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Topic: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR:
THE REALS ARE UNCOUNTABLE!

Replies: 47   Last Post: Jan 12, 2013 11:33 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: FAILURE OF THE DISTINGUISHABILITY ARGUMENT. THE TRIUMPH OF CANTOR: THE REALS ARE UNCOUNTABLE!
Posted: Jan 10, 2013 6:01 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> That is the point. The argument is trivially wrong, but so famaous
> that you cant't believe what you must conclude.

There is something (a considerable amount) wrong with WM but not with
either the first or the second of Cantor's uncountability of the reals
arguments.

When one goes strictly by the formal definitions of countable and
uncountable, Cantor make sense and WM does not.

If one does not go by those definitions, there is no point in talking
about countability at all.

Apparently WM has never learned those definitions because he always
avoids using them.

So here they are again:

Countable set

In mathematics, a countable set is a set with the same cardinality
(number of elements) as some subset of the set of natural numbers. A set
that is not countable is called uncountable. The term was originated by
Georg Cantor. The elements of a countable set can be counted one at a
timealthough the counting may never finish, every element of the set
will eventually be associated with a natural number.
Some authors use countable set to mean a set with the same cardinality
as the set of natural numbers.[1] The difference between the two
definitions is that under the former, finite sets are also considered to
be countable, while under the latter definition, they are not considered
to be countable. To resolve this ambiguity, the term at most countable
is sometimes used for the former notion, and countably infinite for the
latter. The term denumerable can also be used to mean countably
infinite,[2] or countable, in contrast with the term nondenumerable.
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