- Russell's paradox
Part of the

foundations of mathematics ,**Russell's paradox**(also known as**Russell's antinomy**), discovered byBertrand Russell in 1901, showed that thenaive set theory of Frege leads to a contradiction.It might be assumed that, for any formal criterion, a set exists whose members are those objects (and only those objects) that satisfy the criterion; but this assumption is disproved by a set containing exactly the sets that are not members of themselves. If such a set qualifies as a member of itself, it would contradict its own definition as "a set containing sets that are not members of themselves". On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

In 1908, two ways of avoiding the paradox were proposed, Russell's

type theory andErnst Zermelo 's axiomatic set theory, the first consciously constructedaxiomatic set theory . Zermelo's axioms went well beyond Frege's axioms ofextensionality and unlimited set abstraction, and evolved into the now-canonicalZFC set theory.**Informal presentation**Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set of all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".

Now we consider the set of all normal sets – let us give it a name: "R". If "R" were abnormal, that is, if "R" were a member of itself, then "R" would be normal since all its members are. So, "R" cannot be abnormal, which means "R" is normal. Further, since every normal set is a member of "R", "R" itself must be a member of "R", making "R" abnormal. Paradoxically, we are led to the contradiction that "R" is both normal and abnormal.

A longer argument often given for this contradiction, by Russell himself, for example, proceeds by cases. Is "R" a normal set? If it is normal, then it is a member of "R", since "R" contains all normal sets. But if that is the case, then "R" contains itself as a member, and therefore is abnormal. On the other hand, if "R" is abnormal, then it is not a member of "R", since "R" contains only normal sets. But if that is the case, then "R" does not contain itself as a member, and therefore is normal. Clearly, this is a paradox: if we suppose "R" is normal we can prove it is abnormal, and if we suppose "R" is abnormal we can prove it is normal. Hence, "R" is both normal "and" abnormal, which is a contradiction.

**Formal derivation**Let "R" be "the set of all sets that do not contain themselves as members". Formally: "A" is an element of "R" if and only if "A" is not an element of "A". In

set-builder notation :: $R=\{Amid\; A\; otin\; A\}.$

Nothing in the system of Frege's "Grundgesetze der Arithmetik" rules out "R" being a

well-defined set. The problem arises when it is considered whether "R" is an element of itself. If "R" is an element of "R", then according to the definition "R" is not an element of "R". If "R" is not an element of "R", then "R" has to be an element of "R", again by its very definition. The statements "R" is an element of "R" and "R" is not an element of "R" cannot both be true, thus the contradiction.The following fully formal yet elementary derivation of Russell's paradox [

*Adapted from Potter (2004: 24-25).*] makes plain that the paradox requires nothing more thanfirst-order logic with the unrestricted use ofset abstraction . The proof is given in terms of collections (all sets are collections, but not conversely). It invokes neither set theory axioms nor the law ofexcluded middle explicitly or tacitly.**Definition**. The collection $\{x\; mid\; Phi(x)\},!$, in which $Phi(x),!$ is any predicate of first-order logic in which $x,!$ is afree variable , denotes the individual $A,!$ satisfying $forall\; x,\; [x\; in\; A\; Leftrightarrow\; Phi(x)]\; ,!$.**Theorem**. The collection $R=\{x\; mid\; x\; otin\; x\},!$ is contradictory."Proof". Replace $Phi(x),!$ in the definition of collection by $x\; otin\; x,!$, so that the implicit definition of $R,!$ becomes $forall\; x,\; [x\; in\; R\; Leftrightarrow\; x\; otin\; x]\; .,!$ Instantiating $x,!$ by $R,!$ then yields the contradiction $R\; in\; R\; Leftrightarrow\; R\; otin\; R.\; square,!$

"Remark". The above definition and theorem are the first theorem and definition in Potter (2004), consistent with the fact that Russell's paradox requires no set theory whatsoever. Incidentally, the force of this argument cannot be evaded by simply proscribing the substitution of $x\; otin\; x,!$ for $Phi(x),!$. In fact, there are denumerably many formulae $Phi(x),!$ giving rise to the paradox. [

