Non-Well-Founded Sets (Center for the Study of Language and Information - Lecture Notes)

Ever since Frege and Russell, most logicians have had a terrible, irrational fear of self-reference. They wake up screaming in the night and tell their loved ones that they were dreaming about Cretan liars, Spanish barbers or the set of all sets that do not include themselves. You'd think Gödel's theorem, arguably the most significant discovery in logic during the 20th century, would have convinced them that self-reference had a positive side. But when an irrational dread is sufficiently deep-rooted, there's not much to be done. At least, that was the state of affairs until Peter Aczel came along and decided he couldn't take any more pathetic whining. Granted, he says in his foreword, self-reference, when incorrectly used, can result in nasty paradoxes; but there are many self-referential sentences, like this one, which make perfect sense. He then proceeds to develop an elegant theory which assigns straightforward, non-paradoxical meanings to many sentences which directly or indirectly refer to themselves. - from a Goodreads review by Manny