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Re: Cantor's first proof,
Posted:
Nov 15, 2012 5:13 PM


Uirgil <uirgil@uirgil.ur> writes:
> In article > <d0d377c2ee0d4fee9317f705ba75ed9d@p11g2000vbi.googlegroups.com>, > WM <mueckenh@rz.fhaugsburg.de> wrote: > >> On 15 Nov., 16:44, Ben Bacarisse <ben.use...@bsb.me.uk> wrote: >> > WM <mueck...@rz.fhaugsburg.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 nonzero 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.