*See*] For some examples, see reciprocation below.Willard Quine , 1938, "On the theory of types," "Journal of Symbolic Logic 3".**The paradox holds in intuitionistic logic**Often derivations of Russell's paradox employ the law of the excluded middle (Russell's own derivation did).Thus it may be tempting to conclude that the paradox is avoided if the law of excluded middle is disallowed, as with

intuitionistic logic . However, as is clear from the first of the two informal arguments presented above, the law of excluded middle is not needed for the paradoxical argument. Here we show that the paradox can still be generated formally by means of the intuitionistically validlaw of non-contradiction .**Theorem**. The collection $R=\{x\; mid\; x\; otin\; x\},!$ is contradictory even if the background logic is intuitionistic."Proof". From the definition of $R\; ,\; !$, we have that $R\; in\; R\; Leftrightarrow\; R\; otin\; R\; ,\; !$. Then $R\; in\; R\; Rightarrow\; R\; otin\; R\; ,\; !$ (

biconditional elimination ). But also $R\; in\; R\; Rightarrow\; R\; in\; R\; ,\; !$ (thelaw of identity ), so $R\; in\; R\; Rightarrow\; (\; R\; in\; R\; and\; R\; otin\; R\; )\; ,\; !$. But by the law of non-contradiction we know that $eg\; (\; R\; in\; R\; and\; R\; otin\; R\; )\; ,\; !$. Bymodus tollens we conclude $R\; otin\; R\; ,\; !$.But since $R\; in\; R\; Leftrightarrow\; R\; otin\; R\; ,\; !$, we also have that $R\; otin\; R\; Rightarrow\; R\; in\; R\; ,\; !$, and so we also conclude $R\; in\; R\; ,\; !$ by

modus ponens . Hence we have deduced both $R\; in\; R\; ,\; !$ and its negation using only intuitionistically valid methods. $square,!$More simply, it is intuitionistically impossible for a proposition to be equivalent to its negation. Assume nowrap|"P" ⇔ ¬"P". Then nowrap|"P" ⇒ ¬"P". Hence ¬"P". Symmetrically, we can derive ¬¬"P", using nowrap|¬"P" ⇒ "P". So we have inferred both ¬"P" and its negation from our assumption, with no use of excluded middle.

**Reciprocation**Russell's paradox arises from the supposition that one can meaningfully define a class in terms of any well-defined property $Phi(x)$; that is, that we can form the set $P\; =\; \{\; x\; |\; Phi(x)\; mbox\{\; is\; true\; \}\; \}$. When we take $Phi(x)\; =\; x\; otin\; x$, we get Russell's paradox. This is only the simplest of many possible variations of this theme.

For example, if one takes $Phi(x)\; =\; eg(exists\; z:\; xin\; zwedge\; zin\; x)$, one gets a similar paradox; there is no set $P$ of all $x$ with this property. For convenience, let us agree to call a set $S$ "reciprocated" if there is a set $T$ with $Sin\; Twedge\; Tin\; S$; then $P$, the set of all non-reciprocated sets, does not exist. If $Pin\; P$, we would immediately have a contradiction, since $P$ is reciprocated (by itself) and so should not belong to $P$. But if $P\; otin\; P$, then $P$ is reciprocated by some set $Q$, so that we have $Pin\; Qwedge\; Qin\; P$, and then $Q$ is also a reciprocated set, and so $Q\; otin\; P$, another contradiction.

