Editing Axiom of regularity

{{short description|Axiom of set theory}}
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 [[Empty set|non-empty]] [[Set (mathematics)|set]] ''A'' contains an element that is [[Disjoint sets|disjoint]] from ''A''. In [[first-order logic]], the axiom reads:

<math display="block">\forall x\,(x \neq \varnothing \rightarrow (\exists y \in x) (y \cap x = \varnothing)).</math>

The axiom of regularity together with the [[axiom of pairing]] implies that [[Russell paradox|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, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.

The axiom was originally formulated by von Neumann;{{sfn|von Neumann|1925}} it was adopted in a formulation closer to the one found in contemporary textbooks by Zermelo.{{sfn|Zermelo|1930}} Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity.{{sfn|Kunen|1980|loc=ch. 3}} 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 display="inline">\{ (n, \alpha) \mid n \in \omega \land \alpha \text{ is an ordinal } \} \,.</math>

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.

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.

==Elementary implications of regularity==

===No set is an element of itself===
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 <math display="inline">A \cap \{A\} = \varnothing</math>, we cannot have ''A''  an element of ''A'' (by the definition of [[Disjoint sets|disjoint]]).

===No infinite descending sequence of sets exists===
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 (set theory)|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''.

The nonexistence of a set containing itself can be seen as a special case where the sequence is infinite and constant.

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.{{dubious|date=February 2023|reason=They satisfy the first-order Peano axioms, so it seems dubious to claim that they are not actually natural numbers. They presumably do not satisfy the second-order Peano axioms with respect to the subset relation of the "ambient" set theory inside of which the model is constructed. But don't they actually satisfy the second-order Peano axioms with respect to the internal subset relation of the model?}} They are "fake" natural numbers which are "larger" than any actual natural number. This model will contain infinite descending sequences of elements.{{clarification needed|date=February 2023|reason=Is the set membership relation in this infinite descending chain the "internal" set membership relation of the model? (I.e. the model's interpretation of the set membership relation?) Or is what follows referring to the set membership relation of the "ambient" set theory in which the model is constructed? Presumably it can't be the latter, because the fact that the latter has a von Neumann cumulative hierarchy, e.g. V_omega, seems to presuppose that it satisfies regularity, and thus otherwise this section would be describing a contradiction. If so, then this section ideally would clarify that what follows refers to the model's interpretation of the set membership relation, and that this is necessarily distinct from (in particular not the restriction of) the ambient set theory's set membership relation.}} For example, suppose ''n'' is a non-standard natural number, then <math display="inline">(n-1) \in n</math> and <math display="inline">(n-2) \in (n-1)</math>, and so on. For any actual natural number ''k'', <math display="inline">(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.

===Simpler set-theoretic definition of the ordered pair===
The axiom of regularity enables defining the ordered pair (''a'',''b'') as {''a'',{''a'',''b''}}; see [[ordered pair]] for specifics. This definition eliminates one pair of braces from the canonical [[Kuratowski]] definition (''a'',''b'') = <nowiki>{{</nowiki>''a''},{''a'',''b''}}.

=== Every set has an ordinal rank ===
This was actually the original form of the axiom in von Neumann's axiomatization.

Suppose ''x'' is any set. Let ''t'' be the [[transitive closure (set)|transitive closure]] of {''x''}. Let ''u'' be the subset of ''t'' consisting of unranked sets. If ''u'' is empty, then ''x'' is ranked and we are done. Otherwise, apply the axiom of regularity to ''u'' to get an element ''w'' of ''u'' which is disjoint from ''u''. Since ''w'' is in ''u'', ''w'' is unranked. ''w'' is a subset of ''t'' by the definition of transitive closure. Since ''w'' is disjoint from ''u'', every element of ''w'' is ranked. Applying the axioms of replacement and union to combine the ranks of the elements of ''w'', we get an ordinal rank for ''w'', to wit <math display="inline">\textstyle \operatorname{rank} (w) = \cup \{ \operatorname{rank} (z) + 1 \mid z \in w \}</math>. This contradicts the conclusion that ''w'' is unranked. So the assumption that ''u'' was non-empty must be false and ''x'' must have rank.

=== For every two sets, only one can be an element of the other ===
Let ''X'' and ''Y'' be sets. Then apply the axiom of regularity to the set {''X'',''Y''} (which exists by the axiom of pairing). 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.

==The axiom of dependent choice and no infinite descending sequence of sets implies regularity==
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 display="inline">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.

