Re: Cantor Confusion

Virgil schrieb:

In article <1165614850.907256.254500@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

Virgil schrieb:

[concerning Cantor's first proof of uncountability of the real numbers]

In the reals, any subset which has a real upper bound has a real least
upper bound and, similarly , any subset which has a real lower bound has
a real greatest lower bound.

The only subsets of the reals for which there is a similar property are
real intervals.

Thus it is only for the set of all reals, or for real intervals, that
the proof appplies.

Yes. And it does not apply if only one single element of the
investigated real interval is missing. As the uncontability property of
this interval cannot depend on this single element, the whole proof
fails.

The proof does not fail. It merely does not apply to the whole any more,
but it does apply separately to each of the pieces separated by that
removed point.

Invalid arguing, because the proof would fail as well for the remainig
sets after removing one element.

Or is WM arguing that removing a single element from a set can make an
uncountable set countable,

This change is highly improbable. *Therefore* the proof is invalid.

even though its removal real partitions the
remaining reals into two equally uncountable sets.

In particular, the proof would fail for the whole set of reals after
removing the set of rationals (as I explained in my paper) - and there
would not remain any sets which are provably uncountable by this or
Cantor's second proof.

Regards, WM

.

Relevant Pages

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  • Re: infinity
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  • Re: An uncountable countable set
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