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Re: Cantor's diagonal argument.
Posted:
Oct 3, 2001 2:58 PM
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"Giles Redgrave" <g.d.redgrave@elostirion.freeserve.co.uk> wrote in message
news://c645a590.0110030525.199817f7@posting.google.com...
> 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)
What kind of integer has an infinite series of ones? Integers must have a
finite number of digits.
> 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.
You've only included decimal numbers that terminate (that is, they only
contain a finite number of digits after the decimal point). Many real
numbers (for example 1/3 = .333333....) require an infinite number of digits
to represent them.
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