
Re: Cantor's diagonal argument.
Posted:
Oct 3, 2001 10:03 AM


"Jan Kristian Haugland" <jkhaug00@stud.hia.no> wrote in message news://Pine.GSO.4.05.10110031531300.2789100000@svale.hia.no...
> > On 3 Oct 2001, 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. > (...) > > 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. > > No, it is clear that it is _not_ a natural > number, because it is not finite.
Agreed, but I always think it's unwise to mention finiteness here since some people then drag the discussion into a pseudophilosophical quagmire. It is best to note, that if a natural number has decimal digits a_i, then a_i = 0 for all i > some N which we may take to equal a if we like. Proof: induction. Then it's apparent that the "number" with a_i = 1 for all i is no natural number.
Robin Chapman
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