Why does the well-ordering principle exclude $0$?

Question

I was surprised I couldn't find an answer to this question. Simply put, why does the well-ordering principle exclude $0$? Is it just for cleaner notation (since we all recognize that $\mathbb Z^+ > 0$)?

i.e. if a set is a subset of $\{\{0\} \cup\mathbb Z^+\}$, this set still exhibits the well ordering property, right? ...and therefore still contains a least integer.

Edit: I guess the question is really, why doesn't the definition use the language of non-negative integers?

From Wikipedia: "the well-ordering principle states that every non-empty set of positive integers contains a least element"...this definition matches that in my book (Pinter's A Book of Abstract Algebra)

Answer 1

There is a close relationship between the the ideas of:

The well-ordering principle

Mathematical induction

Proof by minimal counterexample

Fermat's method of infinite descent.

It is interesting that all of these tend to be expressed in terms of positive integers. In terms of infinite descent there is a very good reason for this since Fermat was concerned with positive integer solutions to equations. (Note how $a^p+b^p=c^p$ has infinitely many solutions for all primes $p$ if we allow zeroes!)

Given that infinite descent can be regarded as a forerunner of the other ideas it is only to be expected that all the ideas should be consistent regarding the issue of 'positive integers'.