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

ESSENTIAL DISCRIMINATIONS 117
derivable therefrom a doctrinal function, it would be one
having no values, admitting of no interpretation, giving
rise to no true doctrine. For this reason an incompatible
collection of propositional functions is not called a postulate system.
If a collection of propositional functions be compatible,
how may the fact be ascertained? There is but one known
test: if we find a set of constants that we are convinced
verify the functions, then and only then we say the
collection is compatible. If you will examine Hilbert's
book, you will find that he showed, or rather indicated
very briefly how to show, that his postulate system is
compatible by indicating how to show that the postulates
are verified by the pairs and certain systems of pairs of a
specified set of algebraic numbers. In Lecture VI, I have
shown in detail the compatibility of the postulates of
HIF by showing that they are verified by the set of dyads
and certain systems of dyads of the real numbers, and
have indicated the analogous procedure for the postulates
of IAF'. That is one of the reasons why Lecture VI
is so detailed-I desired it should incidentally serve to
exemplify what we may call compatibility proof.