Virgil
Posts:
4,660
Registered:
1/6/11
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Re: Matheology � 203
Posted:
Jan 31, 2013 5:28 PM
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In article
<2e61c07e-7f63-40ed-a7f3-3621af6f11c2@l9g2000yqp.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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.
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