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| In mathematics, the '''axiom of regularity''' (also known as the '''axiom of foundation''') is an axiom of [[Zermelo–Fraenkel set theory]] that states that every non-empty [[Set (mathematics)|set]] ''A'' contains an element that is disjoint from ''A''. In [[first-order logic]] the axiom reads:
| | Many folks are seeking an efficient hemorrhoid treatment because they wish to get rid of their hemorrhoids as soon because possible. They do not wish To see any blood on their toilet papers anymore, plus they do not like to have any pain during constipation anymore either. This is the reason why they are struggling so difficult in purchase to eliminate hemorrhoid in a short period of time.<br><br>If you have decided which lotions are the [http://hemorrhoidtreatmentfix.com/external-hemorrhoids external hemorrhoids] you would like then you would find which they are found inside countless drug stores and pharmacies, and that are fairly inexpensive.<br><br>And then there was the occasional bleeding. When whenever I removed my light colored pants at the end of the day, I noticed a little blood stain place showing found on the outside. How embarrassing! I question how various people saw which plus were too polite to mention anything!<br><br>Having a advantageous bowel movement etiquette is without question 1 of the ideal ways to prevent hemorrhoids. Everytime we really feel the urge, do thus without hindrance. Numerous delays of going to the rest room brings about constipation which causes we to strain hard inside purchase to eliminate the bowels. As quickly because you're finished, instantly receive up. Staying too lengthy found on the toilet seat can amplify stress on the veins. After a bowel movement, we would additionally try sitting inside a tub with lukewarm water for 10 to 15 minutes to relax the rectum.<br><br>Another tip which would offer some immediate relief to the hemorrhoid problem is chatting a good warm bath. The bath will sooth the pain you feel, plus should you add a little salt into your shower, regarding a teaspoon or so, plus massage the hemorrhoid this may even further enable with pain relief.<br><br>So, because it turns out there are a great deal of options. I can't remember all the fancy names, but I remember rubber bands, lasers, plus surgery with REAL scalpels being revealed. Yes, the whole range from medieval torture instruments to Star Wars weapons. And they wanted to employ THOSE elements - inside THAT region!<br><br>Piles could change every aspect of the life, from how commonly we take a seat plus curl up to how much food we eat plus for many; it may feel like you are able to not do something to create a hemorrhoids disappear. While various individuals would have the periodic flare up, a lot of folks have piles that have progressed into a long-term condition. As an example, several fresh mothers suffer from hemorrhoids from labour. The issue is the fact that based on the strength, you may end up suffering with this issue again, and again plus again. |
| : <math>\forall x\,(x \neq \varnothing \rightarrow \exist y \in x\,(y \cap x = \varnothing))</math>.
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| The axiom implies that no set is an element of itself, and that there is no infinite [[sequence]] (''a<sub>n</sub>'') such that ''a<sub>i+1</sub>'' is an element of ''a<sub>i</sub>'' for all ''i''. With the [[axiom of dependent choice]] (which is a weakened form of the [[axiom of choice]]), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.
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| The axiom of regularity was introduced by {{harvtxt|von Neumann|1925}}; it was adopted in a formulation closer to the one found in contemporary textbooks by {{harvtxt|Zermelo|1930}}. Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of {{harvtxt|Kunen|1980}}. However, regularity makes some properties of [[Ordinal number|ordinals]] easier to prove; and it not only allows induction to be done on [[well-ordering|well-ordered sets]] but also on proper classes that are [[well-founded relation|well-founded relational structures]] such as the [[lexicographical ordering]] on <math>\{ (n, \alpha) \vert n \in \omega \land \alpha \text{ is an ordinal } \} \,.</math>
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| Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the [[epsilon-induction|axiom of induction]]. The axiom of induction tends to be used in place of the axiom of regularity in [[intuitionism|intuitionistic]] theories (ones that do not accept the [[law of the excluded middle]]), where the two axioms are not equivalent.
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| In addition to omitting the axiom of regularity, [[Non-well-founded set theory|non-standard set theories]] have indeed postulated the existence of sets that are elements of themselves.
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| ==Elementary implications of regularity==
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| ===No set is an element of itself===
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| Let ''A'' be a set, and apply the axiom of regularity to {''A''}, which is a set by the [[axiom of pairing]]. We see that there must be an element of {''A''} which is disjoint from {''A''}. Since the only element of {''A''} is ''A'', it must be that ''A'' is disjoint from {''A''}. So, since ''A'' ∈ {''A''}, we cannot have ''A'' ∈ ''A'' (by the definition of [[Disjoint sets|disjoint]]).
