# Worse than a stupid joke …

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 $A$ is well-founded if it is either empty or $\exists b \in A: A \cap b = \emptyset$. A (probably more intuitive) characterization is that there does not exist an infinite sequence $A_0, A_1, \ldots$, not necessarily distinct, such that $A_0 = A$, and $\forall i: A_{i+1} \in A_i$. So a simple example of a non-wellfounded set is a set $A$ which is equal to $\{A\}$ (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 $A = \{b,c\}$ such that $A \in b$ and $A \in c$. Now you see what I meant in my last post? Shri Tiruchi Sankaran was in his elements!!