Re: Cantor Confusion
- From: Virgil <virgil@xxxxxxxxxxx>
- Date: Wed, 01 Nov 2006 13:14:34 -0700
In article <1162382668.224018.252560@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:
MoeBlee schrieb:
mueckenh@xxxxxxxxxxxxxxxxx wrote:
All entries of the list have a finite number of letters. An infinite
sequence is larger than any finite sequence. The diagonal of a list
cannot have more letters than the lines.
According to your logic the list can have infinitely many lines. But
even if that was correct it would not facilitate an infinte diagonal.
The number of diagonal elements is the minimum of columns and lines.
0
1 2
3 4 5
6 7 8 9
...............
.................
...................
infinitely downward for an infinite list of finite lists.
The diagonal is 0 2 5 9 14 ... infinitely across.
All entries in the infinite list are finite lists.
better say finite sequences or numbers or entries
The infinite list is
longer than any finite list.
The entries surpass every finite entry. Nevertheless you call all of
them finite.
The diagonal of the list is infinite.
That is your assertion. But obviously the diagonal elements are
simultaneously elements of the entries.
Each position in the diagonal is determined by a position in one of the
listed elements, but if that position number is an unbounded function of
the list number, with each listed entry at least that long, then there
is no problem with having an infinite diagonal.
And that be formalized easily in set theory.
That may be, therefore it is no wonder that set theory yields
selfcontradictions.
That set theory yields results that WM does not like, and which
cotradict things that WM chooses to assume, but nothing which
contradicts any of its own axioms, which is the only relevant criterion
for being self-contradictory.
So that WM makes false claims.
The diagonal elements are simultaneously elements of the entries.
Therefore the diagonal elements cannot sum up to a number which is
larger than any natural number unless also the elements of list entries
sum up to a number which is larger than any natural.
WM deliberately blinds himself by assuming that if every string of
characters in an infinite list of strings is finite that there must be a
UNIFORM limit on their lengths.
But a trivial example proves him wrong:
For each n in the infinite completed set of naturals Net f(n) be a
string of length at least n, then there is no maximally long string in
the image of f, as for each m in N there is n in N with n > Length(f(m)).
Or put it so: Every segment of the diagonal is covered by an entry.
There is no segment which is not covered. If all entries are finite,
then the diagonal cannot be infinite (if infinite omega is larger than
any finite n).
Note that the "diagonal" in my example above MUST be greater than any
finite n, so that either there is a number greater than every finite n
and simultaneously less than omega or WM is wrong again.
.
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