itex]2^{\aleph_0}[/itex] numbers you are reffering to.
     
  19. Dec 5, 2003 #18
    And how do we check if ALL R numbers are countable XOR uncountable?

    We try to create a bijection (1-1 and onto) among all N numbers and all R numbers, for example:

    1 <--> 0.14321...
    2 <--> 0.37130...
    3 <--> 0.24031...
    4 <--> 0.28850...
    5 <--> 0.99210...
    ...

    Now let us use Cantor's diagonal method on this list.


    Cantor's method:

    Let us use some function that get infinitely many digits as input,
    and define some number as an output, according to some rule.

    Let us say that the rule is:

    A) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.

    B) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.

    (we can define any rule that we like)

    Now, the input for Cantor's function is:

    The first digit of the first number, which in the list,
    The second digit of the second number, which in the list,
    and so on.

    1 <--> 0.14321...
    2 <--> 0.37130...
    3 <--> 0.24031...
    4 <--> 0.28850...
    5 <--> 0.99210...
    ...

    In this example the input R number is 0.17050...
    and the output R number is 0.00101... (which depends on the arbitrary R list, and the rule that have been used by Cantor's function).

    The output number is not in the list because it is different from any R number (which is in the list) by at least one digit.

    If we add this new number to the list, then we can find a new input
    of infinitely many digits for Cantor's function, and create another R number, which is not in the list, for example:

    1 <--> 0.143211...
    2 <--> 0.371301...
    3 <--> 0.240311...
    4 <--> 0.288501...
    5 <--> 0.992101...
    6 <--> 0.001011...
    ...

    In this example the input R number is 0.170501...
    and the output R number (which is not in the list) is 0.001010...

    So, no matter how infinitely many Cantor's functions outputs we add
    to the list, we still can find at list one R number which is not in the list.

    Therefore we can conclude that there is no bijection between ALL N numbers and ALL R numbers and we can conclude that |N|<|R| or aleph0 < 2^aleph0.

    The information's point of view on this proof:

    I clime that Cantor cannot conclude that aleph0 < 2^aleph0 because
    of the simple reason, which is:

    No matter what representation we use to define some input for Cantor's function, first we have to define if there are an invariant signatures of R numbers that stays in any representation's form.

    And by today's math language we can recognize two signatures, which are (in this case we shall use the decimal form, but again, it does not matter, we can use another base instead of 10, or the equivalence classes of increasing sequences of rational numbers*):

    a) periodic repetitions for rational numbers (i.e. 5.000..., 0.101010...)

    b) non-periodic repetitions for irrational numbers (i.e. .4589203...)

    Now we come to the heart of my argument:

    If we still can find some rational(periodic repetitions) or irrational(non-periodic repetitions)number which is not in the list (Q or R list), WE CAN CONCLUDE that our Q(all rational numbers) or R(all rational + all irrational numbers) list IS NOT COMPLETE.

    In the case of a complete Q list, no rational number can be found as an input for Cantor's function.

    In the case of a complete R list, no rational(periodic repetitions) or irrational(non-periodic repetitions) number can be found as an input for Cantor's function.

    If R is not complete then we cannot conclude anything about the bijection (1-1 and onto) between N numbers and some NOT COMPLETE list of R numbers.







     
    Last edited: Dec 5, 2003
  20. Dec 5, 2003 #19
    so do you think that Cantor wrote a list with all the natural numbers and their real corespondents?????
    sorry, but you clearly don't understand the proof by contradiction
    the ideea of the proof is:
    1) a countable list of all real numbers exist (doesn't matter how we found it....)
    2) based on 1) show that there is a real number who's not in the list
    3) thus we conclude that the countable list of real numbers does not exist an furthermore that there are more real numbers than the list can handle.

    please explain what do you mean by invariant signature of a number

    what you call equivalence class of increasing sequences of rational numbers is still a rational number. You can multiply all the rational numbers you want the result is still a rational number. So this representation does not cover the irrational numbers
     
  21. Dec 5, 2003 #20
    No, you missed the point.

    Let us say that we want to mark any R number by any N number.

