Date: Jan 31, 2013 5:28 PM
Subject: Re: Matheology � 203
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.