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.