Re: abundance of irrationals!)

In article <fb701d3c.0504191055.3d0db276@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> "Randy Poe" <poespam-trap@xxxxxxxxx> wrote in message
> news:<1113846213.793230.306610@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>...
> > Randy Poe wrote:
> > > W. Mueckenheim wrote:
> > > > Please repeat your proof
> > > > for the infinite Sum 1/2^k = 1, and you will see where you fail.
> > >
> > > The definition of this infinite sum is a sequence, and
> > > you said "sequences cause no problems".
> >
> > Sigh. The definition of this infinite sum is the
> > LIMIT of a sequence, as I stated correctly here:
> >
> > > By definition, S is the limit of the sequence S_1, S_2,
> > > ... where S_n = sum(k=1,n) 1/2^k.
>
> But the limit of the sequence is not a member of the sequence (in
> general). Hence Cantor's diagonal is not the limit.

Non-sequitur.

The only issue is whether an infinite sequence of decimal digits
following a decimal point represents a real number. Mathematicians say
it does.


If that is the case, then for every list of such infinite sequences
representing a list of reals, there is a sequence of such digits
representing a real which can be shown not to be in the list.

One method of showing this, due to Cantor, is to show how to construct
such a sequence which does not represent the same real number as any of
the listed sequences.

None of WM's double-talk so far has managed to discredit the Cantor
method in the opinion of anybody except WM.
.

Relevant Pages

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  • Re: Cantor Confusion
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  • Re: Cantor Confusion
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  • Re: abundance of irrationals!)
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