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.

Further reading:

• Algorithmic information theory
• Gödel's proof

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).