Any of the variations of Russell's paradox described above can be reformulated to use this new paradoxical property. For example, the reformulation of the

Grelling paradox is as follows. Let us agree to call an adjective $P$ "nonreciprocated" if and only if there is no adjective $Q$ such that both $P$ describes $Q$ and $Q$ describes $P$. Then one obtains a paradox when one asks if the adjective "nonreciprocated" is itself nonreciprocated.This can also be extended to longer chains of mutual inclusion. We may call sets $A\_1,A\_2,...,A\_n$ a chain of set $A\_1$ if $A\_\{i+1\}\; in\; A\_i$ for "i"=1,2,...,"n"-1. A chain can be infinite (in which case each $A\_i$ has an infinite chain). Then we take the set "P" of all sets which have no infinite chain, from which it follows that "P" itself has no infinite chain. But then $P\; in\; P$, so in fact "P" has the infinite chain "P","P","P",... which is a contradiction. This is known as

Mirimanoff 's paradox.**et-theoretic responses**In 1908,

Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as hisaxiom of separation ("Aussonderung"). Modifications to this axiomatic theory proposed in the 1920s byAbraham Fraenkel ,Thoralf Skolem , and by Zermelo himself resulted in the axiomatic set theory calledZFC . This theory became widely accepted once Zermelo'saxiom of choice ceased to be controversial, andZFC has remained the canonicalaxiomatic set theory 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 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 es. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presupposes a background type theory.Through the work of Zermelo and others, especially

John von Neumann , the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of thevon Neumann universe , "V", built up from theempty set by transfinitely iterating thepower set 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 thephilosophy of mathematics .Other resolutions to Russell's paradox, more in the spirit of

type theory , include the axiomatic set theoriesNew Foundations andScott-Potter set theory .**History**Exactly when Russell discovered the paradox is not known. It seems to have been May or June 1901, probably as a result of his work on

Cantor's theorem that the number of entities in a certain domain is smaller than the number of subclasses of those entities. [*In modern terminology, the*] He first mentioned the paradox in a 1901 paper in the "International Monthly", entitled "Recent work in the philosophy of mathematics." He also mentioned Cantor's proof that there is no greatest cardinal, adding that "the master" had been guilty of a subtle fallacy that he would discuss later. Russell also mentioned the paradox in his "Principles of Mathematics" (not to be confused with the later "cardinality of a set is strictly less than that of itspower set .Principia Mathematica "), calling it "The Contradiction." [*cite book |last=Russell |first=Bertrand |authorlink=Bertrand Russell |title=Principles of Mathematics |year=1903 |publisher=Cambridge University Press |location=Cambridge |isbn=0-393-31404-9 |pages=Chapter X, section 100*] Again, he said that he was led to it by analyzing Cantor's "no greatest cardinal" proof.Famously, Russell wrote to Frege about the paradox in June 1902, just as Frege was preparing the second volume of his "Grundgesetze der Arithmetik". [

*Russell's letter and Frege's reply are translated in*] Frege hurriedly wrote an appendix admitting to the paradox, and proposed a solution that was later proved unsatisfactory. In any event, after publishing the second volume of the "Grundgesetze", Frege wrote little onJean van Heijenoort , 1967, and in Frege’s "Philosophical and Mathematical Correspondence."mathematical logic and thephilosophy of mathematics .Zermelo , while working on the axiomatic set theory he published in 1908, also noticed the paradox but thought it beneath notice, and so never published anything about it.In 1923,

Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:"The reason why a function cannot be its own argument is that the signfor a function already contains the prototype of its argument, and itcannot contain itself. For let us suppose that the function F(fx) could beits own argument: in that case there would be a proposition 'F(F(fx))', inwhich the outer function F and the inner function F must have differentmeanings, since the inner one has the form O(f(x)) and the outer one hasthe form Y(O(fx)). Only the letter 'F' is common to the two functions, butthe letter by itself signifies nothing. This immediately becomes clear ifinstead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes ofRussell's paradox." ("

Tractatus Logico-Philosophicus ", 3.333)Russell and

Alfred North Whitehead wrote their three-volume "Principia Mathematica " ("PM") hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes ofnaive set theory 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 "PM" 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 Gödel in 1930–31 proved that while the logic of much of "PM", now known asfirst-order logic , is complete, Peano arithmetic is necessarilyincomplete if it isconsistent . This is very widely – though not universally – regarded as having shown thelogicist program of Frege to be impossible to complete.**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, the

Barber paradox supposes a barber who shaves men if and only if they do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.As another example, consider five lists of

encyclopedia entries within the same encyclopedia:If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.

