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

*101. ON 0 AND 1 AND 2.

Sunzmary of *101.
In the present number, we have to show that 0 and 1 and 2 as previously
defined are cardinal numbers in the sense defined in *100, and to add a few
elementary propositions to those already given concerning them. We prove
(*101'12'241) that 0 and 1 are not null, which cannot be proved, with our
axioms, for any other cardinal, except (in the case of finite cardinals) when
the type is specified as a sufficiently high one. Thus we prove (*101'42'43)
that 2cis and 2Rel exist; this follows from A V and A: V. We prove
(*101'22'34) that 0 and 1 and 2 are all different from each other. We prove
(*101'15-28) that sm"0 = 0 and sm"1= 1, but we cannot prove sm"2=2
unless we assume the existence of at least two individuals, or define the first
2 in " sm"2 = 2 " as a 2 of some type other than 2indiv, where " Indiv " stands
for the type of individuals.
It should be observed that, since 0 and 1 and 2 are typically ambiguous,
their properties are analogous to those of " Nc'a " rather than to those of p,
where p e NC. For example, we have
*100-511.: g! Nc',8. 3. sm"Nc', = Nc',8
but we shall not have p e NC.. sm"D j = p unless the "sm " concerned
is homogeneous, since in other cases the symbols do not express a significant
proposition. But in *100'511 we may substitute 0 or 1 or 2, and the
proposition remains significant and true. In fact we have (*101'1'2'31)
F. 0 = Nc'A. 1 = Nc'x. 2 = Nc'('t'x v tA),
where 0 and 1 and 2 have an ambiguity corresponding to that of "Nc."
*101'1. F. 0 = Nc'A [*73'48. *100-1]
~101'11. -. 0 e NC [*101'1. *100-4]
*101'12. F. a! 0 [*51'161. (*54-01)]
*10113.. Fj! 0 n Cl'a. A e 0 n Cl'a [*51'16 *60'3]
2-2