The standard ternary Cantor set C is defined as C = \bigcap_(n=0)^(∞) Cₙ, where C₀ = [0,1] and Cₙ is obtained from C_(n-1) by removing the open middle third of each interval. It is well-known that C is uncountable — it has a bijection with [0,1] via ternary expansions using only digits 0 and 2. Yet its Lebesgue measure is λ(C) = \lim_(n→∞) (2/3)ⁿ = 0. How can a set be simultaneously uncountable and of measure zero? I am looking for an intuitive explanation of how the Cantor set manages to have the cardinality of the continuum while occupying "no length" in the real line. Bonus: are there generalizations to higher dimensions?