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Richard paradox

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

Consider all the English sentences that uniquely specify a real number. One example sentence would be "That positive real number whose square is two." These sentences can be ordered alphabetically, and thus each sentence gets a number, its position in this sequence. Now define a real number as follows: its 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 sentence in our list is 0, and equal to zero otherwise. We have just defined a real number in English, so the previous sentence should occur somewhere in our list; but it cannot, since the defined number differs from the number defined by sentence 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).



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