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| '''Non-well-founded set theories''' are variants of [[axiomatic set theory]] that allow sets to contain themselves and otherwise violate the rule of [[well-foundedness]]. In non-well-founded set theories, the [[axiom of regularity|foundation axiom]] of [[ZFC]] is replaced by axioms implying its negation.
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| The study of non-well-founded sets was initiated by [[Dmitry Mirimanoff]] in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets; he did not regard well-foundedness as an [[axiom]]. Although a number of axiomatic systems of non-well-founded sets were proposed afterwards, they did not find much in the way of applications until [[Peter Aczel]]’s hyperset theory in 1988.<ref>Pakkan and Akman (1994, [http://tinf2.vub.ac.be/~dvermeir/mirrors/www.cs.bilkent.edu.tr/%257Eakman/jour-papers/air/node8.html section link]); Rathjen (2004); Sangiorgi (2011) pp. 17–19 and 26</ref>
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| The theory of non-well-founded sets has been applied in the [[logic]]al [[model (abstract)|modelling]] of non-terminating [[Computing|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, [[non-standard analysis]].{{sfnp|Ballard|Hrbáček|1992|p=}}
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| == Details ==
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| In 1917, Dmitry Mirimanoff introduced<ref>Levy (2002), p. 68; Hallett (1986), [http://books.google.com/books?id=TM3AKPYdQVgC&pg=PA186 p. 186]; Aczel (1988) p. 105 all citing Mirimanoff (1917)</ref> the concept of [[well-founded set|well-foundedness]] of a set:
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| : ''A set, x<sub>0</sub>, is well-founded ''[[iff]]'' it has no infinite descending membership sequence'':
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| :: · · · <math> \in x_2 \in x_1 \in x_0. </math>
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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<sup>−</sup> (that is, ZFC without the axiom of regularity) that well-foundedness implies regularity.
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| In variants of ZFC without the [[axiom of regularity]], the possibility of non-well-founded sets with set-like ∈-chains arises. For example, a set ''A'' such that ''A'' ∈ ''A'' is non-well-founded.
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| Although Mirimanoff also introduced a notion of isomorphism between possibly non-well-founded sets, he considered neither an axiom of foundation nor of anti-foundation.<ref>Aczel (1988) p. 105</ref> In 1926 Paul Finsler introduced the first axiom that allowed non-well-founded sets. After Zermelo adopted Foundation into his own system in 1930 (from previous work of [[von Neumann]] 1925–1929) interest in non-well-founded sets waned for decades.{{sfnp|Aczel|1988|p=107}} An early non-well-founded set theory was [[Willard Van Orman Quine]]’s [[New Foundations]], although it is not merely ZF with a replacement for Foundation.
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| Several proofs of the independence of Foundation from the rest of ZF were published in 1950s particularly by [[Paul Bernays]] (1954), following an announcement of the result in earlier paper of his from 1941, and by [[Ernst Specker]] who gave a different proof in his [[Habilitationschrift]] of 1951, proof which was published in 1957. Then in 1957 [[Rieger's theorem|Rieger’s theorem]] was published, which gave a general method for such proof to be carried out, rekindling some interest in non-well-founded axiomatic systems.{{sfnp|Aczel|1988|pp=107–108}} The next axiom proposal came in a 1960 congress talk of [[Dana Scott]] (never published as a paper), proposing an alternative axiom now called SAFA.{{sfnp|Aczel|1988|pp=108–109}} Another axiom proposed in the late 1960s was [[Maurice Boffa]]’s axiom of [[superuniversality]], described by Aczel as the highpoint of research of its decade.{{sfnp|Aczel|1988|p=110}} Boffa’s idea was to make foundation fail as badly as it can (or rather, as extensionality permits): Boffa’s axiom implies that every [[extensionality|extensional]] [[binary relation|set-like]] relation is isomorphic to the elementhood predicate on a transitive class.
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| A more recent approach to non-well-founded set theory, pioneered by M. Forti and F. Honsell in the 1980s, 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 '''anti-foundation axioms''', and a set that is not necessarily well-founded is called a '''hyperset'''.
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| Four mutually [[Independence (mathematical logic)|independent]] anti-foundation axioms are well-known, sometimes abbreviated by the first letter in the following list:
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| # '''A'''FA (‘Anti-Foundation Axiom’) – due to M. Forti and F. Honsell (this is also known as [[Aczel's anti-foundation axiom|Aczel’s anti-foundation axiom]]);
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| # '''S'''AFA (‘Scott’s AFA’) – due to [[Dana Scott]],
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| # '''F'''AFA (‘Finsler’s AFA’) – due to [[Paul Finsler]],
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| # '''B'''AFA (‘Boffa’s AFA’) – due to [[Maurice Boffa]].
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| They essentially correspond to four different notions of equality for non-well-founded sets. The first of these, AFA, is based on [[accessible pointed graph]]s (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 so-called [[Quine atom]], formally defined by Q={Q}, exists and is unique.
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| Each of the axioms given above extends the universe of the previous, so that: [[Von Neumann universe|V]] ⊆ A ⊆ S ⊆ F ⊆ B. In the Boffa universe, the distinct Quine atoms form a proper class.<ref>Nitta,Okada,Tsouvaras (2003)</ref>
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| It is worth emphasizing that hyperset theory is an extension of classical set theory rather than a replacement: the well-founded sets within a hyperset domain conform to classical set theory.
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| == Applications ==
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| {{expand section|date=November 2012}}
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| Aczel’s hypersets were extensively used by [[Jon Barwise]] and [[John Etchemendy]] in their 1987 book ''The Liar'', on the [[liar's paradox|liar’s paradox]]; The book is also good introduction to the topic of non-well-founded sets.
