Countability Proof Examples

Set Theory · Axiom Academy

Proving that ℤ, ℚ, ℕ × ℕ, and (0,1) are all countable sets Excellent work! You've explored several countability proofs. Here's what we learned: Zigzag enumeration: Clever ordering can enumerate infinite sets systematically (like the integers). Cantor pairing: Two-dimensional arrays of countable sets can be flattened into one dimension via diagonal traversal. Countable products: The product of countable sets is countable (ℤ × ℕ, hence ℚ). Schröder-Bernstein theorem: If there exist injections both ways between sets A and B, then there exists a bijection, so |A| = |B|. Surprising results: ℚ is countable despite being dense in ℝ, while ℝ is uncountable! These techniques are fundamental tools in set theory and help us understand the structure of infinite sets!

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