Date: Feb 16, 2013 5:07 PM
Author: fom
Subject: Re: Matheology § 222 Back to the root<br> s

On 2/16/2013 4:00 PM, Virgil wrote:
> In article
> <3da6693e-4a13-446c-b5cb-4802c039be5d@l13g2000yqe.googlegroups.com>,
> WM <mueckenh@rz.fh-augsburg.de> wrote:
>

>> On 15 Feb., 23:58, William Hughes <wpihug...@gmail.com> wrote:
>>> On Feb 15, 10:30 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>>>

>>>> On 15 Feb., 00:44, William Hughes <wpihug...@gmail.com> wrote:
>>>
>>>>>>> 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

>>>
>>>>> So we note that it makes perfect sense to ask
>>>>> if potentially infinite sequences x and y are equal,

>>>
>>>> and to answer that they can be equal if they are actually infinite.
>>>> But this answer does not make sense.
>>>> You cannot prove equality without having an end, a q.e.d..

>>>
>>> A very strange statement. Anyway there is no reason to
>>> claim equality. Let us define the term coFIS
>>>
>>> Two potentially infinite sequences x and y are said to be
>>> coFIS iff for every natural number n, the
>>> nth FIS of x is equal to the nth FIS of y.
>>>

>>
>> So for every natural number the list
>> 1
>> 12
>> 123
>> ...
>> is coFIS with its diagonal.

>
>
> WRONG! for your list, L
> FIS1(L) = 1,
> FIS2(L) = 1, 12,
> FIS3(L) = 1, 12, 123
> and so on
> whereas for the diagonal, D = 123...
> FIS1(D) = 1
> FIS2(D) = 12
> FIS3(D) = 123
> and so on
>
> So the list of FISs of an endless list like D can never be the same as
> the the list itself.
>


Very nice. This is how to use the impredicativity
of his definition of number to distinguish from
what it means to be a finite initial segment of
a list.