In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 31 Jan., 16:15, William Hughes <wpihug...@gmail.com> wrote: > > > > > > Would you say that a line that is not in the list is in the list? > > > > Nope. But you did. > > Yes, but for an actually infinite list. Cantor's actually infinite > list of all terminating decimals
Cantor never used any actually infinite list of all terminating decimals. What he did use was an endless list of endless binaries.
> has the property that > every line of the list differs from the antidiagonal > but since the antidiagonal (or at least that initial segment that > contains digits that can be compared with digits of the listed > numbers) is a terminating decimal too, it must be in the list. > Therefore it cannot differ from all entries.
Since none of the finite initial segments of that anti-diagonal are the complete anti-diagonal, the presence of finite initial segments of that anti-diagnal being in the list is irrelevant. > > > Both of the above statements are statements > > you have said are true. > > > > My claim is that the first is true and the second is false. > > *No* segment (1, 2, ..., 3) of the potentially infinite set |N is > larger than every natural number. Outside of every such segment there > are infinitely many naturals.
Only if there is an "actually" infinite set of naturals to draw from. --