```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 knowsThis 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
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