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 just 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.

Attempting to avoid antinomies such as this motivated the development of several versions of axiomatic set theory.

For example, the best-known system ZFC of axiomatic set theory only includes (relatively) restrictive methods of generating new sets by using properties. In particular, the axiom of subsets only allows us to use properties to define subsets of already-existing sets, and the axiom of replacement only uses those properties which define functions. This restriction makes 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.

He discovered it in 1901.