Cardinal arithmetic defines operations on cardinals: addition κ + λ = |K ⊔ L| (disjoint union), multiplication κ · λ = |K × L| (Cartesian product), and exponentiation κ^λ = |K^L| (all functions from L to K). For infinite cardinals under AC, both addition and multiplication simplify dramatically: κ + λ = κ · λ = max(κ, λ) for any infinite cardinals κ, λ. Cardinal exponentiation, however, is far less trivial — the value of 2^ℵ₀ cannot be determined from ZFC alone and is the subject of the continuum hypothesis. These operations behave very differently from their ordinal arithmetic counterparts.
Prove κ + κ = κ and κ · κ = κ for infinite cardinals (using well-ordering to exhibit explicit bijections). Compute 2^ℵ₀ = |ℝ| = |P(ℕ)| via binary representations of reals. Then contrast with ordinal arithmetic: ω + ω > ω in ordinals, but ℵ₀ + ℵ₀ = ℵ₀ in cardinals — the same symbol behaves differently in the two systems.
You know from infinite cardinal numbers that cardinals measure the "size" of sets, and Cantor's theorem guarantees infinitely many distinct infinite cardinals: ℵ₀ < ℵ₁ < ℵ₂ < ... You also know that two sets have the same cardinality exactly when a bijection between them exists. Cardinal arithmetic extends this framework by defining operations on cardinalities. The definitions are natural — but the behavior of infinite cardinals is shockingly different from finite arithmetic.
Cardinal addition is defined via disjoint union: κ + λ = |K ⊔ L|, where K and L are disjoint sets of cardinalities κ and λ. Cardinal multiplication is the cardinality of the Cartesian product: κ · λ = |K × L|. For finite cardinals, these agree with ordinary arithmetic. For infinite cardinals, both operations collapse: if κ and λ are infinite and κ ≥ λ, then κ + λ = κ · λ = κ. Intuitively: ℕ ∪ ℕ and ℕ × ℕ are both countable, so ℵ₀ + ℵ₀ = ℵ₀ · ℵ₀ = ℵ₀. The general proof uses the axiom of choice to well-order κ and exhibit an explicit bijection κ × κ → κ by a transfinite diagonal enumeration. The conceptual consequence is that infinity absorbs: adding or multiplying an infinite cardinal by anything no larger leaves the cardinal unchanged. There is nothing analogous in finite arithmetic.
Cardinal exponentiation κ^λ is the cardinality of the set of all functions from L to K — equivalently, K^L. This operation does not collapse. The most important case is 2^ℵ₀: the cardinality of all functions ℕ → {0, 1}, equivalently the cardinality of all subsets of ℕ (by binary representation), equivalently the cardinality of ℝ (by the decimal expansion bijection). Cantor's theorem guarantees 2^ℵ₀ > ℵ₀. But the precise value of 2^ℵ₀ in the ℵ-hierarchy cannot be determined from ZFC alone — the assertion 2^ℵ₀ = ℵ₁ is the continuum hypothesis, which Gödel showed cannot be disproved from ZFC and Cohen showed cannot be proved. It is genuinely independent, meaning there are models of ZFC where 2^ℵ₀ = ℵ₁ and models where 2^ℵ₀ = ℵ₂₃₇.
Comparing cardinal and ordinal arithmetic reveals how different they are. In ordinal arithmetic, ω + ω > ω: the first copy of ω finishes before the second one begins, producing a strictly larger well-order. In cardinal arithmetic, ℵ₀ + ℵ₀ = ℵ₀: cardinality ignores order and cares only about bijective matching — the two copies of ℕ can be interleaved into one. The symbols ω and ℵ₀ name the same underlying set (the natural numbers), but they represent different mathematical structures: ω is that set viewed as a well-ordered type, ℵ₀ is that set viewed as a cardinality class. Operating on them under different arithmetic rules is not a contradiction — it is a consequence of measuring different things.