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