Years ago a professor explained that the Cantor Set is uncountable the following way:
C0 has two endpoints, C1 has four endpoints, C2 has eight endpoints, etc.
Therefor Cn has 2^n endpoints and thus the set of all endpoints is 2^N. Now, since 2^N is uncountable, and the endpoints is a subset of the full Cantor set, therefor the Cantor Set is uncountable.
At first glance this made sense to me but then I realized that the endpoints are all Rational!
Obviously this "proof" is flawed but my question is, where is the flaw?