== Regularity and the rest of ZF(C) axioms ==
Regularity was shown to be relatively consistent with the rest of ZF by Skolem{{sfn|Skolem|1923}} and von Neumann,{{sfn|von Neumann|1929}} meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent.<ref>For his{{ambiguous|reason=Skolem or Von Neumann?|date=December 2024}} proof in modern notation, see {{harvtxt|Vaught|2001|loc=§10.1}} for instance.</ref>

The axiom of regularity was also shown to be [[Independence (mathematical logic)|independent]] from the other axioms of ZFC, 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 model]]s (or method), which were used for other proofs of independence for non-well-founded systems.{{sfn|Rathjen|2004|p=193}}{{sfn|Forster|2003|pp=210–212}}

== Regularity and Russell's paradox ==
[[Naive set theory]] (the axiom schema of [[unrestricted comprehension]] and the [[axiom of extensionality]]) is inconsistent due to [[Russell's paradox]]. In early formalizations of sets, mathematicians and logicians have avoided that contradiction by replacing the axiom schema of comprehension with the much weaker [[axiom schema of separation]]. However, this step alone takes one to theories of sets which are considered too weak.{{clarification needed|date=January 2023|reason=This seems to have in mind a specific result or interpretation, however what that might be is not stated. Ideally that would be given, along with a citation of at least one reference stating/explaining the corresponding result/interpretation.}}{{citation needed|date=January 2023}} 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.{{citation needed|date=January 2023}}{{clarification needed|date=January 2023|reason=Interpreting this literally, if these axioms were all special cases of the particular comprehension axiom, then adding them back would neither strengthen nor weaken the theory. So clearly something slightly different is what the author had in mind, and expressed it this way heuristically. Fine. But a reference to a more precise explanation/result could still be helpful.}} 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.

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 together with the axiom of pairing also prohibit such a universal set. However, Russell's paradox yields a proof that there is no "set of all sets" using the axiom schema of separation alone, without any additional axioms. In particular, ZF without the axiom of regularity already prohibits such a universal set.

If a theory is extended by adding an axiom or axioms, then any (possibly undesirable) consequences of the original theory remain consequences of the extended theory. In particular, if ZF without regularity is extended by adding regularity to get ZF, then any contradiction (such as Russell's paradox) which followed from the original theory would still follow in the extended theory.

The existence of [[Quine atom]]s (sets that satisfy the formula equation ''x''&nbsp;=&nbsp;{''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.{{sfn|Rieger|2011|pp=175,178}}

== Regularity, the cumulative hierarchy, and types ==
In ZF it can be proven that the class <math display="inline"> \bigcup_{\alpha} V_\alpha </math>, called the [[von Neumann universe]], 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 the axiom of regularity, a model which satisfies it can be constructed by taking only sets in <math display="inline"> \bigcup_{\alpha} V_\alpha </math>.

Herbert Enderton{{sfn|Enderton|1977|loc=p. 206}} 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.<ref>The details of this implicit typing are spelled out in {{harvnb|Zermelo|1930}}, and again in {{harvnb|Boolos|1971}}.</ref>{{sfn|Urquhart|2003|p=305}}

Dana Scott{{sfn|Scott|1974}} went further and claimed that:

{{Blockquote|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&mdash;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.{{sfn|Lévy|2002|p=73}}

== History ==

The concept of well-foundedness and [[Von Neumann universe|rank]] of a set were both introduced by [[Dmitry Mirimanoff]].{{sfn|Mirimanoff|1917}}<ref>cf. {{harvnb|Lévy|2002|p=68}} and {{harvnb|Hallett|1996|loc=§4.4, esp. p. 186, 188}}.</ref> Mirimanoff called a set ''x'' "regular" ({{langx|fr|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;{{sfn|Halbeisen|2012|pp=62–63}} in later papers Mirimanoff also explored what are now called [[non-well-founded set theory|non-well-founded sets]] ({{lang|fr|extraordinaire}} in Mirimanoff's terminology).{{sfn|Sangiorgi|2011|pp=17–19, 26}}

Skolem{{sfn|Skolem|1923}} and von Neumann{{sfn|von Neumann|1925}} pointed out that non-well-founded sets are superfluous{{sfn|van Heijenoort|1967|p=404}} and in the same publication von Neumann gives an axiom{{sfn|van Heijenoort|1967|p=412}} which excludes some, but not all, non-well-founded sets.{{sfn|Rieger|2011|p=179}} In a subsequent publication, von Neumann{{sfn|von Neumann|1929|p=231}} gave an equivalent but more complex version of the axiom of class foundation:<ref>cf. {{harvnb|Suppes|1972|p=53}} and {{harvnb|Lévy|2002|p=72}}</ref>

<math display="block"> A \neq \emptyset \rightarrow \exists x \in A\,(x \cap A = \emptyset).</math>

The contemporary and final form of the axiom is due to Zermelo.{{sfn|Zermelo|1930}}

== Regularity in the presence of urelements ==

[[Urelements]] are objects that are not sets, but which can be elements of sets. In ZF set theory, there are no urelements, but in some other set theories such as [[Urelement#Urelements in set theory|ZFA]], there are. In these theories, the axiom of regularity must be modified. The statement "<math display="inline">x \neq \emptyset</math>" needs to be replaced with a statement that <math display="inline">x</math> is not empty and is not an urelement. One suitable replacement is <math display="inline">(\exists y)[y \in x]</math>, which states that ''x'' is [[inhabited set|inhabited]].

