Ordinal addition, multiplication, and exponentiation are defined via transfinite recursion and differ fundamentally from cardinal arithmetic. Order matters: 1 + ω = ω but ω + 1 > ω. Commutativity fails; properties reveal deep structure about order type and the ordinal hierarchy.
From your study of von Neumann ordinals and successor/limit ordinals, you know that each ordinal is the set of all smaller ordinals, and that the ordinal ω is the first infinite ordinal — the set {0, 1, 2, 3, …}. Ordinal arithmetic extends this structure by defining operations that respect order type: what matters is not just how many elements are in a set but how those elements are arranged. This is why ordinal arithmetic diverges sharply from the arithmetic you learned for natural numbers.
Ordinal addition α + β means: take a copy of α, then append a copy of β after it. Concretely, 1 + ω means: put one element, then put a copy of ω after it. But a single element followed by {0, 1, 2, 3, …} still has order type ω — there's a least element (the original 1), then an infinite ascending chain, so the order type is just ω. But ω + 1 means: put ω first, then append one element after it. Now you have {0, 1, 2, 3, …, ω} — an infinite ascending chain with a new last element stuck on the end. This is strictly larger than ω because ω has no greatest element but ω + 1 does. The asymmetry is not a trick; it reflects that order type is sensitive to *where* the new elements are placed relative to the existing ones.
Ordinal multiplication α × β means: take β many copies of α, laid end-to-end. So ω × 2 means two copies of ω concatenated: {0, 1, 2, …, ω, ω+1, ω+2, …}, which has order type ω × 2. But 2 × ω means ω many copies of 2 laid end-to-end: {0, 1 | 0, 1 | 0, 1 | …}, which is just ω pairs — and that has order type ω, since each pair {0, 1} is finite and the whole sequence is still a simple infinite ascending chain. So ω × 2 ≠ 2 × ω; multiplication is non-commutative for infinite ordinals. The intuition is that right-multiplication "scales" the left argument, but left-multiplication "repeats" it in a way that collapses.
Ordinal exponentiation α^β is defined by transfinite recursion: α^0 = 1, α^(β+1) = α^β × α, and at limit ordinals, α^λ is the supremum of all α^β for β < λ. The most important case is ω^ω, which represents the ordinal corresponding to sequences of natural numbers listed in a specific lexicographic order — it is a countable ordinal but already far beyond ω × ω. The famous ordinal ε₀ = ω^(ω^(ω^⋯)) is the first ordinal satisfying ω^ε₀ = ε₀, and it plays a central role in proof theory. Every operation here builds on transfinite recursion — the same tool you'll use extensively when studying transfinite induction — because the definitions must handle successor cases and limit cases separately.