"|-|ercules" <radgray123@yahoo.com> wrote in message news:87ov0tFu9lU1@mid.individual.net... > "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote >> "|-|ercules" <radgray123@yahoo.com> wrote in message >> news:87om34FahrU1@mid.individual.net... >>> "Peter Webb" <webbfamily@DIESPAMDIEoptusnet.com.au> wrote >>>> "|-|ercules" <radgray123@yahoo.com> wrote in message >>>> news:87ocucFrn3U1@mid.individual.net... >>>>> Consider the list of increasing lengths of finite prefixes of pi >>>>> >>>>> 3 >>>>> 31 >>>>> 314 >>>>> 3141 >>>>> .... >>>>> >>>>> Everyone agrees that: >>>>> this list contains every digit of pi (1) >>>>> >>>> >>>> Sloppy terminology, but I agree with what I think you are trying to >>>> say. >>>> >>>>> as pi is an infinite digit sequence, this means >>>>> >>>>> this list contains every digit of an infinite digit sequence (2) >>>>> >>>> >>>> Again sloppy, but basically true. >>>> >>>>> similarly, as computable digit sequences contain increasing lengths of >>>>> ALL possible finite prefixes >>>>> >>>> >>>> Not "similarly", but if you are claiming that all Reals which have >>>> finite decimal expansions can be listed, this is correct. >>> >>> You didn't follow the similarity. >>> >>> Given the increasing finite prefixes of pi >>> >>> 3 >>> 31 >>> 314 >>> .. >>> >>> This list contains every digit of the infinite expansion of pi. >>> >> >> But pi doesn't appear on the list. >> >> So? > > > that doesn't matter, because that's a convergent sequence. >
So what? If this has something to do with Cantor, you don't need to construct pi as some limit of a sequence, just whack pi in as the first in your list of Reals.
Yes, Reals form the limit of sequences of rationals, and you can draw up a list of terminating rationals as Cantor's list, and any Real can be derived as the limit of some subsequence of these Rationals, but that still doesn't change the fact that the specific Real does not appear on the list. Cantors proof only applies to a list of Reals, not a sequence of Rational approximations, which are themselves uncountable.
> This is what matters. >
No, that sequences in list of Reals can define other Reals (as there limit) is interesting, but nothing to do with Cantor's diagonal proof.
