Naive Set Theory - Paradoxes

Paradoxes

We referred earlier to the need for a formal, axiomatic approach. What problems arise in the treatment we have given? The problems relate to the formation of sets. One's first intuition might be that we can form any set we want, but this view leads to inconsistencies. For any set x we can ask whether x is a member of itself. Define

Z = {x : x is not a member of x}.

Now for the problem: is Z a member of Z? If yes, then by the defining quality of Z, Z is not a member of itself, i.e., Z is not a member of Z. This forces us to declare that Z is not a member of Z. Then Z is not a member of itself and so, again by definition of Z, Z is a member of Z. Thus both options lead us to a contradiction and we have an inconsistent theory. More succinctly, one says that Z is a member of Z if and only if Z is not a member of Z. Axiomatic developments place restrictions on the sort of sets we are allowed to form and thus prevent problems like our set Z from arising. This particular paradox is Russell's paradox.

The penalty is that one must take more care with one's development, as one must in any rigorous mathematical argument. In particular, it is problematic to speak of a set of everything, or to be (possibly) a bit less ambitious, even a set of all sets. In fact, in the standard axiomatisation of set theory, there is no set of all sets. In areas of mathematics that seem to require a set of all sets (such as category theory), one can sometimes make do with a universal set so large that all of ordinary mathematics can be done within it (see universe). Alternatively, one can make use of proper classes. Or, one can use a different axiomatisation of set theory, such as W. V. Quine's New Foundations, which allows for a set of all sets and avoids Russell's paradox in another way.