Diagonal Argument Examples

Set Theory · Axiom Academy

EXAMPLE Cantor's Diagonal Argument A proof that the real numbers are uncountable using diagonalization Excellent work! You've completed Cantor's diagonal argument. Here's what we learned: Proof by contradiction: We assume the opposite of what we want to prove and show it leads to a contradiction. Diagonalization technique: By systematically differing from each element in a list, we construct something not in the list. The reals are uncountable: There is no bijection between ℕ and ℝ, so |ℝ| > |ℕ|. Different infinities exist! Power of the method: This technique extends to prove Cantor's theorem: |A| < |𝒫(A)| for any set A. This is one of the most elegant and powerful results in mathematics, revealing the rich structure of infinite sets!

This is the written version of the interactive lesson above. See the full Set Theory course.