Worse than a stupid joke is a stupid joke explained in detail. But anyway, what else did you expect from me? Here goes …

A set is **well-founded** if it is either empty or . A (probably more intuitive) characterization is that there does not exist an infinite sequence , not necessarily distinct, such that , and . So a simple example of a non-wellfounded set is a set which is equal to (whatever such a set might be!). In fact, non-wellfounded objects are quite bizarre, and we do not have a need for them anyway, so one of the axioms of Zermelo-Frankael set theory is that every set is wellfounded. This is the **axiom of regularity**.

Anyway, another example of a non-wellfounded set is such that and . Now you see what I meant in my last post? Shri Tiruchi Sankaran was in his elements!!

/me blinketh, waiteth…

Okay, I confess to being quite amused, with some help :D