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

POSTULATE PROPERTIES 121
having such a certainty respecting any practical affair, yet
refrained, for lack of certainty, from action where action
was called for, would be rightly judged, not necessarily
stupid, but foolish or morbid or insane.
The Hilbert Postulate System Not Intrinsically Superior
to Others.-In our discussions this particular system has
been the subject of so many commentaries that, despite
the precautions explicitly stated in Lecture II, it may
seem to you to be the subject of our study instead of being
merely one of the instruments employed. Perhaps I
need not remind you that what we have been mainly
examining is the nature of the important general conception denoted by the term " postulate system," and that,
instead of beginning with an abstract definition of the
concept, we have preferred to study it by means of a
specific representative, or typical, example. For such an
example, we have chosen the Hilbert system because of its
familiarity, accessibility and fame. Our purpose had
been served equally well, however, had we employed
some other system, whether logically equivalent or nonequivalent to that of Hilbert. Systems of both kinds
abound, and I shall presently refer you to some of
them.
Equivalence of Postulate Systems and Identity of Their
Doctrinal Functions.-Two postulate systems, S and S',
are said to be equivalent if, and only if, every postulate
in S is in S' or is deducible as a theorem from those in
S' and every postulate in S' is in S or is deducible from
those in S. The same conception may be approached
and viewed as follows: S, we know, gives rise to a doctrinal function, say, AF, composed of the postulates in S
and the theorems deducible therefrom. Similarly, S'
yields a doctrinal function, say, AzF'. Let us agree to