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

SECTION E]

INDUCTIVE RELATIONS

571

one of the two senses which the word finite may have) belong to the class
R*'O, and they belong to it by definition, being defined as the class
/ 3 {. [a a. * a+ 1 e L: O()L: D),.3 fe/},
4 -i.e. as RE'O in the above sense. To infinite numbers, inductive proofs of
this kind starting from 0 cannot be applied.
The study of RB will occupy *90. The relation RB holds between x and y
if x (I r C'R) y or xRy or xR2y or etc. The study of this "etc. " occupies *91,
" on the powers of a relation." We may, for many technical purposes, regard
I r C'R as the 0th power of R; the other powers are R, R2, etc. If S is
a power of R, so is S I R. Now S R is i R'S, according to the definition in
*38. Thus if we have
R e /: Se fJs. SI Re pL: D,). P ep,
P must be a power of R, because the class of powers of R is a value of pu
which satisfies the hypothesis
Re: Sep. Ds. S R ep.
Conversely, if P is a power of' R, then P is reached by repetitions of the
process of turning S into S R, starting this process with R. Hence if P is
a power of R, we shall have
Re /:Se /L. SS SRe p: ),..P ep.
Consequently, if we denote the class of powers of R by Pot'R, we have
P e Pot'R. -:. Re ~: Se p. s. S R Lep: D. P e p.
We might use this as the definition of Pot'R; but we can get a somewhat
simpler form. For the above is shown, without much difficulty, to be
equivalent to
P e Pot'R.. P (I R)* R,
that is, P belongs to the ancestry of R with respect to R, in other words,
P is reached from R by proceeding along the series
R, R'R, IR'IR'R, etc.
which is the same as the series
R, R2, R3, etc.
The relation (I R)B is important on its own account. We put
Rts=( R) Df,
and then we put
Pot'R = RS'R Df.
We often want to include I C'R among the powers of R; the class
consisting of Pot'R together with I C'R we call Potid'R. The definition is
Potid'R = Rt'(I[ C'R),