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Re: Matheology § 152
Posted:
Nov 17, 2012 1:13 PM
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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
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