##  [Transfinite Induction](/transfinite-induction) 

  ##  [Transfinite Induction](https://mathlogic.quantumdictionary.io/transfinite-induction-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An extension of mathematical induction to well-ordered sets of ordinals: to prove a property P(α) for every ordinal α one shows P(0), proves P(α+1) assuming P(α) for each successor α+1, and proves P(λ) for each limit ordinal λ assuming P(β) for all β<λ.

 

 

 

 

 





 

 



 ##  [Transfinite Induction](https://puremath.quantumdictionary.io/transfinite-induction-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An extension of mathematical induction to well-ordered sets indexed by ordinals: to prove a property P(α) for all ordinals α one shows P(0), proves that P(β) implies P(β+1) for any successor β+1, and for any limit ordinal λ shows that P(γ) holds for all γ < λ implies P(λ).