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Re: Cantor's diagonal argument.
Posted:
Oct 3, 2001 9:39 AM
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Giles Redgrave wrote...
>I'm having a problem understanding Cantor's diagonal argument (CDA).
>Specifically it's use in proving the uncountability of the reals from
>0 to 1.
>
>I don't understand why you can't apply CDA to the natural numbers
>themselves. If we list the natural numbers padding to the left with
>zeros like so:
>
>...000
>...001
>...002
>...003
>.
>.
>.
>
>and apply CDA by adding one to the nth digit (from the right) of n and
>constructing our new number from these digits.
>
>We then have a number (consisting of an infinite series of ones) that
>is not in the original list because it is different from each n in
>it's nth digit.
>
>But it is clear that this number *is* in the list because it is a
>natural number.
>
>To put it another way, what is wrong with mapping of the natural
>numbers to the reals by reversing the digits of the natural numbers
>and placing them after a decimal point like so:
>
>0 -> 0
>1 -> 0.1
>2 -> 0.2
>...
>10 -> 0.01
>11 -> 0.11
>...
>123456 -> 0.654321
>...
>
>This is driving me mad. Can someone point out what's wrong with this
>argument because I can't think of a real that can not be generated by
>the above mapping and I can't see why applying CDA to the natural
>numbers does not lead to a contradiction.
>
>My degree is in physics not maths, an explanation in terms I can
>understand would be appreciated.
Which natural number generates the real number 1/3 ?
--
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
http://www.clivetooth.dk/
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