Transfinite Induction

Set Theory · Axiom Academy

Extending mathematical induction beyond finite numbers to prove properties of all ordinals. To prove a property P(α) holds for all ordinals α, we must establish three cases: The animation below illustrates how transfinite induction climbs through the ordinals, handling each type of ordinal appropriately. Every transfinite induction begins by proving the property holds for the empty set, which represents the ordinal 0. This is analogous to proving P(0) in ordinary induction on natural numbers. Watch as we establish P(0) as our foundation: 3. The Successor Case: P(α) → P(α⁺) For successor ordinals, we assume P(α) holds and prove P(α⁺) must hold. Since α⁺ = α ∪ α , we can often use facts about α directly. The animation shows how the property propagates from α to its successor: 4. The Limit Case: (∀β < λ, P(β)) → P(λ) The limit case is unique to transfinite induction. For a limit ordinal λ, we assume P(β) holds for all β < λ and prove P(λ) holds. Observe how the limit ordinal inherits the property from all its predecessors: 5. Example: All Ordinals are Well-Ordered Let's prove that every ordinal α is well-ordered by ∈ using transfinite induction. The animation demonstrates the complete proof structure:

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