Transfinite induction

Transfinite induction is a method used to prove properties about very large, well-ordered sets—called ordinal numbers—that extend beyond ordinary counting. It works by showing that a property holds for the smallest element, and that if it holds for any element, it also holds for the next. Then, because the set is well-ordered (every subset has a smallest element), this step-by-step logic proves the property for all elements, including infinitely large ones. It’s an extension of ordinary induction, tailored to handle infinite and transfinite (beyond infinity) sequences systematically.