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| ===No infinite descending sequence of sets exists===
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| Suppose, to the contrary, that there is a [[function (mathematics)|function]], ''f'', on the [[natural number]]s with ''f''(''n''+1) an element of ''f''(''n'') for each ''n''. Define ''S'' = {''f''(''n''): ''n'' a natural number}, the range of ''f'', which can be seen to be a set from the [[axiom schema of replacement]]. Applying the axiom of regularity to ''S'', let ''B'' be an element of ''S'' which is disjoint from ''S''. By the definition of ''S'', ''B'' must be ''f''(''k'') for some natural number ''k''. However, we are given that ''f''(''k'') contains ''f''(''k''+1) which is also an element of ''S''. So ''f''(''k''+1) is in the intersection of ''f''(''k'') and ''S''. This contradicts the fact that they are disjoint sets. Since our supposition led to a contradiction, there must not be any such function, ''f''.
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| The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.
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| Notice that this argument only applies to functions ''f'' that can be represented as sets as opposed to undefinable classes. The [[hereditarily finite set]]s, V<sub>ω</sub>, satisfy the axiom of regularity (and all other axioms of [[ZFC]] except the [[axiom of infinity]]). So if one forms a non-trivial [[ultraproduct|ultrapower]] of V<sub>ω</sub>, then it will also satisfy the axiom of regularity. The resulting [[model (logic)|model]] <!--WHAT model?--> will contain elements, called non-standard natural numbers, that satisfy the definition of natural numbers in that model but are not really natural numbers. They are fake natural numbers which are "larger" than any actual natural number. This model will contain infinite descending sequences of elements. For example, suppose ''n'' is a non-standard natural number, then <math>(n-1) \in n</math> and <math>(n-2) \in (n-1)</math>, and so on. For any actual natural number ''k'', <math>(n-k-1) \in (n-k)</math>. This is an unending descending sequence of elements. But this sequence is not definable in the model and thus not a set. So no contradiction to regularity can be proved.
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| ===Simpler set-theoretic definition of the ordered pair===
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| The axiom of regularity enables defining the ordered pair (''a'',''b'') as <nowiki>{</nowiki>''a'',<nowiki>{</nowiki>''a'',''b''<nowiki>}}</nowiki>. See [[ordered pair]] for specifics. This definition eliminates one pair of braces from the canonical [[Kuratowski]] definition (''a'',''b'') = <nowiki>{{</nowiki>''a''<nowiki>}</nowiki>,<nowiki>{</nowiki>''a'',''b''<nowiki>}}</nowiki>.
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| === Every set has an ordinal rank ===
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| This was actually the original form of von Neumann's axiomatization.
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| === For every two sets, only one can be an element of the other ===
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| Let ''X'' and ''Y'' be sets. Then apply the axiom of regularity to the set {''X'',''Y''}. We see there must be an element of {''X'',''Y''} which is also disjoint from it. It must be either ''X'' or ''Y''. By the definition of disjoint then, we must have either ''Y'' is not an element of ''X'' or vice versa.
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| ==The axiom of dependent choice and no infinite descending sequence of sets implies regularity==
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| Let the non-empty set ''S'' be a counter-example to the axiom of regularity; that is, every element of ''S'' has a non-empty intersection with ''S''. We define a binary relation ''R'' on ''S'' by <math>aRb :\Leftrightarrow b \in S \cap a</math>, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (''a<sub>n</sub>'') in ''S'' satisfying ''a<sub>n</sub>Ra<sub>n+1</sub>'' for all ''n'' in '''N'''. As this is an infinite descending chain, we arrive at a contradiction and so, no such ''S'' exists.
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| == Regularity and the rest of ZF(C) axioms ==
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| Regularity was shown to be relatively consistent with the rest of ZF by {{harvtxt|von Neumann|1929}}, meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation see {{harvtxt|Vaught|2001|loc=§10.1}} for instance.
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| The axiom of regularity was also shown to be [[Independence (mathematical logic)|independent]] from the other axioms of ZF(C), assuming they are consistent. The result was announced by [[Paul Bernays]] in 1941, although he did not publish a proof until 1954. The proof involves (and led to the study of) [[Rieger-Bernays permutation]] models (or method), which were used for other proofs of independence for non-well-founded systems ({{harvnb|Rathjen|2004|p=193}} and {{harvnb|Forster|2003|pp=210–212}}).
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| == Regularity and Russell's paradox ==
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| [[Naive set theory]] (the axiom schema of [[unrestricted comprehension]] and the [[axiom of extensionality]]) is inconsistent due to [[Russell's paradox]]. Set theorists have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker [[axiom schema of separation]]. However, this makes set theory too weak. So some of the power of comprehension was added back via the other existence axioms of ZF set theory (pairing, union, powerset, replacement, and infinity) which may be regarded as special cases of comprehension. So far, these axioms do not seem to lead to any contradiction. Subsequently, the axiom of choice and the axiom of regularity were added to exclude models with some undesirable properties. These two axioms are known to be relatively consistent.
