In

^{9} gave to the set-theoretic antinomies could have persuaded them . KÃ¶nig, Jourdain, F. Bernsteinthat the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set". Footnote 9 is where he stakes his claim:
Frege sent a copy of his ''Grundgesetze der Arithmetik'' to Hilbert; as noted above, Frege's last volume mentioned the paradox that Russell had communicated to Frege. After receiving Frege's last volume, on 7 November 1903, Hilbert wrote a letter to Frege in which he said, referring to Russell's paradox, "I believe Dr. Zermelo discovered it three or four years ago". A written account of Zermelo's actual argument was discovered in the ''Nachlass'' of

Russell's Paradox

at

Russell's (Barber) paradox explanation in Python

{{DEFAULTSORT:Russell's Paradox Bertrand Russell Paradoxes of naive set theory 1901 in science Self-referential paradoxes

mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...

philosopher
A philosopher is a person who practices or investigates philosophy. The term ''philosopher'' comes from the grc, Ï†Î¹Î»ÏŒÏƒÎ¿Ï†Î¿Ï‚, , translit=philosophos, meaning 'lover of wisdom'. The coining of the term has been attributed to the Greek th ...

and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History ...

Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 â€“ 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...

in 1901. Russell's paradox shows that every set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

that contains an unrestricted comprehension principle leads to contradictions. The paradox had already been discovered independently in 1899 by the German mathematician Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermeloâ€“Fraenkel axiomatic se ...

. However, Zermelo did not publish the idea, which remained known only to David Hilbert
David Hilbert (; ; 23 January 1862 â€“ 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...

, Edmund Husserl
, thesis1_title = BeitrÃ¤ge zur Variationsrechnung (Contributions to the Calculus of Variations)
, thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view
, thesis1_year = 1883
, thesis2_title ...

, and other academics at the University of GÃ¶ttingen
The University of GÃ¶ttingen, officially the Georg August University of GÃ¶ttingen, (german: Georg-August-UniversitÃ¤t GÃ¶ttingen, known informally as Georgia Augusta) is a public research university in the city of GÃ¶ttingen, Germany. Founded ...

. At the end of the 1890s, Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( , ; â€“ January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance o ...

â€“ considered the founder of modern set theory â€“ had already realized that his theory would lead to a contradiction, which he told Hilbert and Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 â€“ 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...

by letter.
According to the unrestricted comprehension principle, for any sufficiently well-defined property
Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, ...

, there is the 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 ...

of all and only the objects that have that property. Let ''R'' be the set of all sets that are not members of themselves. If ''R'' is not a member of itself, then its definition entails that it is a member of itself; if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:
:$\backslash text\; R\; =\; \backslash \; \backslash text\; R\; \backslash in\; R\; \backslash iff\; R\; \backslash not\; \backslash in\; R$
Russell also showed that a version of the paradox could be derived in the axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...

constructed by the German philosopher and mathematician Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 â€“ 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phi ...

, hence undermining Frege's attempt to reduce mathematics to logic and questioning the logicist programme. Two influential ways of avoiding the paradox were both proposed in 1908: Russell's own type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

and the Zermelo set theory
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermeloâ€“Fraenkel set theory (ZF) and its extensions, such as von Neumannâ€“Bernaysâ€“GÃ¶del set theory (NBG). It b ...

. In particular, Zermelo's axioms restricted the unlimited comprehension principle. With the additional contributions of Abraham Fraenkel, Zermelo set theory developed into the now-standard Zermeloâ€“Fraenkel set theory
In set theory, Zermeloâ€“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 ...

(commonly known as ZFC when including 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 o ...

). The main difference between Russell's and Zermelo's solution to the paradox is that Zermelo modified the axioms of set theory while maintaining a standard logical language, while Russell modified the logical language itself. The language of ZFC, with the help of Thoralf Skolem, turned out to be that of first-order logic
First-order logicâ€”also known as predicate logic, quantificational logic, and first-order predicate calculusâ€”is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...

.
Informal presentation

Most sets commonly encountered are not members of themselves. For example, consider the set of allsquare
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, Ï€/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...

s in a plane. This set is not itself a square in the plane, thus it is not a member of itself. Let us call a set "normal" if it is not a member of itself, and "abnormal" if it is a member of itself. Clearly every set must be either normal or abnormal. The set of squares in the plane is normal. In contrast, the complementary set that contains everything which is not a square in the plane is itself not a square in the plane, and so it is one of its own members and is therefore abnormal.
Now we consider the set of all normal sets, ''R'', and try to determine whether ''R'' is normal or abnormal. If ''R'' were normal, it would be contained in the set of all normal sets (itself), and therefore be abnormal; on the other hand if ''R'' were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that ''R'' is neither normal nor abnormal: Russell's paradox.
Formal presentation

The term "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 ...

