What are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization? Are there any based on dynamical systems?
What are the ways of proving that the Cantor set is uncountable apart from Cantor diagonalization?
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elementary-set-theory
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8I 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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0Some proofs are listed in the book "Measure and Category" by Oxtoby. – 2012-12-04
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2@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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0I 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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5@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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0Oh, I see now what you mean...hehe. Yes, of course – 2012-12-05