Date: Feb 14, 2013 2:26 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 13 Feb., 23:22, William Hughes <wpihug...@gmail.com> wrote:
> On Feb 13, 9:03 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> <snip>
>

> > You cannot discern that two potentially infinity sequences are equal.
> > When will you understand that such a result requires completeness?

>
> Nope
>
> Two potentially infinite sequences x and y are
> equal iff for every natural number n, the
> nth FIS of x is equal to the nth FIS of y


And just this criterion is satisfied for the system

1
12
123
...

For every n all FISs of d are identical with all FISs of line n.

> No concept of completeness is needed or used.

That means we have to go to n only.
>
> E.G,
>
> we can use induction to show
>
> x={1,1+2,1+2+3,...,1+2+...+n,...}
>
> is equal to
>
> y={(1)(2)/2,(2)(3)/2,(3)(4)/2,...,n(n+1)/2,...}


And the three points stand for every finite number, but not for all.
>
> Consider the list of potentially infinite sequence
> L1=
> 1000...
> 11000...
> 111000...
> ...
>
> L2=
> 111...
> 11000...
> 111000...
> ...
>
> The diagonals are both
> d=111...


And again you confuse every with all.
>
> It makes perfect sense to say that there
> is no line in L1 that is equal
> to d


Perfect sense?
Do you claim that the list
1
12
123
...
does not contain every FIS of d?
Do you claim that there are two or more FISs of d that require more
than one line for their accomodation?

We can use induction to show that!

Ponder about this question and then try to make perfect sense.

Regards, WM