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What are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?

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    I don't know to what Cantor diagonalization you're referring here: the only proof I know that the Cantor set is uncountable uses writing elements in base 3 and then an onto function. Cantor Diagonalization is used to show that the set of all real numbers in $\,[0,1]\,$ is uncountable.2012-12-04
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    Some proofs are listed in the book "Measure and Category" by Oxtoby.2012-12-04
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    @DonAntonio given the "base 3" definition of the Cantor set, isn't a (direct) diagonalization the easiest way to show that it is uncountable?2012-12-04
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    I can't say, @TrevorWilson, but I don't see it that clear right now. It surely would surprise me if it were "simply" that because I haven't seen this, but of course I can't say is impossible.2012-12-05
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    @DonAntonio I just mean that the diagonal argument showing that the set of $\{0,2\}$-sequences is uncountable is exactly the same as the one showing that the set of $\{0,1\}$-sequences is uncountable. So introducing the interval $[0,1]$ only complicates things (as far as diagonal arguments are concerned.)2012-12-05
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    Oh, I see now what you mean...hehe. Yes, of course2012-12-05

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