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 nth digit after the decimal point is equal to one if the nth digit after the decimal point of the real number described by the nth 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.
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