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Transfinite recursion allows the definition of functions on all ordinals by specifying: F(0) = base value, F(α+1) = g(F(α)) at successors, and F(λ) = h({F(β) : β < λ}) at limit ordinals. The axiom of replacement is needed to ensure that the partial functions at each stage form a set. The theorem on transfinite recursion guarantees a unique such F exists given any valid specifications. Key applications include defining ordinal arithmetic (+, ·, exponentiation), constructing the cumulative hierarchy V_α, and building the aleph sequence ℵ₀, ℵ₁, ℵ₂, ....
Define ordinal addition α+β by recursion on β: (α+0) = α, (α+(β+1)) = (α+β)+1, (α+λ) = sup{α+β : β < λ}. Then define ordinal multiplication and exponentiation similarly. In each case explicitly state all three cases. Separately define the cumulative hierarchy V_α by recursion on α and compute V₀, V₁, V₂, V_ω.
You already understand ordinary recursion on the natural numbers: define F(0) as a base case, and define F(n+1) in terms of F(n). Transfinite recursion extends this idea to all ordinals, but with one important new wrinkle — ordinals come in two flavors beyond zero, not one. There are successor ordinals of the form α+1, and there are limit ordinals like ω (the first infinite ordinal), ω+ω, or ω₁, which have no immediate predecessor. Because a limit ordinal has no predecessor, you cannot define F(λ) as "F of the thing before λ" — there is no such thing. Instead, at limit ordinals you take a supremum or union over all the values F(β) for β < λ. This three-case structure — base, successor, limit — is the signature of every transfinite recursion.
Why does this require the axiom of replacement? Consider building up the sequence F(0), F(1), F(2), ..., F(ω). To form F(ω) as the union or sup of all prior values, you first need all those prior values to constitute a *set* — a completed collection you can hand to a function. Without replacement, the class {F(n) : n < ω} might exist as a proper class that cannot be collected into a set. Replacement guarantees: if you have a set A and a definable function sending each element of A to some set, the image is also a set. In the recursion context, this ensures that the partial approximations at every stage can be assembled into a set, which is then used to define the next stage.
A canonical example is ordinal arithmetic. Define ordinal addition α + β by recursion on β: α + 0 = α (base), α + (β+1) = (α + β) + 1 (successor step), and α + λ = sup{α + β : β < λ} for limit λ (limit step). This three-clause definition is precise, unambiguous, and mirrors the recursive definition of addition on natural numbers — except the third clause has no analogue in the finite case. Ordinal multiplication and exponentiation follow the same pattern. Notice that ordinal addition is not commutative: ω + 1 ≠ 1 + ω, which you can verify from the definitions.
The cumulative hierarchy V_α is another essential application. Define V_0 = ∅, V_{α+1} = P(V_α) (the power set), and V_λ = ⋃_{β < λ} V_β for limit λ. Again, three clauses. Computing the first few stages builds intuition: V_1 = {∅}, V_2 = {∅, {∅}}, V_3 has four elements, V_4 has 16, and V_ω = ⋃_{n<ω} V_n is the set of all hereditarily finite sets. The transfinite recursion theorem guarantees that this construction produces a unique, well-defined function on all ordinals — every set in the universe (under the regularity axiom) appears somewhere in this hierarchy.
The deep connection to transfinite induction is this: recursion and induction are dual. Transfinite induction proves properties of all ordinals by the same three-case argument; transfinite recursion *constructs* functions on all ordinals by the same three-case argument. The proof that transfinite recursion works is itself a transfinite induction — you prove that the partial approximation built up to stage α is uniquely determined, for all α, by induction on α. Once you have internalized both tools, you can define essentially any well-behaved function on ordinals and immediately inherit a powerful proof principle for its properties.