Date: Nov 17, 2012 10:34 PM
Author: Virgil
Subject: Re: Matheology � 152
In article

<126c3310-d023-4f33-9b13-6cac84751832@o8g2000yqh.googlegroups.com>,

WM <mueckenh@rz.fh-augsburg.de> wrote:

> 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?

There is no number which is larger than any number.

>

> >

> > > 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.

Every numeral being a number does not limit the number of numbers,

it only, at most, limits the number of numerals.

So that, as usual, LV has things backwards.

--