" is used in various ways. In one usage, naive set theory is a formal theory, that is formulated in a first-order language with a binary non-logical predicate
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...

$\backslash in$, and that includes 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 Zermeloâ€“Fraenkel set theory. It says that sets having the same elemen ...

:
:$\backslash forall\; x\; \backslash ,\; \backslash forall\; y\; \backslash ,\; (\; \backslash forall\; z\; \backslash ,\; (z\; \backslash in\; x\; \backslash iff\; z\; \backslash in\; y)\; \backslash implies\; x\; =\; y)$
and the axiom schema of unrestricted comprehension:
:$\backslash exists\; y\; \backslash forall\; x\; (x\; \backslash in\; y\; \backslash iff\; \backslash varphi(x))$
for any formula $\backslash varphi$ with the variable as a free variable inside $\backslash varphi$. Substitute $x\; \backslash notin\; x$ for $\backslash varphi(x)$. Then by existential instantiation (reusing the symbol $y$) and universal instantiation we have
:$y\; \backslash in\; y\; \backslash iff\; y\; \backslash notin\; y$
a contradiction. Therefore, this naive set theory is inconsistent.
Set-theoretic responses

From theprinciple of explosion
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a ...

of classical logic
Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy.
Characteristics
Each logical system in this clas ...

, ''any'' proposition can be proved from a contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...

. Therefore, the presence of contradictions like Russell's paradox in an axiomatic set theory is disastrous; since if any formula can be proven true it destroys the conventional meaning of truth and falsity. Further, since set theory was seen as the basis for an axiomatic development of all other branches of mathematics, Russell's paradox threatened the foundations of mathematics as a whole. This motivated a great deal of research around the turn of the 20th century to develop a consistent (contradiction-free) set theory.
In 1908, Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermeloâ€“Fraenkel axiomatic se ...

proposed an axiomatization
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...

of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom 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 an ...

(''Aussonderung''). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem
Thoralf Albert Skolem (; 23 May 1887 â€“ 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory.
Life
Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skolem ...

, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's 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 o ...

ceased to be controversial, and ZFC has remained the canonical axiomatic set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...

down to the present day.
ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set ''X'', any subset of ''X'' definable using first-order logic
First-order logicâ€”also known as predicate logic, quantificational logic, and first-order predicate calculusâ€”is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...

exists. The object ''R'' discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like ''R'' are called proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map f ...

es.
ZFC is silent about types, although the cumulative hierarchy has a notion of layers that resemble types. Zermelo himself never accepted Skolem's formulation of ZFC using the language of first-order logic. As JosÃ© FerreirÃ³s notes, Zermelo insisted instead that "propositional functions (conditions or predicates) used for separating off subsets, as well as the replacement functions, can be 'entirely ''arbitrary anz ''beliebig''" the modern interpretation given to this statement is that Zermelo wanted to include higher-order quantification in order to avoid Skolem's paradox. Around 1930, Zermelo also introduced (apparently independently of von Neumann), the axiom of foundation, thusâ€”as FerreirÃ³s observesâ€” "by forbidding 'circular' and 'ungrounded' sets, it FCincorporated one of the crucial motivations of TT ype theory€”the principle of the types of arguments". This 2nd order ZFC preferred by Zermelo, including axiom of foundation, allowed a rich cumulative hierarchy. FerreirÃ³s writes that "Zermelo's 'layers' are essentially the same as the types in the contemporary versions of simple TT ype theoryoffered by GÃ¶del and Tarski. One can describe the cumulative hierarchy into which Zermelo developed his models as the universe of a cumulative TT in which transfinite types are allowed. (Once we have adopted an impredicative standpoint, abandoning the idea that classes are constructed, it is not unnatural to accept transfinite types.) Thus, simple TT and ZFC could now be regarded as systems that 'talk' essentially about the same intended objects. The main difference is that TT relies on a strong higher-order logic, while Zermelo employed second-order logic, and ZFC can also be given a first-order formulation. The first-order 'description' of the cumulative hierarchy is much weaker, as is shown by the existence of denumerable models (Skolem paradox), but it enjoys some important advantages."
In ZFC, given a set ''A'', it is possible to define a set ''B'' that consists of exactly the sets in ''A'' that are not members of themselves. ''B'' cannot be in ''A'' by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.
Through the work of Zermelo and others, especially John von Neumann
John von Neumann (; hu, Neumann JÃ¡nos Lajos, ; December 28, 1903 â€“ February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cov ...

, the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe, ''V'', built up from the empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

by transfinitely iterating the power set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...

operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of ''V''. Whether it is ''appropriate'' to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...

