Peter Aczel writes in his An Introduction to Inductive Definitions:
Let $\prec$ be a binary relation on a set $A$. The well-founded part of $\prec$ is the set $W(\prec)$ of $a\in A$ such that there is no infinite descending sequence $a\succ a_0\succ a_1\succ\ldots$ .
Am I right that if I take $\mathbb{Z}$, the set of integers with the usual less-than relation, then the well-founded part of it, $W(\prec)$ is the empty set?
Yes - for any integer $z$, consider the sequence $$z>z-1>z-2>z-3>...$$
By contrast, the well-founded part of $\mathbb{N}$ (with the usual ordering) is exactly all of $\mathbb{N}$ - this is equivalent, in a precise sense, to proof by induction being valid!
As a third example - between $\mathbb{Z}$ and $\mathbb{N}$ - consider the set of real numbers which are either $> 1$ or are of the form $1-{1\over n}$ for some nonzero natural number $n$. Then the well-founded part of this set (ordered in the usual way) is exactly $$\{1-{1\over n}: n\in\mathbb{N}\}.$$