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November 1st, 2011, 12:16 PM   #1
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Diagonalization argument

Hi,

When working with the diagonalization argument, we construct a function which is different
from all other functions in the enumeration by looking at the diagonal. Why dont
we walk straight down or right?

any reasons?
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November 1st, 2011, 12:24 PM   #2
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Re: Diagonalization argument

If you go straight down, you are changing the first digit over and over again. If you go straight across, you are constructing something different from one number.
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November 1st, 2011, 12:35 PM   #3
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Re: Diagonalization argument

Quote:
Originally Posted by mathman
If you go straight down, you are changing the first digit over and over again.
(and this doesn't guarantee that you are finding a "new" number)

Quote:
If you go straight across, you are constructing something different from one number.
(and to show that this new number is not in a given enumeration of the reals in [0, 1), you need to show it's different from ALL the numbers)
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November 1st, 2011, 12:38 PM   #4
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Re: Diagonalization argument

Thank you for the response,

But how can i visualize whether it works i take two down and one right and then continue?

(or)

whether it works if i take one down and two right and then continue ?
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November 1st, 2011, 12:40 PM   #5
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Re: Diagonalization argument

Take a look at, for instance, this resource: http://en.wikipedia.org/wiki/Cantor%27s ... l_argument

The line "in short, by definition, s_0 is not in the countable sequence S" is key here.
Can you guarantee that your method will yield this ^^ result? If so, then you're fine!
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November 1st, 2011, 01:07 PM   #6
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Re: Diagonalization argument

Thank you,

I have one more question,

List(A), (Where A is enumerable)
I just want to know whether the set above is enumerable?
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November 1st, 2011, 01:18 PM   #7
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Re: Diagonalization argument

I don't know what List(A) is.
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November 1st, 2011, 01:29 PM   #8
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Re: Diagonalization argument

May be i can say List(enumerable set).
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November 1st, 2011, 04:22 PM   #9
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Re: Diagonalization argument

Quote:
Originally Posted by RandomNumber
Thank you for the response,

But how can i visualize whether it works i take two down and one right and then continue?

(or)

whether it works if i take one down and two right and then continue ?
If the process makes at least one digit of the final number different from each number on the list, it works. If you're using the same up/down-left/right placement I am, that means two right, one down works but one right, two down does not.
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