◀ ▲ ▶Branches / Number-systems-arithmetics / Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers

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.

Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers

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 $(\mathbb N, \le ).$

Strict order version:

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 $(\mathbb N, < ).$

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References

Bibliography

1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013