Jules Richard - Richard's Paradox

Richard's Paradox

The paradox was first stated in 1905 in a letter to Louis Olivier, director of the Revue générale des sciences pures et appliquées. It was published in 1905 in the article Les Principes des mathématiques et le problème des ensembles. The Principia Mathematica by Alfred North Whitehead and Bertrand Russell quote it together with six other paradoxes concerning the problem of self-reference. In one of the most important compendia of mathematical logic, compiled by Jean van Heijenoort, Richard's article is translated into English. The paradox can be interpreted as an application of Cantor's diagonal argument. It inspired Kurt Gödel and Alan Turing to their famous works. Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows:

Let E be the set of real numbers that can be defined by a finite number of words. This set is denumerable. Let p be the nth decimal of the nth number of the set E; we form a number N having zero for the integral part and p + 1 for the nth decimal, if p is not equal either to 8 or 9, and unity in the contrary case. This number N does not belong to the set E because it differs from any number of this set, namely from the nth number by the nth digit. But N has been defined by a finite number of words. It should therefore belong to the set E. That is a contradiction.

Richard never presented his paradox in another form, but meanwhile there exist several different versions, some of which being only very loosely connected to the original. For the sake of completeness they may be stated here.