```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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