Bertrand Russell discovered a paradox concerning the set of all sets: Is the set of all sets that are not members of themselves a member of itself?

The study of the foundations of mathematics can be traced as far back as Greek antiquity. Since then, three major crises have brought into question previously established areas of mathematics. Each of these crises challenged mathematicians either to revise their position or to leave mathematics resting on unstable ground. The first major crisis occurred in the fifth century b.c.e. and was brought about by the discovery that not all geometrical magnitudes of the same kind are commensurable with one another. It was resolved around 370 b.c.e. by Eudoxus of Cnidus, whose revised theory of proportion and magnitude may be found in the fifth book of Euclid’s Stoicheia (Elements). Elements (Euclid)
Mathematics;set theory
Set theory
Russell’s paradox[Russells paradox]
Paradox, Russell’s[Paradox, Russells]
[kw]Russell Discovers the “Great Paradox” (June 16, 1902)
[kw]”Great Paradox,” Russell Discovers the (June 16, 1902)[Great Paradox, Russell Discovers the (June 16, 1902)]
[kw]Paradox,” Russell Discovers the “Great (June 16, 1902)[Paradox, Russell Discovers the Great (June 16, 1902)]
Mathematics;set theory
Set theory
Russell’s paradox[Russells paradox]
Paradox, Russell’s[Paradox, Russells]
[g]England;June 16, 1902: Russell Discovers the “Great Paradox”[00520]
[c]Mathematics;June 16, 1902: Russell Discovers the “Great Paradox”[00520]
Russell, Bertrand
Cantor, Georg
Frege, Gottlob
Burali-Forti, Cesare
Poincaré, Henri

More than two millennia passed before the second crisis in the foundations of mathematics arose. After the discovery of the calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century, weaknesses in the foundations of analysis soon became evident. Only after the arithmetization of analysis in the nineteenth century was the second crisis thought to be resolved. After the resolution of this crisis, the whole structure of mathematics seemed to be redeemed and placed on unshakable ground. By 1900, mathematicians had seemingly imparted to their subject the ideal structure that Euclid had delineated in his Elements. Several branches were founded on rigorous axiomatic bases, the terminology had been subjected to close scrutiny, and deductive proofs replaced intuitively or empirically based conclusions. When the third crisis struck, its suddenness was surprising.

The third crisis in the foundations of mathematics was brought about by the discovery of paradoxes, or antinomies, in the fringe of Georg Cantor’s general theory of sets. Because so much of mathematics is permeated by set concepts and, for that matter, can actually be made to rest on set theory as a foundation, the discovery of paradoxes in set theory naturally cast into doubt the validity of the whole foundational structure of mathematics.

Bertrand Russell.

(The Nobel Foundation)

In 1897, the Italian mathematician Cesare Burali-Forti brought to light the first publicized paradox of set theory. The essence of the paradox can be conveyed by a description of a very similar paradox found by Cantor two years later. In attempting to assign a number to the set whose members are all possible sets and to the set of all ordinals, Cantor discovered a fundamental paradox about sets. The basic idea is that for any given set, there is always a larger one; the set of all subsets in a given set is larger than the original one. Cantor showed that there are larger and larger transfinite sets and corresponding transfinite numbers. Cantor’s theory of sets proved that, for any given transfinite number, there is always a greater transfinite number, so that just as there is no greatest natural number, there is no greatest transfinite number. How can there be a transfinite number greater than the transfinite number of the set whose members are all possible sets, given that no set can have more members than this set? By 1899, Cantor thought that one could not consider the set of all sets or its number. This caused mathematicians to recognize that they had been using similar concepts not only in their newer creations but also in the supposedly well-established older mathematics. They preferred to call these contradictions “paradoxes” because a paradox can be resolved, and the mathematicians wanted to believe these could be resolved. The technical word commonly used now is “antinomy.”

When Bertrand Russell first discovered Cantor’s conclusion about the set of all sets, he did not believe it. In an essay published in 1901, he wrote:

There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in sublimated form. But in this one point, the master must have been guilty of a very subtle fallacy, which I hope to explain in some future work.

Russell meditated on this matter and added to the problems of the times his own “paradox.” Some years later, when Russell reprinted his essay in Mysticism and Logic (1918), Mysticism and Logic (Russell) he added a footnote apologizing for his mistake, stating that Cantor’s proof that there is no greatest number is valid and that the solution of the puzzle is complicated and depends on a theory of types, which is explained in Alfred North Whitehead and Russell’s Principia Mathematica (1910-1913). Principia Mathematica (Whitehead and Russell) Russell had studied the paradox of Cantor and, instead of finding a fallacy in it, had generated his own version.

Whereas the Burali-Forti and Cantor paradoxes involve results of set theory, the “great paradox,” which Russell discovered in 1902, depends on nothing more than the concept of set itself. In order to understand Russell’s paradox, one must realize that sets either are or are not members of themselves. For example, the set of all sets is itself a set, but the set of all books is not itself a book. Likewise, the set of all comprehensible things is itself a comprehensible thing, whereas the set of all persons is not itself a person. Take the set of all sets that are members of themselves and call it S, and take the set of all sets that are not members of themselves and call it N. Is N a member of itself? If N is not a member of itself, then N is a member of N and not of S, and N is a member of itself. On the other hand, if N is a member of itself, then N is a member of S and not of N, and N is not a member of itself. Russell’s great paradox lies in the fact that, in either case, there is a contradiction. When Russell first discovered this contradiction, he thought the difficulty lay somewhere in the logic rather than in the mathematics. At first, he did not realize that this paradox strikes at the very notion of a set, a notion used throughout mathematics. It was the first of a number of troublesome mathematical paradoxes.

