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Re: Simple Refutation of Cantor's Proof
Posted:
Dec 24, 2012 6:29 AM
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In article
<68bb591d-e426-4a8a-a2f8-da6880aecf09@6g2000pbh.googlegroups.com>,
Graham Cooper <grahamcooper7@gmail.com> writes:
> +----->
> | 0. 542..
> | 0. 983..
> | 0. 143..
> | 0. 543..
> | ...
> v
>
>
> OK - THINK - don't back explain to me.
>
> You run down the Diagonal 5 8 3 ...
>
> IN YOUR MIND - you change each digit ONE AT A TIME
>
> 0.694...
>
> but this process NEVER STOPS
>
> and you NEVER CONSTRUCT A NEW DIGIT SEQUENCE!
>
> There are INFINITE PATHS occupying each row and collumn
> that make 5 8 3 and 6 9 4
>
> There is NOTHING SPECIAL ABOUT THOSE DIGITS OR THAT SEQUENCE!
You are missing the point. The whole idea of the diagonal argument is
that one starts with an infinite list which is presumably exhaustive,
i.e. contains all numbers. The diagonal argument shows that from any
such list it is possible to construct a number not on the list, thus
proving that the list can't be exhaustive.
Using decimals is not very good because it is not clear how you could
order the list to be exhaustive. Use fractions. First, all in which
numerator and denominator add up to 2, then all in which the sum is 3,
and so on. Sure, some numbers will be in the list more than once
(infinitely many times, in fact) but the point is that this constructs
an ordered list of rational numbers. After ordering this way, one can
write the decimal expansion then use the diagonal argument.
> Herc
> --
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