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Re: Cantor's first proof,
Posted:
Nov 15, 2012 5:13 PM
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Uirgil <uirgil@uirgil.ur> writes:
> In article
> <d0d377c2-ee0d-4fee-9317-f705ba75ed9d@p11g2000vbi.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>
>> On 15 Nov., 16:44, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>> > WM <mueck...@rz.fh-augsburg.de> writes:
>> > > On 15 Nov., 12:41, Ben Bacarisse <ben.use...@bsb.me.uk> wrote:
>> >
>> > >> Ah, simple. N is an "impossible set" (not a term he defines but why get
>> > >> bogged down in detail like that). The argument is simple: if you try to
>> > >> enumerate N you can construct a diagonal that is not in N; indeed it's
>> > >> not even a natural number.
>> >
>> > > A number is identified by its digits. Which digit shows you that the
>> > > diagonal is not a natural number?
>> >
>> > Not all sequences of digits correspond to natural numbers, in particular
>> > infinite ones such as the diagonal your construction creates.
>>
>> What digits do you need to know that that the number is unnatural? Can
>> you know these digits?
>>
>> Regards, WM
>
> One can know that any non-zero digits following a decimal point (or
> other radix point) in a number make it not a natural number.
There's a limit to how much time should be sent on stuff like this, but
that part I was referring to was where WM applies a diagonalisation
argument to a list of natural numbers, modifying digits from right
adding zeros as needed. The result is therefore not a natural number.
--
Ben.
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