The well-ordering principle ensures the existence of a minimum in every non-empty subset of natural numbers. But how about the existence of a maximum? It turns out that such a maximum exists if we, in addition, require that the subset is finite.
Every non-empty and finite subset \(M\subseteq\mathbb N\) of natural numbers contains a unique greatest element $m_0 \ge m$ for all $m\in M$ with respect to their order relation $(\mathbb N, \le ).$
Every non-empty and finite subset \(M\subseteq\mathbb N\) of natural numbers contains a unique maximal element $m_0 > m$ for all $m\in M$ with respect to their order relation $(\mathbb N, < ).$
Proofs: 1
Proofs: 1