Russell's Paradox

Theorem
The comprehension principle leads to a contradiction.

Proof
Sets have elements.

Some of those elements may themselves be sets.

So, given two sets $S$ and $T$, we can ask the question: Is $S$ an element of $T$? The answer will either be yes or no.

In particular, given any set $S$, we can ask the question: Is $S$ an element of $S$? Again, the answer will either be yes or no.

Thus, $P \left({S}\right) = S \in S$ is a property on which we can use the comprehension principle to build this set:


 * $T = \left\{{S: S \in S}\right\}$

... which is the set of all sets which contain themselves.

Alternatively, we can apply the comprehension principle to build this set:


 * $R = \left\{{S: S \notin S}\right\}$

(R for Russell, of course.)

We ask the question: Is $R$ itself an element of $R$?

There are two possible answers: yes or no.

If $R \in R$, then $R$ must satisfy the property that $R \notin R$, so from that contradiction we know that $R \in R$ does not hold.

So the only other answer, $R \notin R$, must hold instead. But now we see that $R$ satisfies the conditions of the property that $R \in R$, so we can see that $R \notin R$ doesn't hold either.

Thus we have generated a contradiction from the comprehension principle.

Comment
This paradox is one of a series of antinomies arising from the inherent contradiction in allowing unrestricted use of the comprehension principle. In this case, it is being used to obtain a set which contains itself if and only if it does not contain itself.

A desire to avoid such antinomies was a motivation towards the development of various systems of axiomatic set theory.

The best-known system ZFC of axiomatic set theory includes only (relatively) restrictive methods of generating new sets by using properties. In particular:
 * The axiom of subsets allows us to use properties to define only subsets of sets which have already been demonstrated as being allowed to exist.
 * The axiom of replacement uses only those properties which define functions.

These restrictions both make the above argument invalid in ZFC, since the justification for the existence of the set $R$ is removed.

Some authors, for example, sidestep this issue entirely:
 * This logical impasse can be avoided by restricting the notion of set, so that 'very large' collections ... are not counted as sets. However this is done at some cost in simplicity, and in this book we shall do no more than keep to sets which appear to be harmless, and hope that paradoxes will not appear.

Again, from :
 * It is well known that an informal point of view in the theory of sets leads to contradictions. These difficulties all arise in operations with very large sets. We shall never need to deal with any sets large enough to cause trouble in this way, and, consequently, we may put aside all such worries.

He devised it in 1901.