Circularity is obviously a common means of commonsense knowledge representation. Thus we conclude that a possible commonsense set theory should also allow circular sets to be expressed. However, such formalizations which also capture the cumulative hierarchy principle are not very common. Among theories revised so far, Aczel's theory is the only one which allows circularity. By proposing his Anti-Foundation Axiom, Aczel overrode the FA of ZF which prohibits circular sets, but preserved the hierarchical nature of the original axiomatization.
Situations are parts of the reality that can enter into relations with other parts.
Their internal structures are sets of facts and hence they can be modeled by sets. Circular situations are common in our daily life. If announcements are assumed to be situations, then this one surely contains itself. The authors make use of Aczel's theory for this purpose. This will be our next subject.