Non-Well-Founded Sets (Volume 14)

by Peter Aczel

Non-well-founded structures arise in a variety of ways in the semantics of both natural and formal languages. Two examples are non-well-founded situations and non-terminating computational processes. A natural modelling of such structures in set theory requires the use of non-well-founded sets. This text presents the mathematical background to the anti-foundation axiom and related axioms that imply the existence of non-well-founded sets when used in place of the axiom of foundation in axiomatic set theory.

Genres: Logic, Mathematics

131 pages, Paperback

First published January 1, 1988

Reviews

[Manny · Rating 3 out of 5]

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.

Okay... I'll come clean. I just noticed, to my surprise, that Goodreads now allows you to vote for your own review. I wouldn't normally stoop so low as to do so. But surely, if ever there were a book where the idea was appropriate, this must be it?