So I'm a little confused by the following theorem:
$X$ is well ordered iff every proper initial segment is determined by an element.
My confusion lies in applying this to $\mathbb{Z}$. It seem's to me that every proper initial segment of $\mathbb{Z}$ is determined by an element in $\mathbb{Z}$, which would imply that $\mathbb{Z}$ is well-ordered which we know is false.
So my question is where have I made a mistake in my reasoning. I feel it must be in saying that every initial segment is determined by an element in $\mathbb{Z}$.
The empty set is a proper initial segment of $\mathbb{Z}$ but there is no $x\in\mathbb{Z}$ such that $\emptyset=\{y\in\mathbb{Z}:y<x\}$, since $\mathbb{Z}$ has no least element.