Noetherian Induction

Let (A,1) be a partially ordered set. The relation a 1 b can be read as “a precedes b”. For elements a and b in A, we write a ≺ b if and only if a 1 b and a 6= b holds. For notational convenience, we also write a o b if and only if a 1 b holds, and a  b if and only if a ≺ b holds. We call (A,1) a well-founded set if and only if every non-empty subset M of A contains at least one minimal element m with respect to the order relation 1. Put differently, any non-empty subset M of a well-founded set A contains an element m such that there does not exist an element m′ in M satisfying m′ ≺ m. An infinite descending chain S in a partially ordered set (A,1) is an totally ordered subset of A without minimal element. In other words, S contains elements a1, a2, . . . such that a1  a2  a3  · · · . Proposition 1. Let (A,1) be a partially ordered set. Then the following two statements are equivalent: (i) (A,1) is a well-founded set. (ii) There does not exist any infinite descending chain in A.