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| In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no [[universal set|set of all sets]]. The axiom of regularity (with the axiom of pairing) also prohibits such a universal set, however this prohibition is redundant when added to the rest of ZF. If the ZF axioms without regularity were already inconsistent, then adding regularity would not make them consistent.
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| The existence of [[Quine atom]]s (sets that satisfy the formula equation ''x'' = {''x''}, i.e. have themselves as their only elements) is consistent with the theory obtained by removing the axiom of regularity from ZFC. Various [[non-well-founded set theory|non-wellfounded set theories]] allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.{{harv|Rieger|2011|pp=175,178}} | |
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| == Regularity, the cumulative hierarchy, and types ==
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| In ZF it can be proven that the class <math> \bigcup_{\alpha} V_\alpha \! </math> (see [[cumulative hierarchy]]) is equal to the class of all sets. This statement is even equivalent to the axiom of regularity (if we work in ZF with this axiom omitted). From any model which does not satisfy axiom of regularity, a model which satisfies it can be constructed by taking only sets in <math> \bigcup_{\alpha} V_\alpha \! </math>.
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| {{harvs|txt|last=Enderton|first=Herbert|year=1977|loc=p. 206|author-link=Herbert Enderton}} wrote that "The idea of rank is a descendant of Russell's concept of ''type''". Comparing ZF with [[type theory]], [[Alasdair Urquhart]] wrote that "Zermelo’s system has the notational advantage of not containing any explicitly typed variables, although in fact it can be seen as having an implicit type structure built into it, at least if the axiom of regularity is included. The details of this implicit typing are spelled out in [[#{{harvid|Zermelo|1930}}|[Zermelo 1930]]], and again in a well-known article of [[George Boolos]] [[#{{harvid|Boolos|1971}}|[Boolos 1971]]]." {{harvtxt|Urquhart|2003|p=305}}
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| {{harvs|txt|last=Scott|first=Dana|year=1974|authorlink=Dana Scott}} went further an claimed that: "The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the ''theory of types''. That was at the basis of both Russell's and Zermelo's intuitions. Indeed the best way to regard Zermelo's theory is as a simplification and extension of Russell's. (We mean Russell's ''simple'' theory of types, of course.) The simplification was to make the types ''cumulative''. Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagine ''extending'' the types into the transfinite—just how far we want to go must necessarily be left open. Now Russell made his types ''explicit'' in his notation and Zermelo left them ''implicit''" (emphasis in original). In the same paper, Scott shows that an axiomatic system based on the inherent properties of the cumulative hierarchy turns out to be equivalent to ZF, including regularity. {{harv|Lévy|2002|p=73}}
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| == History ==
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| {{expand section|date=November 2012}}
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| The concept of well-foundedness and rank of a set were both introduced by [[Dmitry Mirimanoff]] ([[#{{harvid|Mirimanoff|1917}}|1917]]) cf. {{harvtxt|Lévy|2002|p=68}} and {{harvtxt|Hallett|1986|loc=§4.4, esp. p. 186, 188}}. Mirimanoff called a set ''x'' "regular" (French: "ordinaire") if every descending chain ''x'' ∋ ''x<sub>1</sub>'' ∋ ''x<sub>2</sub>'' ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets {{harv|Halbeisen|2012|pp=62–63}}; in later papers Mirimanoff also explored what are now called [[non-well-founded set theory|non-well-founded sets]] ("extraordinaire" in Mirimanoff's terminology) {{harv|Sangiorgi|2011|pp=17–19, 26}}.
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| According to Adam Rieger, {{harvtxt|von Neumann|1925}} describes non-well-founded sets as "superfluous" (on p. 404 in [[#{{harvid|van Heijenoort|1967}}|van Heijenoort 's translation]]) and in the same publication von Neumann gives an axiom (p. 412 in translation) which excludes some, but not all, non-well-founded sets {{harv|Rieger|2011|p=179}}. In a subsequent publication, {{harvtxt|von Neumann|1928}} gave the following axiom (rendered in modern notation by A. Rieger):
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| : <math>\forall x\,(x \neq \emptyset \rightarrow \exist y \in x\,(y \cap x = \emptyset))</math>.
