transfinite induction

Transfinite induction is a generalization of mathematical induction used in mathematics, particularly in set theory. While mathematical induction proves a property for all positive integers by showing it holds for 1 and for the successor of any integer for which it holds, transfinite induction extends this concept to any well-ordered class or domain.1

A domain is well-ordered if its elements can be arranged such that no element precedes itself, the order is transitive, and every non-empty subclass has a first element. Transfinite induction relies on the principle that if the first element of a well-ordered domain has a certain property, and if all members of a class having that property imply that the successor of that class also has the property, then all elements in the domain possess that property. This method is crucial for developing theories of ordinal numbers and understanding the structure of sets.12