Well-Ordering Principle/Proof using Von Neumann Construction

Theorem

Every non-empty subset of $\N$ has a smallest (or first) element.

That is, the relational structure $\struct {\N, \le}$ on the set of natural numbers $\N$ under the usual ordering $\le$ forms a well-ordered set.

This is called the well-ordering principle.

Proof

From Von Neumann Construction of Natural Numbers is Minimally Inductive, $\omega$ is minimally inductive class under the successor mapping.

From Successor Mapping on Natural Numbers is Progressing, this successor mapping is a progressing mapping.

The result is a direct application of Minimally Inductive Class under Progressing Mapping is Well-Ordered under Subset Relation.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 5$ Applications to natural numbers: Theorem $5.5$