Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/88

by "$\scriptstyle {{\hat {z}}(\phi z)}$" the class defined by $\scriptstyle {\phi {\hat {z}}}$, the symbol "$\scriptstyle {\phi \{{\hat {z}}(\phi z)\}}$" must be meaningless. Hence a class neither satisfies nor does not satisfy its defining function, and therefore (as will appear more fully in Chapter III) is neither a member of itself nor not a member of itself. This is an immediate consequence of the limitation to the possible arguments to a function which was explained at the beginning of the present Chapter. Thus if $\scriptstyle {\alpha }$ is a class, the statement "$\scriptstyle {\alpha }$ is not a member of $\scriptstyle {\alpha }$" is always meaningless, and there is therefore no sense in the phrase "the class of those classes which are not members of themselves." Hence the contradiction which results from supposing that there is such a class disappears.

(3) Exactly similar remarks apply to "the relation which holds between $\scriptstyle {R}$ and $\scriptstyle {S}$ whenever $\scriptstyle {R}$ does not have the relation $\scriptstyle {R}$ to $\scriptstyle {S}$." Suppose the relation $\scriptstyle {R}$ is defined by a function $\scriptstyle {\phi (x,y)}$, i.e. $\scriptstyle {R}$ holds between $\scriptstyle {x}$ and $\scriptstyle {y}$ whenever $\scriptstyle {\phi (x,y)}$ is true, but not otherwise. Then in order to interpret "$\scriptstyle {R}$ has the relation $\scriptstyle {R}$ to $\scriptstyle {S}$," we shall have to suppose that $\scriptstyle {R}$ and $\scriptstyle {S}$ can significantly be the arguments to $\scriptstyle {\phi }$. But (assuming, as will appear in Chapter III, that $\scriptstyle {R}$ presupposes its defining function) this would require that $\scriptstyle {\phi }$ should be able to take as argument an object which is defined in terms of $\scriptstyle {\phi }$, and this no function can do, as we saw at the beginning of this Chapter. Hence "$\scriptstyle {R}$ has the relation $\scriptstyle {R}$ to $\scriptstyle {S}$" is meaningless, and the contradiction ceases.

(4) The solution of Burali-Forti's contradiction requires some further developments for its solution. At this stage, it must suffice to observe that a series is a relation, and an ordinal number is a class of series. (These statements are justified in the body of the work.) Hence a series of ordinal numbers is a relation between classes of relations, and is of higher type than any of the series which are members of the ordinal numbers in question. Burali-Forti's "ordinal number of all ordinals" must be the ordinal number of all ordinals of a given type, and must therefore be of higher type than any of these ordinals. Hence it is not one of these ordinals, and there is no contradiction in its being greater than any of them^[1].

(5) The paradox about "the least integer not nameable in fewer than nineteen syllables" embodies, as is at once obvious, a vicious-circle fallacy. For the word "nameable" refers to the totality of names, and yet is allowed to occur in what professes to be one among names. Hence there can be no such thing as a totality of names, in the sense in which the paradox speaks of "names." It is easy to see that, in virtue of the hierarchy of functions, the theory of types renders a totality of "names" impossible. We may, in fact, distinguish names of different orders as follows: (a) Elementary names will be such as are true "proper names," i.e. conventional

1. The solution of Burali-Forti's paradox by means of the theory of types is given in detail in *256.