Exercise 1559Logic, sets and proofs › Countability and cardinalityM1Added 3 Nov 2026

The Cantor set is uncountable

#Cantor set#uncountable#ternary

Statement

Show that the Cantor set $K$ (obtained by removing middle thirds iteratively from $[0,1]$ ) is uncountable.

Hint

Elements of $K$ have a ternary expansion using only digits $0$ and $2$ .

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Ternary. $K=\{x\in[0,1]:x$ has a ternary expansion with digits in $\{0,2\}\}$ .

Bijection. Map each ternary expansion (digits $0,2$ ) to a binary expansion (digits $0,1$ ). This gives a surjection $K\to[0,1]$ .

Conclusion. $K$ has cardinality $2^{\aleph_0}$ ; uncountable (Lebesgue measure $0$ , yet uncountable).