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| Boffa’s superuniversality axiom has found application as a basis for axiomatic [[nonstandard analysis]].{{sfnp|Kanovei|Reeken|2004|p=303}}
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| == See also ==
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| * [[Alternative set theory]]
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| * [[Universal set]]
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| * [[Turtles all the way down]]
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| == Notes ==
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| {{reflist|30em}}
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| == References ==
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| *{{citation |last=Aczel|first= Peter |title=Non-well-founded sets |series= CSLI Lecture Notes |volume=14 |publisher= Stanford University, Center for the Study of Language and Information |place=Stanford, CA |year=1988 |pages=xx+137 | isbn=0-937073-22-9 |url=http://standish.stanford.edu/pdf/00000056.pdf|postscript=. |mr=0940014}}
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| *{{citation |first1=David |last1=Ballard |first2=Karel |last2=Hrbáček |title=Standard foundations for nonstandard analysis |journal=Journal of Symbolic Logic |volume=57 |year=1992 |pages=741–748 |postscript=. |jstor=2275304 |issue=2|doi=10.2307/2275304}}
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| *{{citation |first=Michael |last=Hallett |title=Cantorian set theory and limitation of size |publisher=Oxford University Press |year=1986 |postscript=.}}
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| *{{citation |first=Azriel |last=Levy |title=Basic set theory |publisher=Dover Publications |year=2002 |postscript=.}}
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| * Finsler, P., Über die Grundlagen der Mengenlehre, I. Math. Zeitschrift, 25 (1926), 683–713; translation in {{cite book |last1=Finsler |first=Paul |last2=Booth |first=David |title=Finsler Set Theory: Platonism and Circularity : Translation of Paul Finsler's Papers on Set Theory with Introductory Comments |year=1996 |publisher=Springer |isbn=978-3-7643-5400-8}}
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| * Boffa. M., "Les enesembles extraordinaires." Bulletin de la Societe Mathematique de Belgique. XX:3–15, 1968
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| * Boffa, M., Forcing et négation de l’axiome de Fondement, Memoire Acad. Sci. Belg. tome XL, fasc. 7, (1972).
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| * Scott, Dana. "A different kind of model for set theory." Unpublished paper, talk given at the 1960 Stanford Congress of Logic, Methodology and Philosophy of Science. 1960.
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| *{{citation |last1=Mirimanoff |first1=D. |title=Les antinomies de Russell et de Burali-Forti et le probleme fondamental de la theorie des ensembles | url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1917:19::9&id=hitlist |year=1917 |journal=L’Enseignement Mathématique |volume=19 |pages=37–52 |postscript=.}}
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| *{{citation|last1=Nitta|last2=Okada|last3=Tzouvaras|title=Classification of non-well-founded sets and an application|url=http://users.auth.gr/~tzouvara/Texfiles.htm/non-well.pdf|year=2003}}
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| *{{citation|editor1-first=Godehard |editor1-last=Link|title=One Hundred Years of Russell ́s Paradox: Mathematics, Logic, Philosophy|year=2004|publisher=Walter de Gruyter|isbn=978-3-11-019968-0| chapter =Predicativity, Circularity, and Anti-Foundation | author = M. Rathjen| url=http://www1.maths.leeds.ac.uk/~rathjen/russelle.pdf}}
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| *{{citation |last1=Pakkan |first1=M. J. |last2=Akman |first2=V. |author2-link=Varol Akman |doi=10.1007/BF00849061 |title=Issues in commonsense set theory |journal=Artificial Intelligence Review |volume=8 |issue=4 |pages=279–308 |year=1994–1995}}
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| *{{citation |first1=Jon |last1=Barwise |first2=Lawrence S. |last2=Moss |title=Vicious circles. On the mathematics of non-wellfounded phenomena |series=CSLI Lecture Notes |volume=60 |publisher=CSLI Publications |year=1996 |isbn=1575860090 }}
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| *{{citation |first=Davide |last=Sangiorgi | year=2011 |chapter=Origins of bisimulation and coinduction | editor1-first = Davide | editor1-last = Sangiorgi | editor2-first = Jan |editor2-last=Rutten |title=Advanced Topics in Bisimulation and Coinduction |publisher=Cambridge University Press| isbn=9781107004979}}
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| * {{citation| last1=Kanovei |first1=Vladimir |author1link=[[Vladimir Kanovei]] |last2=Reeken |first2=Michael |title=Nonstandard Analysis, Axiomatically|year=2004 |publisher=Springer |isbn=978-3-540-22243-9}}
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| * {{citation |last1=Barwise |first1=Jon |last2=Etchemendy |first2=John |year=1987 |title=The Liar |publisher=Oxford University Press}}
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| * {{citation |last=Devlin |first=Keith Devlin |title=The Joy of Sets: Fundamentals of Contemporary Set Theory |year=1993 |publisher=Springer |isbn=978-0-387-94094-6edition=2nd}}, §7. Non-Well-Founded Set Theory
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| == Further reading ==
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| *{{cite web |last=Moss |first=Lawrence S. |url=http://plato.stanford.edu/entries/nonwellfounded-set-theory/ | title=Non-wellfounded Set Theory |work=Stanford Encyclopedia of Philosophy |separator=, |postscript=}}
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| == External links ==
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| *[[Metamath]] page on the [http://us.metamath.org/mpegif/axreg.html axiom of Regularity.] Scroll to the bottom to see how few Metamath theorems invoke this axiom.
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| [[Category:Systems of set theory]]
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| [[Category:Wellfoundedness]]
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| [[Category:Self-reference]]
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Royal Votaw is my name but I by no means truly liked that name. Climbing is what love doing. Her family lives in Delaware but she requirements to move because of her family. Meter studying is exactly where my primary earnings arrives from but quickly I'll be on my personal.
Here is my blog post; extended auto warranty