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.
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