Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/78

We may now proceed in exactly the same way to third-order matrices, which will be functions containing second-order functions as arguments, and containing no apparent variables, and no arguments except individuals and first-order functions and second-order functions. Thence we shall proceed, as before, to third-order functions; and so we can proceed indefinitely. If the highest order of variable occurring in a function, whether as argument or as apparent variable, is a function of the $\scriptstyle {n}$th order, then the function in which it occurs is of the $\scriptstyle {n+1}$th order. We do not arrive at functions of an infinite order, because the number of arguments and of apparent variables in a function must be finite, and therefore every function must be of a finite order. Since the orders of functions are only defined step by step, there can be no process of "proceeding to the limit," and functions of an infinite order cannot occur.

We will define a function of one variable as predicative when it is of the next order above that of its argument, i.e. of the lowest order compatible with its having that argument. If a function has several arguments, and the highest order of function occurring among the arguments is the $\scriptstyle {n}$th, we call the function predicative if it is of the $\scriptstyle {n+1}$th order, i.e. again, if it is of the lowest order compatible with its having the arguments it has. A function of several arguments is predicative if there is one of its arguments such that, when the other arguments have values assigned to them, we obtain a predicative function of the one undetermined argument.

It is important to observe that all possible functions in the above hierarchy can be obtained by means of predicative functions and apparent variables. Thus, as we saw, second-order functions of an individual $\scriptstyle {x}$ are of the form

$\scriptstyle {(\phi ).f!(\phi !{\hat {z}},x)}$ or $\scriptstyle {(\exists \phi ).f!(\phi !{\hat {z}},x)}$ or $\scriptstyle {(\phi ,\psi ).f!(\phi !{\hat {z}},\psi !{\hat {z}},x)}$ or etc.,

where $\scriptstyle {f}$ is a second-order predicative function. And speaking generally, a non-predicative function of the $\scriptstyle {n}$th order is obtained from a predicative function of the $\scriptstyle {n}$th order by turning all the arguments of the $\scriptstyle {n-1}$th order into apparent variables. (Other arguments also may be turned into apparent variables.) Thus we need not introduce as variables any functions except predicative functions. Moreover, to obtain any function of one variable $\scriptstyle {x}$, we need not go beyond predicative functions of two variables. For the function $\scriptstyle {(\psi ).f!(\phi !{\hat {z}},x)}$, where $\scriptstyle {f}$ is given, is a function of $\scriptstyle {\phi !{\hat {z}}}$ and $\scriptstyle {x}$, and is predicative. Thus it is of the form $\scriptstyle {F!(\phi !{\hat {z}},x)}$, and therefore $\scriptstyle {(\phi ,\psi ).f!(\phi !{\hat {z}},x)}$ is of the form $\scriptstyle {(\phi ).F!(\phi !{\hat {z}},x)}$. Thus speaking generally, by a succession of steps we find that, if $\scriptstyle {\phi !{\hat {u}}}$ is a predicative function of a sufficiently high order, any assigned non-predicative function of $\scriptstyle {x}$ will be of one of the two forms

$\scriptstyle {(\phi ).F!(\phi !{\hat {u}},x),~(\exists \phi ).F!(\phi !{\hat {u}},x)}$,

where $\scriptstyle {F}$ is a predicative function of $\scriptstyle {\phi !{\hat {u}}}$ and $\scriptstyle {x}$.