Set of Integers can be Well-Ordered

Theorem

The set of integers $\Z$ can be well-ordered with an appropriately chosen ordering.

Proof

Consider the ordering $\preccurlyeq \subseteq \Z \times \Z$ defined as:

$x \preccurlyeq y \iff \left({\left\vert{x}\right\vert < \left\vert{y}\right\vert}\right) \lor \left({\left\vert{x}\right\vert = \left\vert{y}\right\vert \land x \le y}\right)$

This needs considerable tedious hard slog to complete it. In particular: It remains to be shown that $\preccurlyeq$ is an ordering, and also that it is a well-ordering. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: Exercise $15 \ \text{(b)}$