1. The problem statement, all variables and given/known data 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?