Classification of non-well-founded sets and an application

In set theory, the foundation axiom (or regularity)(F) says that the relation ∈ is well-founded, that is, there is no infinite descending ∈-sequence : · · · ∈ x 2 ∈ x 1 ∈ x 0. If we identify ∈ in a set with ← in a graph, the set is identified with a graph. The set 1 is 1 = {φ}, that is, φ ∈ 1. It corresponds to a graph x 1 ← x 0 with nodes x 0 , x 1. In terms of graphs and nodes, the well-foundedness is that there is no − the Zermelo-Fraenkel set theory with choice, which does not satisfy (F). Usually we write (AF) to indicate an anti-foundation axiom, any statement refusing (F). In 1988, Aczel([A]) introdeced non-well-founded ZFC − +(AF) set theory, and studied various kinds of anti-foundation axioms and hence the associated non-well-founded set theories, which include Aczel set theory, Scott set theory, Finsler set theory and Boffa set theory. In fact we know that Zermelo-Fraenkel set theory is a subclass of Aczel set theory, Aczel set theory is a subclass of Scott set theory and Scott set theory is a subclass of Finsler set theory. In 1962, Richard Peddicord([P]) computed the number of Zermelo-Fraenkel sets of finite nodes. In 1990, Booth([Boo]) counted Finsler sets of node one, two and three, while pointing out that it is difficult to construct an algorithm to classify Finsler sets. In 1998, Milito and Zhang([M-Z]) proposed an algorithm to classify Aczel sets, and commented that Booth's list is wrong in the case of node three. In this paper, we not 1