Russell's Paradox

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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, $\map P S = S \in S$ is a property on which we can use the comprehension principle to build this set:

$T = \set {S: S \in S}$

which is the set of all sets which contain themselves.

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

$R = \set {S: S \notin S}$

($R$ 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 $R \notin R$ does not hold either.

Thus we have generated a contradiction from the comprehension principle.

$\blacksquare$


Also presented as

This result can also be presented as:

There is no set $A$ that has every set as its element.

Its proof follows the same lines: by assuming that such an $A$ exists, and considering the set $\set {x \in A: \map R x}$ where $\map R x$ is the property $x \notin x$.

The same conclusion is reached.


Also known as

This result is also known as Russell's antinomy.


Also see


Source of Name

This entry was named for Bertrand Russell.


Historical Note

Russell's Paradox was devised by Bertrand Russell in $1901$.


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:

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 1965: J.A. Green: Sets and Groups, sidestep this issue:

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 1971: Allan Clark: Elements of Abstract Algebra:

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.

1975: T.S. Blyth: Set Theory and Abstract Algebra puts it as:

This is not a serious problem since fortunately most sets encountered in mathematics are of a harmless nature ... It is very fortunate that most of the properties dealt with in mathematics (and indeed all the properties we will deal with) are "set forming" in the sense that there is a set whose elements satisfy the property in question.

1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) remarks:

The paradox is resolved by restricting the ways in which sets can be formed, so that it becomes inadmissible to consider 'the set of all those sets that are not members of themselves'. There are several ways of doing this, but they need not concern us here; they will not play a role in the rather simple set-theoretical arguments we shall meet.


Sources