Richard's paradox Guide, Meaning , Facts, Information and Description

Consider all English phrases that uniquely identify a real number, such as "that positive real number whose square is two", or "the ratio between a circle's diameter and its circumference". For any given positive integer n, there are a finite number of such phrases of length n (this might be zero if there are no meaningful phrases of length n, such as when n is 1 or 2). By listing first all phrases of length 3 in some order (say, alphabetical order), then all phrases of length 4, and so on, we generate an infinite ordered list of all such phrases. Call this the "Richard phrase list".

Now define a real number as follows:

"That positive real number whose integral part is zero, and its i-th digit after the decimal point (for all i) is equal to one if the i-th digit after the decimal point of the real number described by the i-th phrase in the Richard phrase list is 0, and equal to zero otherwise".

This English phrase appears to uniquely identify a real number, therefore it must appear in the Richard phrase list, say at position Q; but according to its own definition, its Q-th digit differs from the Q-th digit of number defined by the Q-th phrase, i.e. from itself.

(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.

See also

• 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).

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