On 2 Feb., 09:57, fom <fomJ...@nyms.net> wrote: > On 2/1/2013 9:48 AM, WM wrote: > > > > > > > On 1 Feb., 16:35, William Hughes <wpihug...@gmail.com> wrote: > >> Let P(n) be > >> 0.111... is not the nth line > >> of > > >> 0.1000... > >> 0.11000... > >> 0.111000... > >> ... > > >> Clearly for every natural number n > >> P(n) is true. > > >> This means there is no natural > >> number m for which P(m) is true. > > >> It is not simply that we cannot find m, > >> we know that m does not exist. > > > More. We know that P(n) = 0.111... = 1/0 does not exist as an > > actually infinite sequence of 1's. > > Hmm.... > > As I watch you make these arguments, it occurs to me... > > What proof do you have that some sequence is not infinitely > long?
Even with no regard to Tristram Shandy who disproves actual infinite, we can say: The sequence for 1/9 = 0.111... cannot have indices that differ from all indices of its finite approximations. So you cannot distinguish 0.111... by looking at digits from its finite approximations. And you cannot use it in any discourse because every message is finite and needs an endoffile signal to be meaningful.
Concluding: The property that a sequence of digits does not end cannot be obtained from its digits. (Remember, 0.111... is not a sequence of digits but is only a rule to construct a sequence of digits. The rule yields the sequence, but the sequence does not yield the rule.)
That means in mathematics, understood as discourse, there is no decimal representation of 1/9. What God may use for his mathematics is not my problem.