Constructing Nonlinear Well-Founded Orders

[dirtybiscuit]

Homework Statement

My teacher has notes online that say:

A Simple Construction Technique for WellFounded Orders
Any function ƒ : S→N defines a wellfounded order on S by
x < y iff ƒ(x) < ƒ(y).

Example:
Lists are wellfounded by length. Binary trees are wellfounded by depth, by number of nodes, or by number of leaves. ℤ is wellfounded by absolute value.
Derivations for a grammar are wellfounded by length. These orders are nonlinear.

I am having trouble understanding how these are non-linear orders. Particularly "ℤ is wellfounded by absolute value". From my understanding a linear order is where each element in the set is comparable to the other elements of the set.

So for the abs of ℤ the order I think we get is:
0, -1, 1, -2, 2, -3, 3, ...

which seems like -1 < -2 < 3 and so on because of the way we defined it and thus they are comparable. Am I misunderstanding something in this?

 

[mfb]

which seems like -1 < -2 < 3 and so on because of the way we defined it and thus they are comparable.

Right.
The usual ordering of Z is not relevant here.