Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

116 MATHEMATICAL PHILOSOPHY
unity, makes it hang together, gives it and the systems
compendence. Suppose you had, as one might have, a
collection of propositional functions some of which talked
of the variables x, y, z,..., and some of which talked of
the variables x', y', z',...., in such a way that the former
symbols were not connected with the latter, then the
undetermined subject-matter of the collection of functions
would lack unity; the collection would not be compendent
and would, therefore, not be a postulate system. If the
subcollection involving x, y, z,... connected them and if
the same were true of the subcollection involving
x', y', z',..., and each of the two subcollections were
moreover, pregnant and compatible, then the original
collection would constitute two postulate systems, but
these would not together constitute one. They would be
independent; neither of them being included in the other,
they would not be related like the systems involved
respectively in IIHF and H/F', for the former of these is
a part of the latter.
In saying that a collection of propositional functions,
if it is to be a postulate system, must be compatible, or
consistent, we mean that its functions must be such as
not to involve contradiction among themselves-they will
be compatible unless at least two of them contradict each
other explicitly or implicitly. The reason for this requirement is obvious. For, if the collection contained two
mutually contradictory functions, the functions of the
collection would admit of no verifiers; whatever set of
admissible constants we might substitute for the variables
in the functions would yield a set of propositions of which
at least two would be mutually contradictory; hence, if
we called the collection a postulate system and then consistently called the system together with the theorems