The Foundations of Science · Henri Poincaré
V
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 the _n_th decimal of _N_ shall be: 1, 2, 3, 4, 5, 6, 7, 8, 1, 1 [15] _Revue générale des sciences_, June 30, 1905. As we see, _N_ is not equal to the _n_th number of _E_, and as _n_ is arbitrary, _N_ does not appertain to _E_ and yet _N_ should belong to this assemblage since we have defined it with a finite number of words. We shall later see that M. Richard has himself given with much sagacity the explanation of his paradox and that this extends, _mutatis mutandis_, to the other like paradoxes. Again, Russell cites another quite amusing paradox: _What is the least whole number which can not be defined by a phrase composed of less than a hundred English words_? This number exists; and in fact the numbers capable of being defined by a like phrase are evidently finite in number since the words of the English language are not infinite in number. Therefore among them will be one less than all the others. And, on the other hand, this number does not exist, because its definition implies contradiction. This number, in fact, is defined by the phrase in italics which is composed of less than a hundred English words; and by definition this number should not be capable of definition by a like phrase.
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