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Question about diagonal method

  1. Aug 10, 2005 #1
    I am slightly confused about the diagonal method. Can anyone say if I am mistaken.
    i is the imaginary unit.
    1|1+i
    2|1+i+i
    3|1+i+i+i
    4|1+i+i+i+i
    5|1+i+i+i+i+i
    6|1+i+i+i+i+i+i
    7|1+i+i+i+i+i+i+i
    8|1+i+i+i+i+i+i+i+i
    I will now use the diagonal method and make the number i+1+1+1+1+1+1 ...
    Therefore the list on the right side isn't complete and is bigger than the naturals.

    The problem is that the list on the right side is populated by numbers a+bi
    where b and a are natural and its size is therefore N^2.

    However N^2 can be put into one to one relation with N and we have a contradiction.
     
    Last edited: Aug 10, 2005
  2. jcsd
  3. Aug 10, 2005 #2

    matt grime

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    and the diagonal method creates a number of the form a+ib does it? given that you're adding 1 an infinite number of times to itself that doesn't worry you at all?

    Anyone can abuse the diagonal method to create a number not in some list like this, but so what? Of what is that list a supposed complete enumeration? Why must the diagonal element so created be an element of that list?


    Your argument is full of holes and is nonsense. Sorry.
     
  4. Aug 10, 2005 #3
    I never said 1 is added an infinite amount of times. The stadard diagonal method is only used until the end of the list. You simply create a number that isn't in the list, if the number is added later on than you have to make a new number. What was said was simply that the number would be added for each element of the list
     
  5. Aug 11, 2005 #4

    matt grime

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    What? I suggest you read your own post properly.

    The diagonal method works thus:

    take a set whose cardinality you wish to deal with. what is the set in your example? you don't state what it is. as far as we can tell the elements in the set are the numbers

    the n'th element in the list is 1+(n-1)i

    now, you're claiming that these are all of the set you care about. fine. it's cardiality is clearly aleph-0.

    now we apply the diagonal method and you yourself state it produces the term

    i+1+1+1+...

    this *does* imply you are adding up 1 an infinite number of times. it doesn't terminate since the list doesn't terminate.

    there is no reason why the diagonal method misapplied needs to produce an element that *must* be in the set you are enumerating.

    so, to say again, what is the set you want to enumerate? why is the element so produced by the diagonal method necessarily an element of this set? you've not stated either of these things.

    oh, and there is no end of the list if the list is enumerated by the natural numbers, which is another hole in your argument.
     
  6. Aug 11, 2005 #5
    A few important points about applying the diagonal method:

    1) The enumeration must be complete. Enumerating only part of a set and then using the diagonal method to find an element of the set that was not enumerated doesn't prove anything interesting.

    2) The enumeration must be arbitrary. If you provide a specific enumeration and then apply the diagonal method all you prove is that your specific enumeration is incomplete, when you need to prove that every enumeration is incomplete.

    3) The number you generate via the method must be an element of the set being enumerated. If the number you generate is not in the set, it clearly does not prove that the enumeration is incomplete.
     
  7. Aug 12, 2005 #6

    HallsofIvy

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    The list, or any list, can't be "bigger than the naturals" since a list, by definition, is a one-to-one correspondence with the naturals. Cantor's diagonal methods starts by assuming that the set of all real can be written as a list and then gets a contradiction by showing that there must be real number not on the list. You haven't said what set of numbers that list is supposed to constitute. It certainly is not the set of Gaussian integers since you only get 1+ ni: the real part is always 1. And the number you make using the diagonal method is certainly is not on your list- it's not even a gaussian integer: Since the list is unbounded, that "n" is not bounded and the "number" you get is not finite.
     
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