Re: Cantor Confusion

In article <1163071764.333438.152230@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:


No. You are confusing the length of the columns with the set of line
indexes contained in each column. These are not the same.

Explain how in a square matrix (squarity proven by the existence of a
diagonal) the columns can be longer than the lines for *every* line
and every column.

Just lucky, I guess.

The set of line indexes contained in the first column is |N.
The length of the first column is the size of |N, aleph_0.

The set of column indexes is |N.

The bijection defined by the diagonal connects the line indexes
with the column indexes. Finding a bijection between |N and |N
is not hard.

Of course not, because you can do that for finite indexes only.

Can WM find any line or any column that has anything OTHER than a finite
index? He must then explain how it came by that impossible to achieve
index when every index except the first must be an immediate successor
of another.



But
finding that there is a column with aleph_0 indexes while there is no
ine with aleph_0 indexes shows that the idea of the existence of
aleph_0 is false.

Except that it is not individual lines or columns but sets of indices
which are to be compared and those sets are identical. Every row index
is also a column index and vice versa. Can WM name any exceptions?

If there were aleph_0 lines with aleph_0 a (non-natural) number, then
we would need a last line.

WM might, but no one else would.



Therefore "the fact" that there are aleph_0
lines is not a fact.

Maybe not in WM's world, but in the world of actual mathematics it is


It is simply the result of quantifier exchange to
say E a A n : n < a instead of A n E a : n < a.

And those who claim these statement logically equivalent are wrong.

One can show that for a and n members of, say, N
"for all n there is an a such that n < a"
is true merely by considering a = n+1.

One can equally show the falsity of
"there is an a such that for all n, n < a"
by considering n = a.
.

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