Different Sizes of Infinity

Set Theory · Axiom Academy

Georg Cantor's shocking discovery: not all infinities are created equal. Meet two of mathematics' most important infinite sets. They both go on forever, but are they the same size? Let's try to create a bijection. Click to place natural numbers on the real number line between 0 and 1. Cantor proved ℝ is larger than ℕ using his famous diagonal argument. Let's see a glimpse of this brilliant proof. Click to reveal the mind-bending truth about infinity. A set is countable if it has the same cardinality as ℕ (or is finite). A set is uncountable if it's infinite but larger than ℕ. The real numbers are uncountable. Cardinal numbers measure the size of sets. Finite cardinals are just regular numbers (1, 2, 3, ...). Infinite cardinals include _0, _1, _2, forming an infinite hierarchy of infinities. Before Cantor, infinity was treated as a vague concept. Cantor gave it mathematical precision, revealing a beautiful hierarchy. His work initially faced harsh criticism but is now foundational to modern mathematics.

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