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| ==See also==
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| *[[Non-well-founded set theory]]
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| == References ==
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| *{{Citation | last=Jech | first= Thomas |year= 2003 |title = Set Theory: The Third Millennium Edition, Revised and Expanded |publisher=Springer| isbn = 3-540-44085-2}}
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| *{{Citation | last=Kunen | first=Kenneth |year = 1980|title = Set Theory: An Introduction to Independence Proofs| publisher=Elsevier| isbn=0-444-86839-9}}
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| *{{citation|last=Boolos|first= George |year= 1971 | title = The iterative conception of set | journal = Journal of Philosophy | volume = 68 |pages= 215–231}} reprinted in {{citation|last=Boolos|first= George |year=1998|title=Logic, Logic and Logic|pages=13–29|publisher=Harvard University Press}}
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| *{{citation | last= Enderton | first = Herbert B. | title = Elements of Set Theory | publisher = Academic Press | year=1977}}
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| *{{citation|last = Urquhart|first = Alasdair | chapter = The Theory of Types | editor-last = Griffin| editor-first =Nicholas | title =The Cambridge Companion to Bertrand Russell | publisher = Cambridge University Press | year=2003}}
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| *{{citation| first= Lorenz J. |last = Halbeisen | title=Combinatorial Set Theory: With a Gentle Introduction to Forcing|year=2012|publisher=Springer}}
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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}}
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| *{{citation| last = Lévy | first =Azriel | isbn = 0486420795 | year =2002 |origyear= first published in 1979 | title=Basic set theory| publisher=Dover Publications}}
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| * {{citation|first=Michael|last=Hallett|title=Cantorian set theory and limitation of size|publisher=Oxford University Press|year=1996|origyear=first published 1984|isbn=0198532830}}
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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 | first = M. | last= Rathjen| url=http://www1.maths.leeds.ac.uk/~rathjen/russelle.pdf}}
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| *{{citation|title = Logic, induction and sets| last = Forster | first = T. | publisher = Cambridge University Press | year = 2003}}
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| * {{Citation
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| | last1 = Rieger | first1 = Adam
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| | chapter = Paradox, ZF, and the Axiom of Foundation
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| | doi = 10.1007/978-94-007-0214-1_9
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| | title = Logic, Mathematics, Philosophy, Vintage Enthusiasms. Essays in Honour of John L. Bell.
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| | editors = David DeVidi, Michael Hallett, Peter Clark
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| | series =
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| | volume =
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| | pages = 171–187
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| | year = 2011
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| | isbn = 978-94-007-0213-4
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| | pmid =
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| | pmc =
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| }}
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| * {{citation|first=Robert L. |last = Vaught|title=Set Theory: An Introduction| year=2001| publisher=Springer| isbn=978-0-8176-4256-3| edition=2nd}}
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| === Primary sources ===
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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}}
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| *{{citation|last=von Neumann|first = J.|year=1925|title=Eine axiomatiserung der Mengenlehre|journal=Journal für die reine und angewandte Mathematik|volume=154|pages=219–240}}; translation in {{citation|last=van Heijenoort | first =Jean | year =1967 | title = From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 | pages = 393–413 }}
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| *{{citation|last = von Neumann|first = J.|year= 1928|title= Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre| journal= Mathematische Annalen|volume = 99 |pages=373–391|doi=10.1007/BF01459102}}
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| *{{citation| last = von Neumann |first = J. | year = 1929 | title= Uber eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre| journal = Journal fur die reine und angewandte Mathematik |volume = 160 |pages = 227–241}}
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| *{{citation |last=Zermelo|first= Ernst |year= 1930 | title = Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre. | journal = Fundamenta Mathematicae | volume = 16 |pages= 29–47|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf}}; translation in {{citation | editor-last= Ewald |editor-first= W.B. | year = 1996| title = From Kant to Hilbert: A Source Book in the Foundations of Mathematics Vol. 2 | publisher= Clarendon Press |pages = 1219–33}}
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| * {{citation |first=P.|last= Bernays| title= A system of axiomatic set theory. Part II |journal= The Journal of Symbolic Logic| volume= 6 | year = 1941 | pages = 1–17}}
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| * {{citation | first= P.|last = Bernays| title= A system of axiomatic set theory. Part VII |journal = The Journal of Symbolic Logic| volume = 19 | year = 1954 | pages = 81–96}}
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| * {{citation | first = L. |last = Riegger | url = http://dml.cz/bitstream/handle/10338.dmlcz/100254/CzechMathJ_07-1957-3_1.pdf | title = A contribution to Gödel's axiomatic set theory | journal = Czechoslovak Mathematical Journal | volume = 7 | year = 1957 | pages = 323–357}}
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| * {{citation | last = Scott | first = D. | year = 1974 | chapter = Axiomatizing set theory | title = Axiomatic set theory. Proceedings of Symposia in Pure Mathematics Volume 13, Part II | pages = 207–214}}
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| ==External links==
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| *http://www.trinity.edu/cbrown/topics_in_logic/sets/sets.html contains an informative description of the axiom of regularity under the section on Zermelo-Fraenkel set theory.
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| *{{planetmath reference|id=3485|title=Axiom of Foundation}}
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| {{Set theory}}
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| {{DEFAULTSORT:Axiom Of Regularity}}
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| [[Category:Axioms of set theory]]
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| [[Category:Wellfoundedness]]
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