Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

64

INTRODUCTION

[CHAP.

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*.
(6) Among transfinite ordinals some can be defined, while others can
not; for the total number of possible definitions is Ko,, while the number of
transfinite ordinals exceeds Ko. 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+.
(7) Richard's paradox~ 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 E be the class of such decimals. Then E has
0o terms; hence its members can be ordered as the 1st, 2nd, 3rd,.... Let
N be a number defined as follows. If the nth figure in the nth decimal is p,
let the nth figure in N be p + 1 (or 0, if p = 9). Then N is different from
all the members of E, since, whatever finite value n may have, the nth figure
in N is different from the nth figure in the nth of the decimals composing E,
and therefore N is different from the nth decimal. Nevertheless we have
defined N in a finite number of words, and therefore Nought to be a member
of E. Thus N both is and is not a member of 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 w, this must also apply to 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,
* This contradiction was suggested to us by Mr G. G. Berry of the Bodleian Library.
t+ o is the number of finite integers. See *123.
+ Cf. Konig, "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.
~ Cf. Poincare, "Les mathematiques et la logique," Revue de Metaphysique et de Morale,
Mai 1906, especially sections vii. and ix.; also Peano, Revista de Mathematica, Vol. vIII. No. 5
(1906), p. 149 ff.
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