 Nonwellfounded set theory

Nonwellfounded set theories are variants of axiomatic set theory which allow sets to contain themselves and otherwise violate the rule of wellfoundedness. In nonwellfounded set theories, the foundation axiom of ZFC is replaced by axioms implying its negation.
The theory of nonwellfounded sets has been applied in the logical modelling of nonterminating computational processes in computer science (process algebra and final semantics), linguistics and natural language semantics (situation theory), philosophy (work on the Liar Paradox), and in a different setting, nonstandard analysis.^{[1]}
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Details
 A set, x_{0}, is wellfounded iff it has no infinite descending membership sequence:
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In ZFC, there is no infinite descending ∈sequence by the axiom of regularity. In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC^{−} (that is, ZFC without the axiom of regularity) that wellfoundedness implies regularity. In variants of ZFC without the axiom of regularity, the possibility of nonwellfounded sets with setlike ∈chains arises. For example, a set A such that A ∈ A is nonwellfounded.
One early nonwellfounded set theory was Willard Van Orman Quine’s New Foundations. Another was introduced by Maurice Boffa,^{[citation needed]} with the idea of making foundation fail as badly as it can (or rather, as extensionality permits): Boffa's axiom implies that every extensional setlike relation is isomorphic to the elementhood predicate on a transitive class.
Another, more recent, approach to nonwellfounded set theory, pioneered by M. Forti and F. Honsell, borrows from computer science the concept of a bisimulation. Bisimilar sets are considered indistinguishable and thus equal, which leads to a strengthening of the axiom of extensionality. In this context, axioms contradicting the axiom of regularity are known as antifoundation axioms, and a set that is not necessarily wellfounded is called a hyperset.
Four nonequivalent antifoundation axioms are wellknown:
 AFA (‘AntiFoundation Axiom’) — due to M. Forti and F. Honsell (this is also known as Aczel's antifoundation axiom);
 SAFA (‘Scott’s AFA’) — due to Dana Scott,
 FAFA (‘Finsler’s AFA’) — due to Paul Finsler,
 BAFA ('Boffa's AFA') — due to Maurice Boffa.
The first of these, AFA, is based on accessible pointed graphs (apg) and states that two hypersets are equal if and only if they can be pictured by the same apg. Within this framework, it can be shown that the socalled Quine atom, formally defined by Q={Q}, exists and is unique.
Each following axiom extends the universe of the previous, so that: V ⊆ A ⊆ S ⊆ F ⊆ B. In the Boffa universe, the distinct Quine atoms form a proper class. ^{[2]}
It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the wellfounded sets within a hyperset domain conform to classical set theory.
Notes
References
 Aczel, Peter (1988), Nonwellfounded sets, CSLI Lecture Notes, 14, Stanford, CA: Stanford University, Center for the Study of Language and Information, pp. xx+137, ISBN 0937073229, MR0940014, http://standish.stanford.edu/pdf/00000056.pdf.
 Ballard, David; Hrbáček, Karel (1992), "Standard foundations for nonstandard analysis", Journal of Symbolic Logic 57 (2): 741–748, doi:10.2307/2275304, JSTOR 2275304.
 Hallett, Michael (1986), Cantorian set theory and limitation of size, Oxford University Press.
 Levy, Azriel (2002), Basic set theory, Dover Publications.
 Mirimanoff, D. (1917), "Les antinomies de Russell et de BuraliForti et le probleme fondamental de la theorie des ensembles", L'Enseignement Mathématique 19: 37–52, http://retro.seals.ch/digbib/view?rid=ensmat001:1917:19::9&id=hitlist.
 Nitta; Okada; Tzouvaras (2003), Classification of nonwellfounded sets and an application, http://users.auth.gr/~tzouvara/Texfiles.htm/nonwell.pdf
See also
External links
 Lawrence S. Moss (20080416), "Nonwellfounded Set Theory", Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/nonwellfoundedsettheory/
 Metamath page on the axiom of Regularity. Scroll to the bottom to see how few Metamath theorems invoke this axiom.
Categories: Systems of set theory
 Wellfoundedness
 Selfreference
 A set, x_{0}, is wellfounded iff it has no infinite descending membership sequence:
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