While appealing, these

layman 's versions of the paradox share a drawback: an easy refutation of the Barber paradox seems to be that such a barber does not exist. 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 "such a set is empty".A notable exception to the above may be the

Grelling-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 catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident.

Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those which don't.

The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.

However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves.

**Applications and related topics**The Barber paradox, in addition to leading to a tidier set theory, has been used twice more with great success:

Kurt Gödel proved his incompleteness theorem by formalizing the paradox, and Turing proved the undecidability of theHalting problem (and with that theEntscheidungsproblem ) by using the same trick.**Russell-like paradoxes**As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

* Atransitive verb , that can be applied to its substantive 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), that "elect"s all that don't "elect" themselves.

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.

* TheGrelling-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".)

* "Groucho's Paradox":Groucho Marx said that he would never belong to a club that would accept him as a member.**Related paradoxes*** The

liar paradox andEpimenides paradox , whose origins are ancient.

* TheKleene-Rosser paradox , showing that the originallambda calculus is inconsistent, by means of a self-negating statement.

*Curry's paradox (named afterHaskell Curry ) which does not requirenegation .

* The smallest uninteresting integer paradox.**See also***

Self-reference

*Universal set

*On Denoting , one of Russell's first attempts at critiquing Frege**Footnotes****References***Potter, Michael, 2004. "Set Theory and its Philosophy". Oxford Univ. Press.

**External links***

Stanford Encyclopedia of Philosophy : " [*http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox*] " -- by A. D. Irvine.

* [*http://www.cut-the-knot.org/selfreference/russell.shtml Russell's Paradox*] atcut-the-knot

* [*http://www.paradoxes.co.uk/ Some paradoxes - an anthology*]

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Russell's paradox**— Math. a paradox of set theory in which an object is defined in terms of a class of objects that contains the object being defined, resulting in a logical contradiction. [1920 25; first proposed by Bertrand RUSSELL] * * * ▪ logic statement… … Universalium**russell's paradox**— ˈrəsəlz noun Usage: usually capitalized R Etymology: after Bertrand Russell : a paradox that discloses itself in forming a class of all classes that are not members of themselves and in observing that the question of whether it is true or false… … Useful english dictionary**Russell's paradox**— The most famous of the paradoxes in the foundations of set theory, discovered by Russell in 1901. Some classes have themselves as members: the class of all abstract objects, for example, is an abstract object. Others do not: the class of donkeys… … Philosophy dictionary**Russell's paradox**— noun The following paradox: Let A be the set of all sets which do not contain themselves. Then does A contain itself? If it does, then by definition it does not; and if it does not, then by definition it does. See Also: Burali Forti paradox … Wiktionary**Russell**— is an English, Irish, or Scottish name derived from old French, the old French word for Red was rouse ; hence the carry over from French the English Russell, the name also derives from the animal, the fox. Its uses include:People*Arthur Russell… … Wikipedia**Russell , Bertrand Arthur William**— Russell , Bertrand Arthur William, third earl Russell (1872–1970) British philosopher and mathematician Russell, who was born at Trelleck, England, was orphaned at an early age and brought up in the home of his grandfather, the politician Lord… … Scientists**Russell, Bertrand**— ▪ British logician and philosopher in full Bertrand Arthur William Russell, 3rd Earl Russell of Kingston Russell, Viscount Amberley of Amberley and of Ardsalla born May 18, 1872, Trelleck, Monmouthshire, Wales died Feb. 2, 1970,… … Universalium**Paradox**— For other uses, see Paradox (disambiguation). Further information: List of paradoxes A paradox is a seemingly true statement or group of statements that lead to a contradiction or a situation which seems to defy logic or intuition. Typically,… … Wikipedia**Russell's teapot**— part of a series on Bertrand Russell … Wikipedia**paradox**— A paradox arises when a set of apparently incontrovertible premises gives unacceptable or contradictory conclusions. To solve a paradox will involve either showing that there is a hidden flaw in the premises, or that the reasoning is erroneous,… … Philosophy dictionary