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What is Russell's paradox?

John T. Baldwin and Olivier Lessmann of the Department of Mathematics, Statistics and Computer Science at the University of Illinois at Chicago offer the following explanation.

Russell's paradox is based on examples like this: Consider a group of barbers who shave only those men who do not shave themselves. Suppose there is a barber in this collection who does not shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave themselves.)

Bertrand Russell confounded
mathematicians when he published his famous paradox in 1903
BERTRAND RUSSELL confounded mathematicians when he published his famous paradox in 1903.

Bertrand Russell's discovery of this paradox in 1901 dealt a blow to one of his fellow mathematicians. In the late 1800s, Gottlob Frege tried to develop a foundation for all of mathematics using symbolic logic. He established a correspondence between formal expressions (such as x=2) and mathematical properties (such as even numbers). In Frege's development, one could freely use any property to define further properties.

Russell's paradox, which he published in Principles of Mathematics in 1903, demonstrated a fundamental limitation of such a system. In modern terms, this sort of system is best described in terms of sets, using so-called set-builder notation. For example, we can describe the collection of numbers 4, 5 and 6 by saying that x is the collection of integers, represented by n, that are greater than 3 and less than 7. We write this description of the set formally as x = { n: n is an integer and 3 < n < 7} . The objects in the set don't have to be numbers. We might let y ={x: x is a male resident of the United States }.

Seemingly, any description of x could fill the space after the colon. But Russell (and independently, Ernst Zermelo) noticed that x = {a: a is not in a} leads to a contradiction in the same way as the description of the collection of barbers. Is x itself in the set x? Either answer leads to a contradiction.

When Russell discovered this paradox, Frege immediately saw that it had a devastating effect on his system. Even so, he was unable to resolve it, and there have been many attempts in the last century to avoid it.

Russell's own answer to the puzzle came in the form of a "theory of types." The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers. So Russell introduced a hierarchy of objects: numbers, sets of numbers, sets of sets of numbers, etc. This system served as vehicle for the first formalizations of the foundations of mathematics; it is still used in some philosophical investigations and in branches of computer science.

Zermelo's solution to Russell's paradox was to replace the axiom "for every formula A(x) there is a set y = {x: A(x)}" by the axiom "for every formula A(x) and every set b there is a set y = {x: x is in b and A(x)}."

What became of the effort to develop a logical foundation for all of mathematics? Mathematicians now recognize that the field can be formalized using so-called Zermelo-Fraenkel set theory. The formal language contains symbols such as e to express "is a member of," = for equality and ¿ to denote the set with no elements. So one can write formulas such as B(x): if y e x then y is empty. In set-builder notation we could write this as y = {x : x = ¿} or more simply as y = {¿}. Russell's paradox becomes: let y = {x: x is not in x}, is y in y?

Russell's and Frege's correspondence on Russell's discovery of the paradox can be found in From Frege to Godel, a Source Book in Mathematical Logic, 1879-1931, edited by Jean van Heijenoort, Harvard University Press, 1967.

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