Although Russell first published this paradox in The Principles of Mathematics (1903), Principles of Mathematics, The (Russell) he had communicated his discovery a bit earlier to the mathematician Gottlob Frege. Russell had sent a letter to Frege on June 16, 1902, in which he told Frege about the paradox. Frege received the letter while the second volume of his Grundgesetze der Arithmetik
Grundgesetze der Arithmetik (Frege) (1893, 1903; the basic laws of arithmetic) was at the printer. In this treatise on the foundations of arithmetic, Frege had used the theory of sets, which involved the very paradox that Russell noted in his letter. In effect, Russell’s letter broke the foundation of Frege’s work just as his treatise was finished—a project on which Frege had labored for more than twelve years.

Russell’s great paradox has been popularized in many forms. One of the best known of these, the “barber paradox,” Barber paradox was introduced by Russell himself some years after his discovery of the great paradox. This paradox concerns the plight of a village barber who has advertised that he does not shave those persons who shave themselves, but he does shave all those persons and only those persons of the village who do not shave themselves. The paradoxical nature of this situation is revealed when the barber asks himself whether he should shave himself. If he does shave himself, then he should not because he has advertised that he shaves all those persons and only those persons of the village who do not shave themselves. If he does not shave himself, then he should, according to his claim to shave all those who do not shave themselves. The barber is in a logical predicament.

Although many mathematicians of the early 1900’s tended to disregard the paradoxes because they involved set theory, which was new and peripheral at the time, others were disturbed, recognizing that the paradoxes affected not only classical mathematics but also general reasoning. Russell’s great paradox had a catastrophic effect on the world of mathematics. Since its discovery, additional paradoxes in set theory have been produced in abundance. The existence of paradoxes in set theory clearly indicates that something is wrong. Much literature on the subject has been published, and numerous attempts at solutions have been offered.

As far as mathematics is concerned, there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude the known antinomies. The first such attempt was made by Ernst Zermelo Zermelo, Ernst in 1908, and subsequent refinements have been made. This procedure has been criticized as merely avoiding the paradoxes; certainly it does not explain them. Moreover, this procedure does not guarantee that other kinds of paradoxes will not surface in the future.

In 1905, Russell thought that the paradoxes of set theory arose from a fallacy he called the “vicious circle principle.” Vicious circle principle Simply stated, the principle says that whatever involves all of a collection must not be one of the collection. In other words, if to define a set of objects one must use the set of objects itself in the definition, then the definition is meaningless. This was accepted by Henri Poincaré in 1906, who called these definitions “impredicative.” An impredicative definition defines an object in terms of a set of objects that contains the object being defined. Such definitions are illegitimate; by restricting them, at least one can avoid the known paradoxes of set theory.

The restrictions of Russell and Poincaré amount to restrictions on the notion of set, and, although heeding these restrictions provides a solution to the known paradoxes of set theory, there is at least one serious objection to this solution: Although some parts of mathematics contain impredicative definitions that may be circumvented, many others contain impredicative definitions that cannot be circumvented, and mathematicians are reluctant to discard these parts. For example, the least upper bound of a given nonempty set of real numbers is the smallest member of the set of all upper bounds of the given set. This is an impredicative definition that mathematicians are reluctant to give up.

Mathematicians have made various attempts to solve the paradoxes of set theory. Some, for example, search for the cause of the paradoxes in logic, and this has brought about a rigorous investigation into the foundations of logic. In effect, Russell’s discovery of the great paradox led to a search for the causes and solutions to such paradoxes in an effort to restore the foundations of mathematics. Mathematics;set theory
Set theory
Russell’s paradox[Russells paradox]
Paradox, Russell’s[Paradox, Russells]

• Jager, Ronald. The Development of Bertrand Russell’s Philosophy. 1972. Reprint. London: Routledge, 2004. A fine survey of the development of Russell’s philosophy. Part of the book is labeled “The Theory of Logic,” and a subsection of this part, titled “Set Theory and Paradoxes,” provides a good general account.
• Kneale, William, and Martha Kneale. The Development of Logic. 1962. Reprint. New York: Oxford University Press, 1985. One of the best historical treatments of its subject. Philosophically oriented and accessible to the general reader. Chapter 11, titled “The Philosophy of Mathematics After Frege,” is an in-depth treatment of the paradoxes of the theory of sets and Russell’s theory of types as well as other solutions to the paradoxes of set theory. Highly recommended as a scholarly treatment of Russell’s paradox.
• Russell, Bertrand. My Philosophical Development. 1959. Rev. ed. New York: Routledge, 1995. An excellent basic account of the events surrounding the discovery of the paradox and of the paradox in general. Russell clearly articulates his influences and his attempts to resolve the paradox. Essential for the general reader.
• _______. The Principles of Mathematics. 1903. Reprint. New York: W. W. Norton, 1996. The work in which Russell first published his paradox. The fundamental thesis is that mathematics and logic are identical; that is, mathematics is merely later deductions from logical premises. The book does not contain much mathematical and logical symbolism and, as such, is accessible to the diligent general reader.
• Schilpp, Paul Arthur, ed. The Philosophy of Bertrand Russell. Evanston, Ill.: Northwestern University Press, 1944. A valuable collection containing essays by distinguished scholars on many aspects of Russell’s work. Useful information about the discovery of the paradox appears in a number of places throughout the volume, including statements by Russell on his discovery of the paradox and his attempts to resolve it.
• Van Heijenoort, Jean. From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931. Cambridge, Mass.: Harvard University Press, 1967. An important collection of classic selections on mathematical logic with introductory notes to each selection. Many are of interest to the general reader. Russell’s 1902 letter to Frege and Frege’s response to Russell are reproduced.

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