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In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all 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. Symbolically:

Failed to parse (Missing texvc executable; please see math/README to configure.): \text{let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R

In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).^[1]

Informal presentation[link]

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 that contains 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, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and 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[link]

Define Naive Set Theory (NST) as the theory of predicate logic with a binary predicate Failed to parse (Missing texvc executable; please see math/README to configure.): \in

and the following axiom schema of unrestricted comprehension:

Failed to parse (Missing texvc executable; please see math/README to configure.): \exists y \forall x (x \in y \iff P(x))

for all formula P with only the variable x free. Substitute Failed to parse (Missing texvc executable; please see math/README to configure.): x \notin x

for Failed to parse (Missing texvc executable; please see math/README to configure.): P(x)

. Then by existential instantiation (reusing the symbol y) and universal instantiation we have

Failed to parse (Missing texvc executable; please see math/README to configure.): y \in y \iff y \notin y

a contradiction. Therefore NST is inconsistent.

Set-theoretic responses[link]

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 his axiom of separation (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic 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 classes. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a background type theory.

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, 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 by transfinitely iterating the power 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 the philosophy of mathematics.

Other resolutions to Russell's paradox, more in the spirit of type theory, include the axiomatic set theories New Foundations and Scott-Potter set theory.

History[link]

Russell discovered the paradox in May or June 1901.^[2] By his own admission in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal".^[3] In a 1902 letter,^[4] he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function; in the following, p. 17 refers to a page in the original Begriffsschrift, and page 23 refers to the same page in van Heijenoort 1967:

There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality.^[5]

Russell would go to cover it at length in his 1903 The Principles of Mathematics where he repeats his first encounter with the paradox:^[6]

Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's proof...."

Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik.^[7] Frege did not waste time responding to Russell, his letter dated 22 June 1902 appears, with van Heijenoort's commentary in Heijenoort 1967:126–127. Frege then wrote an appendix admitting to the paradox,^[8] and proposed a solution that Russell would endorse in his Principles of Mathematics,^[9] but was later considered by some unsatisfactory.^[10] For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.^[11]

For his part, Ernst Zermelo 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")^[12] laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell⁹ gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that 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".^[13] Footnote 9 is where he stakes his claim:

⁹1903, pp. 366–368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.^[14]

A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl.^[15]

It is also known that unpublished discussions of set theoretical paradoxes took place in the mathematical community at the turn of the century. van Heijenoort in his commentary before Russell's 1902 Letter to Frege states that Zermelo "had discovered the paradox independently of Russell and communicated it to Hilbert, among others, prior to its publication by Russell".^[16]

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 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(f(x)) 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. (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 of naive 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 as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely – though not universally – regarded as having shown the logicist program of Frege to be impossible to complete.

Applied versions[link]

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 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.

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, or at least does not shave (a variant of which is that the barber is a woman). 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". It is like saying, "There is no bucket", or "The bucket 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 that 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[link]

Russell-like paradoxes[link]

As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

• A transitive verb <V>, that can be applied to its substantive form.

Form the sentence:

The <V>er that <V>s all (and only those) who don't <V> themselves,

Sometimes the "all" is replaced by "all <V>ers".

An example would be "paint":

The painter that paints all (and only those) that don't paint themselves.

or "elect"

The elector (representative), that elects 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.
• The Grelling–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.)

Related paradoxes[link]

• The liar paradox and Epimenides paradox, whose origins are ancient
• The Kleene–Rosser paradox, showing that the original lambda calculus is inconsistent, by means of a self-negating statement
• Curry's paradox (named after Haskell Curry), which does not require negation
• The smallest uninteresting integer paradox

See also[link]

• Self-reference
• Universal set
• On Denoting, one of Russell's first attempts at critiquing Frege
• Cantor's diagonal argument

Notes[link]

