# Set of Integers can be Well-Ordered

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## 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:

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

## 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 $15 \ \text{(b)}$