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

208

CARDINAL ARITHMETIC

[PART III

*120-211. F: Nc'p c NC induct - t'A. ). p e Cls induct
(We do not have an equivalence here, because, for aught we know, it
might be possible to determine the ambiguity of Nc'p so that Nc'p = A,
even when p e Cls induct. This will not be possible, however, if the axiom
of infinity is assumed.)
*120-212-213. F. A, I'x e Cls induct
*120'214. F:. p sm. ): p esindt. Cs induct s induct
We have a set of propositions applying induction to classes directly, and
not through the intermediary of cardinals. Thus we have
*120 251. F: 77 e Cls induct. ). lt v t'y e Cls induct
*120'26. F:. p e Cls induct:.,. D,,,. ( v t'): A: D. Op
We then state the axiom of infinity, and prove (*120'33) that it is
equivalent to the assumption that if a is an inductive cardinal, a a +, 1.
To prove this, we first prove various propositions about a +c 1, among others
the following:
*120'311. F: { ' a +, 1. a +c 1 = f, +1..a = sm". a ' a
*120322. F:. a NC induct. D: {! a. _. a a+ 1
We then proceed to consider subtraction (*120'41 —418), which only
gives a cardinal number when the subtrahend is an inductive cardinal.
We have
*120'41. F:. v e NC induct. a + v.: a + v = +c v. D. a = stn"I3
We might validly put a = /3 instead of a = sm"/3, since a = f will be true
whenever it is significant.
We have
*120411. F:. v e NC induct. D:
a! y - c v. ). / -cv E NC 7 >.- (y -c v) n t'7 e N0C
*120-4111. F:. v e NC induct. g! smty. D: 7 > v.. (7 -c v) e N,C
Hence we arrive at the conditions requisite for the usual point of view of
subtraction; namely,
*120-412. F: v NC induct. 7rv. j! sm"7y. D.(Q -c v) )= {(a)(a +c v = 7)}t
Also from *120'4111 we deduce
*120'414.: pu e NoC - t0. [! snlmt. *. (a -, 1)e e NoC
And from *120-411.*119-34, we find
*120-416. F: v e NC induct. g! 7 -c v.. (7y-c v) +c v = sm"n
We prove next that no proper part of an inductive class is similar to the
whole (*120'426), i.e. that inductive classes are non-reflexive, and various
connected propositions, e.g.
*120'423. F: a e N,C induct - '0.. (EBB)., e NC induct. a = (8 +c 1),