.
Other solutions to Russell's paradox, with an underlying strategy closer to that of type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

, include Quine
Quine may refer to:
* Quine (surname), people with the surname ''Quine''
* Willard Van Orman Quine, the philosopher, or things named after him:
** Quine (computing), a program that produces its source code as output
** Quineâ€“McCluskey algorithm ...

's New Foundations and Scottâ€“Potter set theory
An approach to the foundations of mathematics that is of relatively recent origin, Scottâ€“Potter set theory is a collection of nested axiomatic set theories set out by the philosopher Michael Potter, building on earlier work by the mathematician ...

. Yet another approach is to define multiple membership relation with appropriately modified comprehension scheme, as in the Double extension set theory.
History

Russell discovered the paradox in May or June 1901. By his own account in his 1919 ''Introduction to Mathematical Philosophy'', he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal". In a 1902 letter, he announced the discovery toGottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 â€“ 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phi ...

of the paradox in Frege's 1879 ''Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept notatio ...

'' and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function:
Russell would go on to cover it at length in his 1903 '' The Principles of Mathematics'', where he repeated his first encounter with the paradox:
Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his ''Grundgesetze der Arithmetik''. Frege responded to Russell very quickly; his letter dated 22 June 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967:126â€“127. Frege then wrote an appendix admitting to the paradox, and proposed a solution that Russell would endorse in his ''Principles of Mathematics'', but was later considered by some to be unsatisfactory. For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo (, ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermeloâ€“Fraenkel axiomatic se ...

in his (1908) ''A new proof of the possibility of a well-ordering'' (published at the same time he published "the first axiomatic set theory") laid claim to prior discovery of the antinomy
Antinomy ( Greek á¼€Î½Ï„Î¯, ''antÃ'', "against, in opposition to", and Î½ÏŒÎ¼Î¿Ï‚, ''nÃ³mos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of ...

in Cantor's naive set theory. He states: "And yet, even the elementary form that RussellEdmund Husserl
, thesis1_title = BeitrÃ¤ge zur Variationsrechnung (Contributions to the Calculus of Variations)
, thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view
, thesis1_year = 1883
, thesis2_title ...

.
In 1923, Ludwig Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 â€“ 29 April 1951) was an Austrians, Austrian-British people, British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy o ...

proposed to "dispose" of Russell's paradox as follows:
The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition F(F(fx)), in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(fx) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of F(Fu) we write (do) : F(Ou) . Ou = Fu. That disposes of Russell's paradox. (''Russell andTractatus Logico-Philosophicus The ''Tractatus Logico-Philosophicus'' (widely abbreviated and cited as TLP) is a book-length philosophical work by the Austrian philosopher Ludwig Wittgenstein which deals with the relationship between language and reality and aims to define th ...'', 3.333)

Alfred North Whitehead
Alfred North Whitehead (15 February 1861 â€“ 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applic ...

wrote their three-volume '' Principia Mathematica'' hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of 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 ...

by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While ''Principia Mathematica'' avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.
In any event, Kurt GÃ¶del
Kurt Friedrich GÃ¶del ( , ; April 28, 1906 â€“ January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, GÃ¶del had an im ...

in 1930â€“31 proved that while the logic of much of ''Principia Mathematica'', now known as first-order logic
First-order logicâ€”also known as predicate logic, quantificational logic, and first-order predicate calculusâ€”is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifi ...

, is complete, Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekindâ€“Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

is necessarily incomplete if it is consistent
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...

. This is very widelyâ€”though not universallyâ€”regarded as having shown the logicist program of Frege to be impossible to complete.
In 2001 A Centenary International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.
Applied versions

There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, thebarber paradox
The barber paradox is a puzzle derived from Russell's paradox. It was used by Bertrand Russell as an illustration of the paradox, though he attributes it to an unnamed person who suggested it to him.''The Philosophy of Logical Atomism'', repr ...

supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.
An easy refutation of the "layman's versions" such as the barber paradox seems to be that no such barber exists, or that the barber has alopecia
Hair loss, also known as alopecia or baldness, refers to a loss of hair from part of the head or body. Typically at least the head is involved. The severity of hair loss can vary from a small area to the entire body. Inflammation or scarrin ...

, or is a woman, and in the latter two cases the barber doesn't shave, and so can exist without paradox. The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory. Note the difference between the statements "such a set does not exist" and "it is an empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in oth ...

