The Crisis of Foundations

The crisis took place in many parts of mathematics but was particularly acute in set theory and logic. In 1874 Cantor developed a theory called transfinite arithmetic which dealt with the arithmetic of infinite sets. It was based on the naive set theory now taught in schools, for example there is an axiom, the axiom of comprehension, which says: given a well-defined property P there is a set S which consists of all objects which satisfy the property P. This is written

Thus the set of all even numbers and the set of all living creatures exist, provided we can always tell whether an object has these properties or not. In this way, sets were also admitted to the Platonist realm.

Cantor's theory was an immensely powerful theory and is still the subject of much research today. However, it led to a number of paradoxes the most famous of which is called Russell's paradox. Russell defines a property P which a set may or may not have. P(S) holds if S is not an element of itself. The set of even numbers, for example, satisfies this property, because the set of even numbers is not, itself, an even number. Most sets in fact have this property, and it is always possible to tell whether a given set satisfies the property or not. So there must be a set R where

Now the paradox. We ask whether R is a member of R. The only objects which belong to R are those which do not belong to themselves, so if R belongs to R then R does not belong to R. Conversely, if R does not belong to R then it follows that R belongs to R.

The conclusion must be drawn that despite having a perfectly clear definition, the set R does not exist. The axiom of comprehension leads to an inconsistency. This paradox marked a major set-back particularly for Frege's Logicist project (below), but led to fruitful developments in axiomatic set theory. It might be added, in defence of Cantor, that he was aware of contradictions like this and was usually very careful in his formulations of the axioms, but others, including Frege, certainly adopted the axiom of comprehension.