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.
