```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 impredicativityof his definition of number to distinguish fromwhat it means to be a finite initial segment ofa list.
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