Well-Ordering Principle/Proof using Naturally Ordered Semigroup
Jump to navigation
Jump to search
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
Consider the natural numbers $\N$ defined as the naturally ordered semigroup $\struct {S, \circ, \preceq}$.
From its definition, $\struct {S, \circ, \preceq}$ is well-ordered by $\preceq$.
The result follows.
As $\N_{\ne 0} = \N \setminus \set 0$, by Set Difference is Subset $\N_{\ne 0} \subseteq \N$.
As $\N$ is well-ordered, by definition, every subset of $\N$ has a smallest element.
$\blacksquare$