Robin Shakal

Journal Article

Russell’s Paradox

In mathematics, inconsistent systems are to be avoided at all costs. A system that is not consistent will yield a truth value for any proposition, and the proposition’s negation. If

this occurs, absolutely everything we know is true and not true at the same time. In 1901, Bertrand Russell discovered an inconsistency while working with set theory. This inconsistency threatened all of mathematics because logically all sentences must follow from a contradiction. The inconsistency is known as Russell’s Paradox and is as follows:

If S is the set of all sets which do not contain themselves as elements, how can we define S? For a logical definition to hold, one of two cases must hold:

1. S contains itself as an element OR

2. S does not contain itself as an element.

If we first approach case one, we immediately see a contradiction. If S contains itself as an element, this goes against the very definition of what S is. Since S is defined as the set of all sets which do not contain themselves as elements, S cannot be in S. Since we have reached a contradiction in this case, logically the definition of S must be in case two.

Case two deals with S being a set which does not contain itself as an element. If this is true, then S cannot be in S. Unfortunately we see that the definition of S not being in S is what it means to be in S. This is a logical contradiction and this case fails as well.

Since neither case holds we reach an impasse. An impasse, however, is not acceptable when the very foundation of mathematics is being challenged. Russell realized the importance of finding a solution to his paradox and at the end of his Principles of Mathematics he called out to all mathematicians and students of logic to try and resolve the paradox. At the writing of this article, there are four responses which are deemed worthy of consideration. Russell himself came up with a response, but we discuss his response later.

Georg Cantor developed the unrestricted comprehension axiom to try and circumvent the paradox. This axiom defines a set by any predicate expression, P(x), that contains x as a free variable. The members of the set are exactly those objects which satisfy P(x) (all x’s which are P) (Stanford Encyclopedia of Philosophy webpage 1999). It appears that Cantor has created a set which does not contain itself as an element, but his approach is generally not accepted at face value. One reason is it would seem that a set of all sets would include the set of all x’s which are not P as well. Therefore, the unrestricted comprehension axiom is deemed too naive to be practical, but as we will see in a future response to the paradox, it is able to be modified

As mentioned before, Russell himself came up with a response to his paradox. His approach is known as the Theory of Types. In this, Russell claimed S, the set of all sets which do not contain themselves, is not a possibility (Pears 1972). To create his theory of types, Russell organized all propositions into a collective hierarchy. The base of this hierarchy is made up of individual thoughts and ideas. The next lowest level is comprised of the sets of the individuals (let’s call it X). The next level then is the sets of the sets of the individuals (we’ll call this set Y, this means Y is the set containing X). Likewise, the next level is the set of sets of the sets of individuals (call it Z, thus is the set containing Y, and therefore X as well). Russell then claimed it is possible to refer to all objects for which a given predicate holds only if they are of the same level (type) or lower. This approach by Russell, however, does not assess how to handle the infinite level. Therefore, the theory of types works on a small scale, but experts criticize this approach as being too short sighted and not comprehensive of a larger scale.

David Hilbert and Luitzen Brouwer while working separately came up with essentially the same response to the paradox. Hilbert and his followers maintained that only finite and well-defined terms could be used to define the paradox. Brouwer claimed that no mathematical object can exist unless it can be constructed with "simple", finite terms. Unlike Russell who decided not to handle the infinite case in his set theory, these two men claimed the infinite case simply cannot exist. Both of these men "solved" the paradox by creating rules which prevented its existence. Since this approach does not really attempt to solve the paradox it is also generally not accepted as a practical response.

The response to the paradox that is generally accepted in today’s world was developed by Ernest Zermelo and modified by Abraham Fraenkel. Zermelo’s approach was like Cantor’s, in that he used an axiomatic approach (the formal term is axiomatizaion of set theory). To create this, Zermelo used axioms to modify and restrict Cantor’s comprehension principle. Fraenkel took Zermelo’s theory and fine-tuned to the approach we generally use today. Axiomatization is not uncommon in mathematics as many subjects are based on axioms. Euclidean geometry for example is completely based on undefined terms (axioms). We know nothing about the meaning of these undefined terms, but we are able to work with them. In respect to Russell’s paradox, the undefined term is set. We don’t know what a set is but we are able to discuss and use the term set.

The understanding of and the attempts to circumvent Russell’s paradox have created a better understanding of formal logic as a whole. The concept of sets is a fundamental and crucial topic on which all of mathematics is built. Obviously then if this foundation is shown to be inconsistent, all of mathematics would be seen as contradictory, and therefore worthless. Though the paradox seemed to spell doom for mathematics, fortunately we have dealt with the issue of Russell’s paradox and its ramifications.

BIBLIOGRAPHY

Chihara, Charles S. (1972) "Russell’s Theory of Types", in Pears, D.F., Bertrand Russell: A Collection of Critical Essays, Garden City, NY, Doubleday and Company Inc., 1967, 251.

Stanford Encyclopedia of Philosophy webpage, http://plato.stanford.edu/entries/russell-paradox/#HOTP, 1999.

c., 1967, 251.

Stanford Encyclopedia of Philosophy