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From: Dan Christensen on 10 Jun 2010 01:26
Hi,

I am looking for a proof that the cardinality of the real numbers
(represented as Dedekind cuts) is greater than that of the natural
numbers -- so, not the usual diagonal proof. Either that or a sketch
outline of how to proceed, or even suggestions on how to begin. Any
help would be appreciated.

Dan



From: Tim Little on 10 Jun 2010 02:18
On 2010-06-10, Dan Christensen <Dan_Christensen(a)sympatico.ca> wrote:
> I am looking for a proof that the cardinality of the real numbers
> (represented as Dedekind cuts) is greater than that of the natural
> numbers -- so, not the usual diagonal proof.

Actually the diagonal proof of |P(N)| > |N| can be applied following a
demonstration of a fairly straightforward injection from P(N) into the
set of Dedekind cuts.


> Either that or a sketch outline of how to proceed, or even
> suggestions on how to begin. Any help would be appreciated.

Cantor's first uncountability proof would apply, though it is
certainly not as obvious in hindsight as his diagonal proof.


- Tim
From: riderofgiraffes on 10 Jun 2010 01:27
In the following construction, details are left for
the interested reader.

Consider any countable collection X of Dedekind cuts.
Divide the line segement [0,1] into four equal sized
pieces and retain only the rightmost of those that do
not touch (i.e. cannot be separated from) the first
element of X.

Now repeat. Divide into four, retain the rightmost
of those that do not touch the second element of X.

And so on.

Now the left-most points of the line-segments you have
retained form a strictly increasing, bounded above
sequence of rationals. This defines a Dedkind cut,
and that Dedekind cut cannot be in X.

Thus every countable set of Dedekind cuts cannot contain
every Dedekind cut, so the set od DC's is uncountable.
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