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

280

CARDINAL ARITHMETIC

[PART III

we make a fresh start. We lave, by hypothesis, a progression R whose
domain is contained in CI'p; hence s'D'R C p. Thus it will suffice to
prove
Ko e NC mult. R e Prog. D'R C Cls induct. D.! N% n s'D'R,
where the conditions of significance require that D'R should consist of
classes.
For this purpose, we prove that no member of D'R can be the last that
has new members wlich have not occurred before. The proof proceeds by
showing that if this were not so, s'D'R would be an inductive class, and
therefore, by *120-75, D'R would be an inductive class. Hence (*124'534)
the members of D'R which introduce new terms form an 0o, by *123'19;
and so therefore do the classes of new terms which they introduce (*124'535).
Hence (*124'536) a selection from these classes of new terms, which is a subclass of s'D'R, is also an 0o, and therefore (*124-54) there is a progression
contained in s'D'R if the selection in question exists. This completes
the proof.
In virtue of *12451.1 and *120'74, we have, without the multiplicative
axiom,
*124 6.: p e Cls induct.. Cl'Cl'p e Cls refl
Hence if it could be shown that Cl'p cannot be reflexive unless p is
reflexive, a double application of this would enable us, by means of *124'6,
to identify the two definitions of the finite without the multiplicative axiom.
*124-01. Cls refl = p {(gR). R e 1 -1. I'R C D'R.;! B'R. p = D'R} Df
An equivalent definition would be
Cls refl = D" {( -1) n a(B - Cnv"('B} 1)f.
*124-02. NC refl = Noc"Cls refl Df
*124-021. Nc'p e NC refl. =. Noc'p e NC refl Df
*124'03. NC mult = NC r a { c e a r Cls ex2 excl.:. a*! ea'K} Df
*124-1.: p e Cls refl. -. (2tR). R 1 -1. d'R C D'R. 5! B'R. p = D'R
[(*124-01)]
*12411.: R e 1-1. 'R C DR. B'R. ). D'R e Cls refl [*124-1]
*124'12. F. No C Cls refl [*123-12. *124-1]

*124'13.: p e CJs refl. ). t! 0 n Cl'p [*124'1. *123-192]