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. You only prove that we can go on an on and on. And that's potential infinity. But it is never proved to be infinite because there is no step inrto the infinite.
> we have cases where they are not equal and cases > where they are equal.
Potentially infinite sequences are never actually infinite. That means: d stretches from d_1 to the d_n of your choice. And exactly the same is in infinitely many lines. Of course you can choose whatever n you like (because that is the meaning of potentially infinite: you can choose whatever n you like).
> We also note that no > concept of completed is needed, so equality can > be demonstrated by induction.
You cannot prove equality because you would need and end of file signal. You can only prove equality up to every desired step. But it can also be stated by induction that d cannot be more than all its FISs which also are in the list. And you agreed to that.
> > 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!