Date: Nov 17, 2012 1:13 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 152
On 17 Nov., 18:57, Uirgil <uir...@uirgil.ur> wrote:

> > > > Consider the following sequence of decimal numbers, consisting of

> > > > digits 0 and 1

>

> > > > 01.

> > > > 0.1

> > > > 010.1

> > > > 01.01

> > > > 0101.01

> > > > 010.101

> > > > 01010.101

> > > > 0101.0101

> > > > ...

>

> > > > which, when indexed by natural numbers, yilooks like this:

>

> > > > 0_2 1_1 .

> > > > 0_2 . 1_1

> > > > 0_4 1_3 0_2 . 1_1

> > > > 0_4 1_3 . 0_2 1_1

> > > > 0_6 1_5 0_4 1_3 . 0_2 1_1

> > > > 0_6 1_5 0_4 . 1_3 0_2 1_1

> > > > 0_8 1_7 0_6 1_5 0_4 . 1_3 0_2 1_1

> > > > 0_8 1_7 0_6 1_5 . 0_4 1_3 0_2 1_1

> > > > ...

> While every real mathematician knows

This sequence grows without limit.

>

> > This can be proved by taking any number n and showing

> > that there is a number k such that all for terms a(j) of the sequence

> > with k > j we have a(j) > n. Proof: For given n take k = n + 10.

>

> ow does that work for the sequence a(j) = 0 for all j?

Is 0 larger than any number n?

>

> > Every set theorist knows that the sequence of sets of indices left of

> > the decimal point has the limit empty set. This is an requirement of

> > set theory.

>

> Then let us see which axiom, or set of axioms, of some set theory which

> actually requires such nonsense. say among the axioms for ZFC, for

> example.

Try to learn it. Look what William Hughes just explains here.

>

>

>

> > And finally everybody knows that decimal numbers, by definition,

> > cannot consist of digits that have no indexs.

>

> Numbers (decimal or otherwise) can exist without any digits of any sort,

> but decimal numerals can not.

But the numbers in above list exist with their digits.

>

> Since a numeral is merely a name for a number,

the set of all numbers is countable.

Regards, WM