". It is like the difference between saying "There is no bucket" and saying "The bucket is empty".
A notable exception to the above may be the Grellingâ€“Nelson paradox, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the barber's paradox by saying that such a barber does not (and ''cannot'') exist, it is impossible to say something similar about a meaningfully defined word.
One way that the paradox has been dramatised is as follows: Suppose that every public library has to compile a catalogue of all its books. Since the catalogue is itself one of the library's books, some librarians include it in the catalogue for completeness; while others leave it out as it being one of the library's books is self-evident. Now imagine that all these catalogues are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master cataloguesâ€”one of all the catalogues that list themselves, and one of all those that don't.
The question is: should these master catalogues list themselves? The 'Catalogue of all catalogues that list themselves' is no problem. If the librarian doesn't include it in its own listing, it remains a true catalogue of those catalogues that do include themselves. If he does include it, it remains a true catalogue of those that list themselves. However, just as the librarian cannot go wrong with the first master catalogue, he is doomed to fail with the second. When it comes to the 'Catalogue of all catalogues that don't list themselves', the librarian cannot include it in its own listing, because then it would include itself, and so belong in the other catalogue, that of catalogues that do include themselves. However, if the librarian leaves it out, the catalogue is incomplete. Either way, it can never be a true master catalogue of catalogues that do not list themselves.
Applications and related topics

Russell-like paradoxes

As illustrated above for the barber paradox, Russell's paradox is not hard to extend. Take: * Atransitive verb
A transitive verb is a verb that accepts one or more objects, for example, 'cleaned' in ''Donald cleaned the window''. This contrasts with intransitive verbs, which do not have objects, for example, 'panicked' in ''Donald panicked''.
Transiti ...

, that can be applied to its substantive
A noun () is a word that generally functions as the name of a specific object or set of objects, such as living creatures, places, actions, qualities, states of existence, or ideas.Example nouns for:
* Living creatures (including people, alive, ...

form.
Form the sentence:
:The er that s all (and only those) who don't themselves,
Sometimes the "all" is replaced by "all ers".
An example would be "paint":
:The ''paint''er that ''paint''s all (and only those) that don't ''paint'' themselves.
or "elect"
:The ''elect''or (representative
Representative may refer to:
Politics
*Representative democracy, type of democracy in which elected officials represent a group of people
*House of Representatives, legislative body in various countries or sub-national entities
*Legislator, someon ...

), that ''elect''s all that don't ''elect'' themselves.
In the Season 8
A season is a division of the year based on changes in weather, ecology, and the number of daylight hours in a given region. On Earth, seasons are the result of the axial parallelism of Earth's tilted orbit around the Sun. In temperate and p ...

episode of ''The Big Bang Theory
''The Big Bang Theory'' is an American television sitcom created by Chuck Lorre and Bill Prady, both of whom served as executive producers on the series, along with Steven Molaro, all of whom also served as head writers. It premiered on ...

'', "The Skywalker Intrusion", Sheldon Cooper analyzes the song " Play That Funky Music", concluding that the lyrics present a musical example of Russell's Paradox.
Paradoxes that fall in this scheme include:
* The barber with "shave".
* The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
* The Grellingâ€“Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves.
* Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that don't denote themselves" is here called ''Richardian''.)
* "I am lying.", namely the liar paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth ...

and Epimenides paradox, whose origins are ancient
* Russellâ€“Myhill paradox
Related paradoxes

* The Burali-Forti paradox, about the order type of allwell-ordering
In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-ord ...

s
* The Kleeneâ€“Rosser paradox
In mathematics, the Kleene–Rosser paradox is a paradox that shows that certain systems of formal logic are inconsistent, in particular the version of Haskell Curry's combinatory logic introduced in 1930, and Alonzo Church's original lambda c ...

, showing that the original lambda calculus
Lambda calculus (also written as ''Î»''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computatio ...

is inconsistent, by means of a self-negating statement
* Curry's paradox
Curry's paradox is a paradox in which an arbitrary claim ''F'' is proved from the mere existence of a sentence ''C'' that says of itself "If ''C'', then ''F''", requiring only a few apparently innocuous logical deduction rules. Since ''F'' is arbi ...

(named after Haskell Curry
Haskell Brooks Curry (; September 12, 1900 â€“ September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper by ...

), which does not require negation
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and fal ...

* The smallest uninteresting integer paradox
* Girard's paradox in type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a founda ...

See also

* Basic Law V * * * " On Denoting" * * Quine's paradox *Self-reference
Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directlyâ€”through some intermediate sentence or formulaâ€”or by means of some encoding. In philosop ...

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Notes

References

Sources

* * *External links

* * *Russell's Paradox

at

Cut-the-Knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and ...

Russell's (Barber) paradox explanation in Python

{{DEFAULTSORT:Russell's Paradox Bertrand Russell Paradoxes of naive set theory 1901 in science Self-referential paradoxes