Re: Cantor Confusion


Franziska Neugebauer wrote:
William Hughes wrote:

You want to say that A has a diagonal, but addition of one element to
each line, each column and the diagonal eliminates this property?

Yes, because adding one element to each column changes the number
of lines, but adding one element to each line does not
change the number of columns.

What did I tell you!?

Yes, but the discussion has reached the point where
the distinction between adding an element to an infinite
sequence and adding an element to a finite sequence
can be made clear. My contention is that an earlier
attempt to clarify this would not have been productive.
(not to say that this attempt will be productive, just
that it is my opinion that it has a better chance of
being productive).

You may disagree, but it's my time that I'm wasting.


[...]
The original matrix has the same number of lines and columns,
however,the lines and columns are different. Doing the same thing to
the lines and columns does not have the same result.

You are *not* doing the same thing!


Yes and no. In both cases we are "adding a single element", in
both cases we are increasing the ordinal represented by a sequence
by one. So in one sense we are "doing the same thing". However,
adding one element to a sequence without an end means we have
to add the element "after" the sequence, while adding one
element to a sequence with an end mean we add the element
"at" the end. So in another sense we are not doing the same thing.
My preferred informal description of this is "we are doing the
same thing to two things that are different and we get
different results".
- William Hughes

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