Cardinal Arithmetic

Set Theory · Axiom Academy

Operations on cardinal numbers: addition, multiplication, and exponentiation For cardinal numbers κ and λ, we define addition as the cardinality of the disjoint union. If A and B are disjoint sets with |A| = κ and |B| = λ, then: The animation shows two disjoint sets being combined to form their union, illustrating that we count all elements from both sets. For cardinal numbers κ and λ, multiplication is defined as the cardinality of the Cartesian product. If |A| = κ and |B| = λ, then: The Cartesian product A × B contains all ordered pairs (a, b) where a ∈ A and b ∈ B. The animation shows how pairs are formed between elements of two sets. For cardinal numbers κ and λ, exponentiation is defined using function spaces. If |A| = κ and |B| = λ, then: Here A^B denotes the set of all functions from B to A. This generalizes our familiar notion that 2^n counts the number of functions from an n-element set to a 2-element set (which is the power set). 4. Infinite Cardinal Properties For infinite cardinals, arithmetic behaves differently than for finite numbers. If κ is an infinite cardinal, then: This surprising fact means that adding an infinite cardinal to itself doesn't make it larger! The animation demonstrates why κ + κ = κ by showing a bijection. 5. The Smallest Infinite Cardinal Let's examine specific examples with ℵ₀, the cardinality of the natural numbers:

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