Transfinite induction extends mathematical induction to all ordinals. To prove a property P holds for every ordinal α, it suffices to verify three cases: (1) the base case P(0); (2) the successor step: P(α) implies P(α+1) for all α; and (3) the limit step: if P(β) holds for all β < λ, then P(λ) holds for every limit ordinal λ. The limit step is the essential addition beyond ordinary induction and captures behavior at stages like ω, ω², and ε₀. The principle is justified by the fact that the ordinals are well-ordered by ∈, so every non-empty class of ordinals has a least element.
Prove simple properties of ordinals by transfinite induction: every ordinal is 0, a successor, or a limit; every ordinal is transitive. Then prove results in ordinal arithmetic. Internalize that the limit case typically takes a union or supremum of all previous values rather than appealing to an immediate predecessor.
Ordinary mathematical induction works by establishing a domino chain: prove the first domino falls (base case), then prove each domino knocks over its immediate successor (inductive step). This is powerful, but it only reaches the natural numbers — finitely many steps from the start. Transfinite induction is what happens when you ask: what if there are infinitely many dominoes laid out, but also infinitely long "rows" beyond them? The ordinals, which you've studied via von Neumann's construction, are exactly this structure: every finite number, then ω, then ω+1, ω+2, ..., then ω·2, and so on through an incomprehensibly large well-ordered hierarchy.
The reason ordinary induction breaks down at ω is that ω has no immediate predecessor — there is no "last finite number" to hand off from. The well-ordering principle you've already encountered guarantees that any non-empty set of ordinals has a least element, which is what makes induction work at all. But to reach limit ordinals like ω, you need a third case: the limit step. This says that if you've already established P(β) for every β smaller than a limit ordinal λ, then P(λ) must also hold. This is sometimes called strong induction or complete induction — instead of inheriting from your single predecessor, you inherit from your entire downward neighborhood.
To internalize the three cases, think of them as covering a partition of all ordinals. Every ordinal is exactly one of: zero (the base case), a successor ordinal of the form α+1 (handled by the successor step), or a limit ordinal with no immediate predecessor like ω, ω·2, or ε₀ (handled by the limit step). A proof that handles all three cases has no gap — every ordinal falls into exactly one bucket. A proof that omits the limit case establishes P only for successor ordinals reachable from 0, which captures all finite ordinals but stops dead at ω.
The limit step typically invokes a supremum or union argument. In ordinal arithmetic, for example, ω+ω is defined as the least upper bound of {ω+n : n ∈ ω}. Properties of ω+ω are proved by showing they follow from the properties of all ω+n. This is the structural reason why transfinite induction is the foundation for transfinite recursion — defining functions on all ordinals — which you'll use next to build ordinal arithmetic and study cardinality. Just as recursion on ℕ produces sequences, transfinite recursion on Ord produces cumulative hierarchies, the von Neumann universe V, and ultimately the entire model-theoretic picture of set theory.