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

*151. ORDINAL SIMILARITY.

Summary of *151.
In this number, we give the definition of ordinal similarity, and various
equivalent forms; we prove that ordinal similarity is reflexive (*151'13),
symmetrical (*151'14) and transitive (*151'15), and we give some particular
cases of ordinal similarity (*151'6 if.). Propositions in this number should be
compared with those in *73, to which they are analogous.
The class of ordinal correlators of P and Q is written P smor Q, where
" smor" stands for " similar ordinally." We put
Psmor Q=SSe 1-l. CGQ='S.P =S;QJ Df.
(We might equally well put
_4 -P smor Q=(1 -1) n 'C'Q t tQ'P Df,
which is an equivalent but more condensed form of the definition.) We then
define "P is ordinally similar to Q" as meaning that there is at least one
ordinal correlator of P and Q, i.e.
smor = PQ (g! P smior Q) Df.
We shall find that if P and Q generate well-ordered series, they have at
most one correlator (*250'6), but this does not hold in general for other
series.
After giving the elementary properties of ordinal similarity, we have
three important propositions on its connection with cardinal similarity,
namely: (*151'18) if P is similar to Q, the field of P is similar to the
field of Q (the converse does not hold in general, but holds if P and Q are
finite serial relations); (*151'19) if C'P is similar to C'Q, there is a relation
R similar to Q and having C'P for its field, and vice versa; (*151'191) S is
an ordinal correlator of P and Q when, and only when, it is a cardinal
correlator of C'P and C'Q and P=S;Q.
We then have a set of propositions on correlators of the form Sr C'Q
(*151'2 —243). Most of the correlators with which we shall be concerned
are of this form. The most useful proposition here is