Date: Mar 26, 2013 4:44 PM
Subject: Re: Matheology § 224
On 26 Mrz., 20:41, Virgil <vir...@ligriv.com> wrote:
> In article
> WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 26 Mrz., 00:08, Virgil <vir...@ligriv.com> wrote:
> > > In article
> > > <efb405f6-8b50-4aff-a689-501d4a3da...@l9g2000yqp.googlegroups.com>,
> > > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > On 25 Mrz., 21:43, Virgil <vir...@ligriv.com> wrote:
> > > > > > That is the question too. Why has never anybody written the complete
> > > > > > decimal- or binary expansion of a periodic rational?
> > > > > Everyone who tried died of boredom before finishing.
> > > > In brief: It is impossible.
> > > It is also unnecessary, which is how the boredom arose.
> > If uncountably many real numbers should be distinguished ort be
> > applied for any mathematical reason, then all infinitely many digits
> > of each of them have to be written
> There are sufficient proofs that there must be more reals than names for
and that there must be Gods
> so that while they must exist, most of them are forever unnameable.
> Such existence is certainly no less believable than that in gods,
> angels, demons, life after death, etc.
No! You got it!! It is precisely as believable.
Here is your most sufficient proof: It is the Cantor diagonal. Have
you ever observed a diagonal that defines a single number?
Every digit and every initial segment of any given diagonal belong to
an infinite set of rational numbers. Even if you start with a list
containing all rational numbers, then the diagonal differs from every
rational number by some digit, but on the other hand, up to every
digit the diagonal does not differ from an infinite set of rationals
the cardinality of which is not less than the cardinality of all
numbers in the list.
So, at no digit of the diagonal, the set of different rationals is
larger than the set of not differing rationals.