1. ^ Set theory paradoxes
2. ^ Godehard Link (2004), One hundred years of Russell's paradox, p. 350, ISBN 978-3-11-017438-0, http://books.google.com/?id=Xg6QpedPpcsC&pg=PA350 
3. ^ Russell 1920:136
4. ^ Gottlob Frege, Michael Beaney (1997), The Frege reader, p. 253, ISBN 978-0-631-19445-3, http://books.google.com/?id=4ktC0UrG4V8C&pg=PA253 . Also van Heijenoort 1967:124–125
5. ^ Remarkably, this letter was unpublished until van Heijenoort 1967 – it appears with van Heijenoort's commentary at van Heijenoort 1967:124–125.
6. ^ Russell 1903:101
7. ^ cf van Heijenoort's commentary before Frege's Letter to Russell in van Heijenoort 1967:126.
8. ^ van Heijenoort's commentary, cf van Heijenoort 1967:126 ; Frege starts his analysis by this exceptionally honest comment : "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion" (Appendix of Grundgesetze der Arithmetik, vol. II, in The Frege Reader, p.279, translation by Michael Beaney
9. ^ cf van Heijenoort's commentary, cf van Heijenoort 1967:126. The added text reads as follows: " Note. The second volume of Gg., which appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction (pp. 253–265), suggesting that the solution is to be found by denying that two propositional functions that determine equal classes must be equivalent. As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (Russell 1903:522); The abbreviation Gg. stands for Frege's Grundgezetze der Arithmetik. Begriffsschriftlich abgeleitet. Vol. I. Jena, 1893. Vol. II. 1903.
10. ^ Livio states that "While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous...." Livio 2009:188. But van Heijenoort in his commentary before Frege's (1902) Letter to Russell describes Frege's proposed "way out" in some detail – the matter has to do with the " 'transformation of the generalization of an equality into an equality of courses-of-values. For Frege a function is something incomplete, 'unsaturated' "; this seems to contradict the contemporary notion of a "function in extension"; see Frege's wording at page 128: "Incidentally, it seems to me that the expession 'a predicate is predicated of itself' is not exact. ...Therefore I would prefer to say that 'a concept is predicated of its own extension' [etc]". But he waffles at the end of his suggestion that a function-as-concept-in-extension can be written as predicated of its function. van Heijenoort cites Quine: "For a late and thorough study of Frege's "way out", see Quine 1955": "On Frege's way out", Mind 64, 145–159; reprinted in Quine 1955b: Appendix. Completeness of quantification theory. Loewenheim's theorem, enclosed as a pamphlet with part of the third printing (1955) of Quine 1950 and incorporated in the revised edition (1959), 253—260" (cf REFERENCES in van Heijenoort 1967:649)
11. ^ Russell mentions this fact to Frege, cf van Heijenoort's commentary before Frege's (1902) Letter to Russell in van Heijenoort 1967:126
12. ^ van Heijenoort's commentary before Zermelo (1908a) Investigations in the foundations of set theory I in van Heijenoort 1967:199
13. ^ van Heijenoort 1967:190–191. In the section before this he objects strenuously to the notion of impredicativity as defined by Poincaré (and soon to be taken by Russell, too, in his 1908 Mathematical logic as based on the theory of types cf van Heijenoort 1967:150–182).
14. ^ Ernst Zermelo (1908) A new proof of the possibility of a well-ordering in van Heijenoort 1967:183–198. Livio 2009:191 reports that Zermelo "discovered Russell's paradox independently as early as 1900"; Livio in turn cites Ewald 1996 and van Heijenoort 1967 (cf Livio 2009:268).
15. ^ B. Rang and W. Thomas, "Zermelo's discovery of the 'Russell Paradox'", Historia Mathematica, v. 8 n. 1, 1981, pp. 15–22. doi:10.1016/0315-0860(81)90002-1
16. ^ van Heijenoort 1967:124

References[link]

• Potter, Michael (15 January 2004), Set Theory and its Philosophy, Clarendon Press (Oxford University Press), ISBN 978-0-19-926973-0 
• van Heijenoort, Jean (1967, third printing 1976), From Frege to Gödel: A Source Book in Mathematical Logic, 1979-1931, Cambridge, Massachusetts: Harvard University Press, ISBN 0-674-32449-8 
• Livio, Mario (6 January 2009), Is God a Mathematician?, Simon & Schuster, ISBN 978-0-7432-9405-8 

External links[link]

• Russell's Paradox at Cut-the-Knot
• Stanford Encyclopedia of Philosophy: "Russell's Paradox" – by A. D. Irvine.