Proof that the Cantor Set is uncountable. Find the error:

We assert without proof that (0,1) is uncountable. We show [0,1] is uncountable by contraction. We assume there exists some bijective f:N->[0,1]. Then there exists some subset N' of N such that f:N'->(0,1). But we know (0,1) is uncountable, so such a function cannot exist. By contradiction, [0,1] is uncountable. We now construct a bijective map g:C->[0,1], simply let g(c)=c. Then C shares the cardinality of [0,1], so C is uncountable.