==See also==
*[[Non-well-founded set theory]]
*[[Scott's trick]]
*[[Epsilon-induction]]

== References ==
<references />

==Sources==
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* {{cite journal | first= Paul Isaac|last = Bernays|author-link=Paul Bernays| title= A system of axiomatic set theory. Part VII |journal = The Journal of Symbolic Logic| volume = 19 |issue = 2| year = 1954 | pages = 81–96 | doi=10.2307/2268864 | jstor=2268864| s2cid=250351655 | url = http://doc.rero.ch/record/301843/files/S0022481200087570.pdf}}
*{{cite journal|last=Boolos|first= George |author-link=George Boolos | year= 1971 | title = The iterative conception of set | journal = Journal of Philosophy | volume = 68 |issue= 8 |pages= 215–231 | doi=10.2307/2025204 | jstor=2025204}} Reprinted in {{cite book|last=Boolos|first= George |year=1998|title=Logic, Logic and Logic|pages=13–29|publisher=Harvard University Press}}
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* {{cite book
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*{{cite book| 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}}
* {{cite book | last = Scott | first = Dana Stewart | author-link=Dana Scott | year = 1974 | chapter = Axiomatizing set theory | title = Axiomatic set theory. Proceedings of Symposia in Pure Mathematics |volume=13 |at=Part II, pp. 207–214}}
* {{cite book | last=Skolem| first=Thoralf|author-link=Thoralf Skolem | year=1923|title=Axiomatized set theory}} Reprinted in ''From Frege to Gödel'', van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp.&nbsp;291–301.
* {{cite book|first=Patrick|last=Suppes|author-link=Patrick Suppes|title=Axiomatic Set Theory|publisher=Dover |year=1972|orig-year=first published 1960|isbn=978-0-486-61630-8}}
*{{cite book|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}}
* {{cite book|first=Robert L. |last = Vaught|title=Set Theory: An Introduction| year=2001| publisher=Springer| isbn=978-0-8176-4256-3| edition=2nd}}
*{{cite journal|last=von Neumann|first = John| author-link=John von Neumann|year=1925|title=Eine Axiomatisierung der Mengenlehre|language=de|journal=Journal für die Reine und Angewandte Mathematik|volume=154|pages=219–240}} Translation in {{cite book|last=van Heijenoort | first =Jean | year =1967 | title = From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 | pages = 393–413 }}
*{{cite journal|last = von Neumann|first = John| author-link=John von Neumann |year= 1928|title= Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre|language=de| journal= Mathematische Annalen|volume = 99 |pages=373–391|doi=10.1007/BF01459102|s2cid = 120784562}}
*{{cite journal| last = von Neumann |first = John| author-link=John von Neumann| year = 1929 | title= Über eine Widerspruchfreiheitsfrage in der axiomatischen Mengenlehre|language=de| journal = Journal für die Reine und Angewandte Mathematik |volume = 1929 |issue = 160|pages = 227–241 | doi=10.1515/crll.1929.160.227|s2cid = 199545822}}
*{{cite journal |last=Zermelo|first= Ernst |author-link=Ernst Zermelo | year= 1930 | title = Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre |language=de| journal = Fundamenta Mathematicae | volume = 16 |pages= 29–47|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://matwbn.icm.edu.pl/ksiazki/fm/fm16/fm1615.pdf |archive-date=2022-10-09 |url-status=live|doi= 10.4064/fm-16-1-29-47 |doi-access= free }} Translation in {{cite book | editor-last= Ewald |editor-first= W. B. | year = 1996| title = From Kant to Hilbert: A Source Book in the Foundations of Mathematics |volume=2 | publisher= Clarendon Press |pages = 1219–1233}}

==External links==
*{{PlanetMath|urlname=axiomoffoundation|title=Axiom of foundation}}
*[https://ncatlab.org/nlab/show/inhabited+set Inhabited set] and [https://ncatlab.org/nlab/show/axiom+of+foundation the axiom of foundation] on nLab

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[[Category:Axioms of set theory]]
[[Category:Wellfoundedness]]