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

SECTION C]

INDUCTIVE CARDINALS

211

have, in that type, 9 =A, and the same would hold of 10, 11,.... This
possibility has to be taken account of in what follows.
In order to give typical definiteness to the axiom of infinity, we write
*120-04. Infin ax (x). =: a e NC induct. Da.! a (x) Df
Then "Infin ax (x)" states that, if a is any inductive cardinal, there are
at least a objects of the same type as x.
*120-1. F: a e NC induct. -. a (+c 1)* 0 [(*120-01)]
*120-101. F:: a eNC induct. -: I e.. +: 0 e:Oe:D. aetz
[*120-1. *90-131. *38'12]
The right-hand side of the above equivalence gives the usual formula for
mathematical induction. Observe that the conditions of significance require
that +cl should be taken in the same type as:. This fact is specially
relevant in the proof of *120'15.
The symbol " NC induct" is of ambiguous type not necessarily the same
in different occurrences; also, according to the convention explained in the
prefatory statement as holding for NC and NC induct, "a,,8 e NC induct" will
iiot imply that a and,3 are of the same type. Accordingly to avoid error in
connection with *120'1'101 typical definiteness is required as in the three
following propositions.
*120-102.: a e NC induct. _. a (+c 1)* 0, [(*120-011)]
*120-103.:: a e NC induct. _:.! e,u. 2t. (: + 1)c, e: ~, e p: ),. a e
[*120-101]
*120-11. F:. a e N,C induct:.:. D. (I +c 1 ): q0,: ). >a
[*120-102. *90-112]
*12012. F.0 e NC induct *l120-101 ~1
L a]
*120'121. F: a e NC induct. D. (a -+ 1) e NtC induct [*90-172. *120'102]
By means of this proposition and *120'12, any assigned cardinal in the
series of natural numbers can be shown to be an inductive cardinal; thus e.g.
to show that 27 is an inductive cardinal, we shall only have to use *120'121
twenty-seven times in succession.
*120'122.. 1 e NC induct [*120-12-121. *110641]
*120'123. F. 2 e NC induct. etc. [*120'122'121. *110-643]
*120-124. F.a+c 1 0
Dem.. 110-4. Transp. ) F: a ~ NC.. a+ 1 = A.
[*101'12] D.a + 1 0 (1)
F. *110632. F:. a eNC. D: ea+cl I. 3. g! 4:
[*24'63]: A~ea+cl:
[*54-102]: a +c 1 + 0 (2)
F.(1). (2). ). Prop
14-2