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!!

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