Beth Cherryman

In what follows I will put forward a formulation of Russell’s paradox, which highlights the inconsistency of set theory.  I will outline the axiom of specification and revise the paradox accordingly.  Emphasizing the importance of set theory for mathematics, I shall state some of the implications of Russell’s paradox.  One implication being that there is no set that contains everything, that is, there is no universe.

In the sense that in mathematics everything is a set, set theory can be thought of as the foundation of mathematics.  The argument for claiming everything is a set is as follows:

-If x is an object that is not a set then x has no members

-x is equal to the empty set

-So, x is a set

-We have derived a contradiction

-Everything is a set

This proof does not seem entirely satisfactory.  When we say that x is equal to the empty set, what we mean is that if x is a set then it is equal to the empty set, which makes the proof circular.

In naïve set theory, we assume that given any predicate , the objects to which  applies form a set: A = x: (x) is a set.  Under this assumption it happens that a set is an element of itself.  For instance, the set of all non-England cricketers is itself a non-England cricketer and as such a member of itself.  A more important example is the set of all sets, the set of all sets is itself a set and so is a member of itself, that is, (x) = x is a set and A = the set of all sets.  As such A is a member of A.  Philosophers like Russell are uncomfortable with this occurrence; Russell only wants to consider a good set, that is sets that are not members of themselves.  This gives rise to Russell’s paradox.  Let S be the set of good sets, that is, the set of sets that do not contain themselves as elements.  Russell asks if S is a good set, or more precisely whether S is a member of itself.  Clearly, there are two possibilities, either S is an element of itself or it is not an element of itself.  If S is an element of itself, then it is a bad set because it is a set that is an element of itself, so it cannot be in S, which contains only good sets, so S is not an element of itself.  This is a contradiction.  If S is not an element of itself, then it is a good set, but the good sets are collected in S, so S is in S, so it is an element of itself.  This too is a contradiction.  Hence, the paradox.  Russell’s paradox has far reaching consequences.  As set theory underlies all branches of mathematics, if (as Russell’s paradox suggests) it is inconsistent then no mathematical truth can be trusted.

In an attempt to avoid the paradox of set theory certain axioms were introduced.  In particular for this discussion the axiom of specification.  The axiom of specification states that every set A and every property  there exists a set B whose elements are exactly those elements x of A for which A holds.  B = x is a member of A: (x).  When we combine the axiom of specification with Russell’s paradox we get the following:

-Let A be an arbitrary set and B be defined as x being a member of A: x not being a member of x

-We make the claim that B is not a member of A

-We prove this indirectly, by assuming B is a member of A

oIf B is a member of A then either B is a member of B or B is not a member of B

oB is a member of B and B is a member of A entails by our definition of B that B is not a member of B – Contradiction

oB is not a member of B and B is a member of A entails by our definition of B that B is a member of B – Contradiction

We have just found that for any set A there is a set B that is not in A on pain of contradiction.  This means that nothing contains everything, further there is no universe of discourse in set theory, that is, one cannot assume that the collection of all the objects in discourse of set theory is itself an object of discussion.  The paradox is unresolved, we still cannot be sure that any (mathematical) property is safe or leads to contradiction and since a contradiction entails anything, the whole of mathematics is in danger.

In sum set theory underlies all branches of mathematics.  Russell’s paradox concerns the set of all good sets, that is, the set of all sets that are not members of themselves.  Assuming that the set of all good sets is a good set leads to a contradiction as does assuming the set of all good sets is a bad set.  The axiom of specification says for every set A and every property  there exists a set B whose elements are exactly those elements x of A for which A holds.  B = x is a member of A: (x).  In Russell’s paradox (x) = x is not a member of x.  Russell’s paradox shows us that for any set A there is a set B that is not in A.  There is nothing that contains everything.  There is no universe.