One reason that well-founded sets are interesting is because a version of transfinite induction can be used on them: if (X, <=) is a well-founded set and P(x) is some property of elements of X and you want to show that P(x) holds for all elements of X, you can proceed as follows:
Examples of well-founded sets which are not totally ordered are:
If (X, ≤) is a well-founded set and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: for every positive integer n, let Xn be a totally ordered set with n elements. Let X be the disjoint union of the Xn together with a single new element x which is bigger than all other elements. Then X is a well-founded set, and there are descending chains starting at x of arbitrary (finite) length.
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