Volume 1 / Part 3 — *101. ON 0 AND 1 AND 2.
Reading edition
*101. ON 0 AND 1 AND 2.
Summary 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 and exist; this follows from and . We prove (*101·22·34) that 0 and 1 and 2 are all different from each other. We prove (*101·15·28) that and , but we cannot prove unless we assume the existence of at least two individuals, or define the first 2 in "" as a 2 of some type other than , where "" 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 "" rather than to those of , where . For example, we have
*100·511.
but we shall not have unless the "" 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) where 0 and 1 and 2 have an ambiguity corresponding to that of "."
*101·1.
*101·11.
*101·12.
*101·13.
*101·14.
Dem.
*101·15.
Dem.
*101·16.
Dem.
*101·17.
Dem.
*101·2.
*101·21.
*101·22.
Dem.
*101·23.
Dem.
*101·24.
Dem.
*101·241.
*101·25.
Dem.
*101·26.
Dem.
*101·27.
Dem.
*101·28.
Dem.
*101·29.
Dem.
*101·3.
Dem.
*101·301.
In comparing *101·31 with *101·1·2·3, it should be observed that and are both classes, whereas in *101·1·2·3 there was no typical limitation beyond what was imposed by the conditions of significance.
*101·31.
Dem.
*101·32.
*101·33.
*101·34.
Dem.
*101·35.
*101·36.
Dem.
*101·37.
*101·38.
Dem.
*101·4.
Dem.
When we are considering the lowest type occurring in a context, our premisses do not suffice to prove . For every other type, this can be proved. Thus and give the required result for classes and relations respectively.
*101·41.
Dem.
*101·42.
Dem.
*101·43.