Russell and Paradoxes: The Crisis of the Foundations of Mathematics

Mathematics on Shaky Ground

At the end of the 19th century, mathematicians thought they had found a solid foundation for all of mathematics through Cantor's set theory. Then, in 1902, the young Bertrand Russell wrote a letter to Gottlob Frege, who was working on the "Foundations of Arithmetic": at the very root of Frege's system there was a contradiction.

Russell's paradox: consider the set of all sets that are not members of themselves. Is this set a member of itself? If yes, it must not be in the set. If no, then it must be there. This logical contradiction destroyed Frege's system. Upon receiving Russell's letter, Frege wrote: "The worst that can befall a scientific author is to discover a refutation of his work after its completion." He was devastated.

Type Theory and the "Principia Mathematica"

Russell proposed a solution—the theory of types: a ban on self-reference through a hierarchy of levels. Statements about objects are of one type, statements about statements are of another type. Therefore, the "set of all sets" is simply forbidden to construct.

Together with Whitehead he wrote "Principia Mathematica" (1910–1913)—three volumes logically deriving mathematics from logic. The proof that $1+1=2$ takes hundreds of pages and is contained in Volume 2. This was a grand attempt—and it showed how complex "obvious" mathematics is.

Hilbert's program: is it possible to formalize all of mathematics, proving its completeness (every true statement is provable) and consistency (no contradiction is provable)? This seemed an achievable goal—until in 1931 Gödel showed that it was impossible.

Question for reflection: Russell's paradox showed that the self-evident foundation collapses under analysis. Which "self-evident" foundations in your professional field or organization may contain hidden contradictions?