Around 1896 Cantor discovered the Burali-Forti paradox and the Cantor paradox. In the Burali-Forti paradox the assumption that all transfinite ordinals form a set leads, by Cantors results, to the result that there is an ordinal that is less than itself. A similar situation holds for cardinals and is called the Cantor paradox.

The comprehension principle is that given any well defined logical or mathematical property then there exists a set of all objects satisfying that property. In symbols this is given x there exists a set {x|p(x)}, where p represents the logical or mathematical property. Around 1901 Zermelo and Russel discovered a very elementary contradiction. The Zermelo-Russell paradox shows the comprehension principle is contradictory. Using the comprehension principle with the very basic property that p(x) = x Ï x we are led to a contradiction. If R = {x|p(x)} then if R Î R by definition we have R Ï R. Similarly, if R Ï R then, again by definition, R Î R. Thus we have a contradiction from some very basic principles.

In addition to these logical paradoxes there are many other semantic paradoxes formulated by Russell, Richard, Konig, Grelling and others.

These paradoxes caused a turmoil in mathematics. They led Hilbert to claim that to claim a set of mathematical objects exists is tantamount to proving that the corresponding axiom system is consistent or free of contradictions.

In 1903 Poincare used the paradoxes in the books of Frege and Russell to criticise logicism and formalism. He coined an important notion called predicativity. He claimed impredictive definitions should be avoided in mathematics. A definition is impredictive when it introduces an element by reference to a totality that already contains the element. Poincare found examples of impredicative procedures in all of the paradoxes he studied.

The successor function is s(N). Dedekind used this function and impredicativity to define the set of natural numbers. There was no problem or paradox here.

Richard’s paradox introduces definable real numbers. There must be a countable number of these. This leads to a paradox by using an argument similar to Cantor’s diagonal process used to prove that R is not countable.

In the predicativity approach all mathematical objects beyond the natural numbers must be introduced by explicit definitions. The definition must be predicative in the sense of referring only to totalities that have already been established before the object one is defining. Russell and Weyl accepted and developed this idea.

Zermelo pointed to cases where impredicativity was used without problems, including: Dedekind’s defining the set of natural numbers, Cauchy’s proof of the fundamental theorem of algebra and the least upper bound in real analysis. Zermelo said these definitions are innocuous in the sense that the object being defined is not created by these definitions it is merely singled out.

Poincare’s idea of abolishing impredicative definitions was utilized by Russell. He developed the theory of types which require members of sets to be of a similar type. This became the basis for the Principia Mathematica where whitehead and Russell 1910 -1913 developed a foundation of mathematics.

There were further studies of type theory by Chwistek, Ramsey, Russell, Weyl and recently by Feferman (1998). This approach does not fit neatly into the triad of logicism, formalism and intuitionism.

The “axiom of choice”, AC, is the principle that given any infinite family of disjoint nonempty sets, there is a set, known as a choice set, that contains exactly one element from each set in the family. Critics Borel, Baire and Lebesgue had all relied on AC to prove theorems in analysis. The axiom was suggested by Erhard Schmidt a student of Hilbert.

With Zermelo’s proof of the well ordering theorem he was led to develop the axiom system using the ideas of Dedekind and Cantor. With additions by Fraenkel, Von Neumann, Weyl and Skolem the axiom system became the one we know today. The system is called the ZFC system (for Zermelo-Fraenkel with choice). This codifies the key traits of modern mathematical methodology. It embodies the 4 principle of Hilbert and it or the von Neumann-Bernays-Godel version is the system that most mathematicians regard as the working foundation for their discipline.

The acceptance of impredicative and existential methodology became known as “Platonism”. Brouwer started to develop his ideas in 1905 and in his thesis in 1907. In intuitionism individual consciousness is the one and only source of knowledge. Brouwer contributed the fixed point theorem to topology. By the end of World War I he began publishing his foundational ideas. He helped to create the famous crisis. He established a distinction between formalism and intuitionism.

In 1921 Weyl published a paper in Mathematische Zeitschrift espousing intuitionism. Hilbert answered this criticizing Weyl and Brouwer. These ideas spurred Brouwer to a new theory of the continuum which enticed Weyl and and brought him to announce the coming of a new age.