    For doing this we have to represent each N number and each R number
    by some common representation technique (let us say that we are using the base^power representation technique, where the base is 10).

    We have no problem to represent all N numbers (aleph0 numbers), but we dont have any representation technique to represent R numbers (2^aleph0 numbers).

    Through this point of view we find this:

    aleph0 = (2^aleph0)-1, which means that we can mark (2^aleph0)-1 R numbers by (2^aleph0)-1 N numbers.

    And where is the missing R number?

    The missing R number is some result of Cantor's function, which depends on our ability to give some input to this function.

    When we deal with 2^aleph0 elements then we find that all of our representations techniques does not have the power to produce the information for some meaningful conclusion.

    We cannot prove anything without information.

    All we can say that 2^aleph0 objects still cannot be represented by any representations techniques that we developed until today.

    But this is a technical problem and not some deep mathematical true.

    Any information system that constructed by rules is limited by its rules, therefore any extrapolation that does not change the rules is nothing but an information system that closed on itself, that recycling old information as if it is a new information.
    ------------------------------------------------------------------------------

    Let me give you a concrete example by using Cantor's diagonal method
    on decimal representation of the rational numbers.

    The name 'rational numbers' is because each rational number is the ratio between two integers, for example:

    1/2 6/1 567/3452 ... and so on.

    "By using decimal representation method we can represent a list of rational numbers (repetitions over scales, for example: 0.123123123...) where the missing rational number is based on the diagonal rational number used as an input for Cantor's function (the function that defines the rational number, which is not in the list), where the result (the rational number, which is not in the list) depends on some arbitrary order of the list and some rule, which is used by Cantor's function.

    For example:

    0 . 1 7 1 1 3 1 7 1 1 3 1 7 ...
    1 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
    0 . 4 2 1 3 4 2 1 3 4 2 1 3 ...
    0 . 1 0 1 0 1 0 1 0 1 0 1 0 ...
    0 . 3 3 3 3 3 3 3 3 3 3 3 3 ...
    2 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
    0 . 3 5 4 9 5 5 1 3 5 4 9 5 ...
    3 . 0 0 0 0 0 0 0 0 0 0 0 0 ...
    0 . 6 4 1 6 4 1 6 4 1 6 4 1 ...
    0 . 3 0 2 0 3 0 2 0 3 0 2 0 ...
    0 . 6 1 3 6 1 3 6 1 3 6 1 3 ...
    0 . 2 7 1 0 2 7 1 0 2 7 1 0 ...
    ...

    In this case Cantor's function result is 0.0101010101010101.... which is not in the list.


    The rule is:

    a) Every 0 in the original diagonal number is turned to 1 in Cantor's new number.

    b) Every non-zero in the original diagonal number is turned to 0 in Cantor's new number.

    We can add 0.0101010101010101.... to the list, and then rearrenge the list in such a way that give us another rational number as cantor's function result, which means that our list is still not complete.


    my conjecture is:

    When we have the Aleph0 complete list of rational numbers, represented by decimal representation method, then and only then we can find only some irrational diagonal number (no repetitions over scales, for example: 0.123005497762...) as an input for Cantor's function."

    I call this conjecture "Aleph0-1 conjecture" because i clime that if even 1 rational number is not in the list, we shall find it as the result of Cantor's function, but when the list is the complete list of all rational numbers (an aleph0 list), we can find only some irrational number as Contor function result.

    ------------------------------------------------------------------------------

    Now take what we have found on Q numbers (the difference between the inputs of aleph0(irrational input) and aleph0-1(rational or irrational inputs)) and translate it to R numbers.

    when we have (2^aleph0)-1 numbers the input for Cantor's function is some irrational number.

    when we have 2^aleph0 numbers the input for Cantor's function cannot be represented, therefore we have no input For Cantor's function,
    and without input (some information) we cannot conclude any meaningful conclusion on the countability of R numbers.
     
    Last edited: Dec 5, 2003
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