Richard's paradox

Richard's paradox is a paradox found by Jules Richard in 1905, a variant on Cantor's diagonal argument.

Consider all the English statements that uniquely specify a real number. One example statement would be "That positive real number whose square is two." These statements can be ordered alphabetically, and thus each statement gets a number, its position in this sequence. Let's call this sequence the "Richard statement list".

Now define a real number as follows:

"the real number whose integral part is zero, and its n-th digit after the decimal point is equal to one if the n-th digit after the decimal point of the real number described by the n-th statement in the Richard statement list is 0, and equal to zero otherwise".

We appear to have just defined a real number in English, so the previous statement should occur somewhere in the Richard statement list; but it cannot, since the defined number differs from the number defined by statement n at digit position n.

(The standard technical caveat applies: decimal expansions ending only in nines are not allowed.)

The paradox arises because the notion of "definable in English" is not cleanly enough defined; as soon as one picks a clean and detailed definition of this concept, the paradox evaporates.

Compare the Berry paradox, which is another take on numbers definable in English.

References

Jules Richard, "Les Principes des mathématiques et le problème des ensembles", Revue générale des sciences pures et appliquées (1905); translated in Heijenoort J. van (ed.), Source Book in Mathematical Logic 1879-1931 (Cambridge, Mass., 1964).