Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/86

must denote a definite integer; in fact, it denotes 111,777. But "the least integer not nameable in fewer than nineteen syllables" is itself a name consisting of eighteen syllables; hence the least integer not nameable in fewer than nineteen syllables can be named in eighteen syllables, which is a contradiction^[1].

(6) Among transfinite ordinals some can be defined, while others can not; for the total number of possible definitions is $\scriptstyle {\aleph _{0}}$^[2], while the number of transfinite ordinals exceeds $\scriptstyle {\aleph _{0}}$. Hence there must be indefinable ordinals, and among these there must be a least. But this is defined as "the least indefinable ordinal," which is a contradiction^[3].

(7) Richard's paradox^[4] is akin to that of the least indefinable ordinal. It is as follows: Consider all decimals that can be defined by means of a finite number of words; let $\scriptstyle {E}$ be the class of such decimals. Then $\scriptstyle {E}$ has $\scriptstyle {\aleph _{0}}$ terms; hence its members can be ordered as the 1st, 2nd, 3rd,‥‥ Let $\scriptstyle {N}$ be a number defined as follows. If the $\scriptstyle {n}$th figure in the $\scriptstyle {n}$th decimal is $\scriptstyle {p}$, let the $\scriptstyle {n}$th figure in $\scriptstyle {N}$ be $\scriptstyle {p+1}$ (or 0, if $\scriptstyle {p=9}$). Then $\scriptstyle {N}$ is different from all the members of $\scriptstyle {E}$, since, whatever finite value $\scriptstyle {n}$ may have, the $\scriptstyle {n}$th figure in $\scriptstyle {N}$ is different from the $\scriptstyle {n}$th figure in the $\scriptstyle {n}$th of the decimals composing $\scriptstyle {E}$, and therefore $\scriptstyle {N}$ is different from the $\scriptstyle {n}$th decimal. Nevertheless we have defined $\scriptstyle {N}$ in a finite number of words, and therefore $\scriptstyle {N}$ ought to be a member of $\scriptstyle {E}$. Thus $\scriptstyle {N}$ both is and is not a member of $\scriptstyle {E}$.

In all the above contradictions (which are merely selections from an indefinite number) there is a common characteristic, which we may describe as self-reference or reflexiveness. The remark of Epimenides must include itself in its own scope. If all classes, provided they are not members of themselves, are members of $\scriptstyle {w}$, this must also apply to $\scriptstyle {w}$; and similarly for the analogous relational contradiction. In the cases of names and definitions, the paradoxes result from considering non-nameability and indefinability as elements in names and definitions. In the case of Burali-Forti's paradox, the series whose ordinal number causes the difficulty is the series of all ordinal numbers. In each contradiction something is said about all cases of some kind, and from what is said a new case seems to be generated,

1. This contradiction was suggested to us by Mr G. G. Berry of the Bodleian Library.
2. $\scriptstyle {\aleph _{0}}$ is the number of finite integers. See *123.
3. Cf. König, "Ueber die Grundlagen der Mengenlehre und das Kontinuumproblem," Math. Annalen, Vol. lxi. (1905); A. C. Dixon, "On 'well-ordered' aggregates," Proc. London Math. Soc. Series 2, Vol. iv. Part i. (1906); and E. W. Hobson, "On the Arithmetic Continuum," ibid. The solution offered in the last of these papers depends upon the variation of the "apparatus of definition," and is thus in outline in agreement with the solution adopted here. But it does not invalidate the statement in the text, if "definition" is given a constant meaning.
4. Cf. Poincaré, "Les mathématiques et la logique," Revue de Métaphysique et de Morale, Mai 1906, especially sections vii. and ix.; also Peano, Revista de Mathematica, Vol. viii. No.5 (1906), p. 149 ff.