Re: An uncountable countable set


Virgil schrieb:

In article <1151845298.388602.107330@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Virgil schrieb:

If you claim that any member of any list is not differentiated from the
number created for that list by any of uncountably many variations of
Cantor's constructions, then you must be declaring there there exists a
first number in some list which is not differentiated according to some
version of Cantor's rules.

So Give us an example of this alleged situation. If you can.

Nothing easier than that. Exchange 0 by 1 in:

0.0
0.1
0.11
0.111
...

The diagonal up to any line number n is contained in line number n+1.

If you think 0.111... is not in the list, remember that any number the
digits of which can be indexd by finite numbers is in the list.

Since every number in your list terminates, butyour "diagonal" 0.111...
does not, it differes from each list member in infinitely many digit
positions, and it is clearly not in your list. So YOU have provided a
nonmember to prove your own error.



If
0.111.. is not in the list, then it must have more digits than can be
indexed (and hence, can exist).

They are satisfactorily indexed by the infinite set of finite natural
numbers, N.

You just proved that there are infinitely digit positions which are not
indexed by natural numbers (*all* of which are given in the list).

As the list is supposed to be endlessly long and the decimal (or other
base) representation of each member is supposed to be endlessly "wide",
why should either endlessness be greater than the other?

Why should it not? Remember, there must be an exact equality between
width and length (better than +/- 1).

If "mueckenh" insists on exact equality and "mueckenh" simultaneously
insists on a difference, "mueckenh" is in trouble.

Therefore he does not insist on a difference, but doubts that this
precision can be achieved.

Regards, WM

.

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