Re: abundance of irrationals

mueckenh_at_rz.fh-augsburg.de
Date: 01/12/05 

Date: 12 Jan 2005 04:09:55 -0800

Virgil wrote:
> In article <1105458371.912296.225750@f14g2000cwb.googlegroups.com>,
> mueckenh@rz.fh-augsburg.de wrote:
> > If you compare
> > Cantor's work you will find that this was his position too.
>
> Decimal expansions are also "just names", not numbers, too.

No, the natural numbers (in digits or bits for instance) make it to be
more than a name. Numbers are required for counting, counting requires
numbers. That is their primary duty. Rational numbers count in the unit
of their denominators. Irrational numbers don't count at all.

There is
> always a distinction between the number and any of its
representations.
> That some representations are more useful thatn others for certain
> purposes does not mean that they are anything more than
representations.

But you must be able to distinguish a number from any other one. Call
pi by c or by p, as Euler did, before he adapted Jone's pi. None of
these names gives you any information other than that there is a goal,
which can be aimed however cannot be hit.
>
> Depending on one's model, a real number is either a certain type of
> partitioning of the rationals (Dedekind) or a collection of infinite
> sequences of rationals (Cauchy sequences whose differences are null
> sequences). It is not, in any truly mathematical model, merely a
decimal
> expansion.
> >
> > Could you decide whether sqlmn < sqrlp, unless you have rational
> > approximations for those names? That would be the minimum
requirement
> > for a number. For sqrt(2) you know a lot of rational approximations
> > which enable you to decide the <, =, or > -question for a lot of
other
> > rational numbers. But you can't leave the rational domain.
> > Regards, WM
>
> Given either the partition form or the family of sequences form, as
> referred to above, of both numbers, such questions are trivial.

The sequential form like Sum 1/n! does not enable you to compare e with
a slightly deviating number represented in decimal form, for instance,
because you will not be able compare them for very large n. In the
universe not more than 2^10^100 numbers can be realized
simultaneuously. By far too less, to give e with "any" desired
precision.

Regards, WM

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