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I'm familiar with several proofs that the real numbers are uncountable (Cantor's initial proof, a proof by diagonalization, etc.). However, I've never seen a proof that the reals are uncountable that proceeds by showing that the set of Dedekind cuts of the rationals are uncountable. I'm aware that the set of all subsets of the rationals is uncountable, but not all of these sets are Dedekind cuts.

Is there a simple proof of the uncountability of $\mathbb{R}$ that works by showing the set of Dedekind cuts is uncountable?

Thanks!

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    You show that the set of Dedekind cuts fulfills the premises used in Cantor's proof, i.e. cuts are totally and densely ordered and "cuts of cuts" are already cuts.2012-09-25
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    @Hagen: But this is not really to use the Dedekind cuts to prove uncountability, this is to show that Dedekind cuts form a field which behaves like the real numbers, and therefore is uncountable (by other proofs, e.g. the diagonal argument).2012-09-25
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    @AsafKaragila: Agreed (though "field" is not needed, only order properties, which is cut-speak for subset relations)2012-09-25

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