Date: Jan 31, 2013 10:34 AM
Subject: Re: Matheology § 203
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 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.
> 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.
In order prove the second statement false, you have to check complete
0/1 sequences. Compare: In order to prove that card(|N) is larger than
every natural number, you need the complete set. *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