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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 In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such a ...
that states that every non-empty
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
''A'' contains an element that is disjoint from ''A''. In
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...
, the axiom reads: : \forall x\,(x \neq \varnothing \rightarrow \exists y(y \in x\ \land y \cap x = \varnothing)). The axiom of regularity together with the
axiom of pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary set ...
implies that no set is an element of itself, and that there is no infinite sequence (''an'') such that ''ai+1'' is an element of ''ai'' for all ''i''. With the
axiom of dependent choice In mathematics, the axiom of dependent choice, denoted by \mathsf , is a weak form of the axiom of choice ( \mathsf ) that is still sufficient to develop most of real analysis. It was introduced by Paul Bernays in a 1942 article that explores whi ...
(which is a weakened form of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
), 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 introduced by ; it was adopted in a formulation closer to the one found in contemporary textbooks by . Virtually all results in the branches of mathematics based on set theory hold even in the absence of regularity; see chapter 3 of . However, regularity makes some properties of ordinals easier to prove; and it not only allows induction to be done on well-ordered sets but also on proper classes that are well-founded relational structures such as the lexicographical ordering on \ \,. Given the other axioms of Zermelo–Fraenkel set theory, the axiom of regularity is equivalent to the
axiom of induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
. The axiom of induction tends to be used in place of the axiom of regularity in
intuitionistic In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of f ...
theories (ones that do not accept the
law of the excluded middle In logic, the law of excluded middle (or the principle of excluded middle) states that for every proposition, either this proposition or its negation is true. It is one of the so-called three laws of thought, along with the law of noncontradi ...
), where the two axioms are not equivalent. In addition to omitting the axiom of regularity, 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 , which is a set by the
axiom of pairing In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary set ...
. We see that there must be an element of which is disjoint from . Since the only element of is ''A'', it must be that ''A'' is disjoint from . So, since A \cap \ = \varnothing, we cannot have ''A'' ∈ ''A'' (by the definition of disjoint).


No infinite descending sequence of sets exists

Suppose, to the contrary, that there is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
, ''f'', on the natural numbers with ''f''(''n''+1) an element of ''f''(''n'') for each ''n''. Define ''S'' = , the range of ''f'', which can be seen to be a set from the
axiom schema of replacement In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
. 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''. 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 In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to ...
s, Vω, satisfy the axiom of regularity (and all other axioms of ZFC except the
axiom of infinity In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing th ...
). So if one forms a non-trivial ultrapower of Vω, then it will also satisfy the axiom of regularity. The resulting
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
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 (n-1) \in n and (n-2) \in (n-1), and so on. For any actual natural number ''k'', (n-k-1) \in (n-k). 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 ; see ordered pair for specifics. This definition eliminates one pair of braces from the canonical Kuratowski definition (''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 In mathematics, the transitive closure of a binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinit ...
of . 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 \textstyle \operatorname (w) = \cup \. 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 (which exists by the axiom of pairing). We see there must be an element of 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 aRb :\Leftrightarrow b \in S \cap a, which is entire by assumption. Thus, by the axiom of dependent choice, there is some sequence (''an'') in ''S'' satisfying ''anRan+1'' 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 and , meaning that if ZF without regularity is consistent, then ZF (with regularity) is also consistent. For his proof in modern notation see for instance. The axiom of regularity was also shown to be 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 ( and ).


Regularity and Russell's paradox

Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It des ...
(the axiom schema of unrestricted comprehension and the
axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same eleme ...
) 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 In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any ...
. However, this step alone takes one to theories of sets which are considered 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. In the presence of the axiom schema of separation, Russell's paradox becomes a proof that there is no
set of all sets In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
. 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 In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory The ...
s (sets that satisfy the formula equation ''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-wellfounded set theories allow "safe" circular sets, such as Quine atoms, without becoming inconsistent by means of Russell's paradox.


Regularity, the cumulative hierarchy, and types

In ZF it can be proven that the class \bigcup_ V_\alpha , 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 axiom of regularity, a model which satisfies it can be constructed by taking only sets in \bigcup_ V_\alpha . wrote that "The idea of rank is a descendant of Russell's concept of ''type''". Comparing ZF with type theory,
Alasdair Urquhart Alasdair Ian Fenton Urquhart (; born 20 December 1945) is a Scottish–Canadian philosopher and emeritus professor of philosophy at the University of Toronto. He has made contributions to the field of logic, especially non-classical logic. On ...
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 ermelo 1930 and again in a well-known article of George Boolos oolos 1971" went further and claimed that: 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.


History

The concept of well-foundedness and
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * H ...
of a set were both introduced by Dmitry Mirimanoff (
1917 Events Below, the events of World War I have the "WWI" prefix. January * January 9 – WWI – Battle of Rafa: The last substantial Ottoman Army garrison on the Sinai Peninsula is captured by the Egyptian Expeditionary Force's ...
) cf. and . Mirimanoff called a set ''x'' "regular" (French: "ordinaire") if every descending chain ''x'' ∋ ''x''1 ∋ ''x''2 ∋ ... is finite. Mirimanoff however did not consider his notion of regularity (and well-foundedness) as an axiom to be observed by all sets; in later papers Mirimanoff also explored what are now called non-well-founded sets ("extraordinaire" in Mirimanoff's terminology). and pointed out that non-well-founded sets are superfluous (on p. 404 in 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. In a subsequent publication, gave the following axiom (rendered in modern notation by A. Rieger): : \forall x\,(x \neq \emptyset \rightarrow \exists y \in x\,(y \cap x = \emptyset)).


Regularity in the presence of urelements

Urelements In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory Th ...
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 ZFA, there are. In these theories, the axiom of regularity must be modified. The statement "x \neq \emptyset" needs to be replaced with a statement that x is not empty and is not an urelement. One suitable replacement is (\exists y) \in x/math>, which states that ''x'' is inhabited.


See also

* Non-well-founded set theory * Scott's trick * Epsilon-induction


References


Sources

* * * reprinted in * * * * * * * * * * * * * * Reprinted in ''From Frege to Gödel'', van Heijenoort, 1967, in English translation by Stefan Bauer-Mengelberg, pp. 291–301. * * *; translation in * * *; translation in


External links

*
Inhabited set
an
the axiom of foundation
on nLab {{DEFAULTSORT:Axiom Of Regularity Axioms of set theory Wellfoundedness