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

POSTULATE PROPERTIES 129
and the ci's, the c2's and C2"',..., in such a way that,
if two or more c's be related by some r, the corresponding
c"s are related by the corresponding r', then and only
then we say that the system S is categorical, or sufficient.
Two sets of verifiers that are transformable into each other
in the manner indicated are said to be of the same type.
It is easy to see the dictional propriety of the adjectives
" categorical" and " sufficient " as thus used. For if S
be categorical, or sufficient, it determines a category, or
class, of sets of verifiers, which sets are all of them of the
same type, and it (S) is " sufficient " to do that. The
Hilbert system is categorical, as are some of the other
systems above listed. For an interesting discussion of the
term categorical, of the advantages and disadvantages of
categoricalness, together with detailed proofs that certain
systems are, and certain others are not, categorical, 1
may refer you to the previously mentioned Fundamental
Concepts of Young and to Huntington's article in the
above-cited Monographs.
Euclid's Postulate System Defective.-In a previous
lecture I stated that Euclid's postulate system is defective
-mainly by omission-and promised to prove the fact
at a later stage. Owing, however, to time limitations and
to the insistence of many other topics which remain to be
considered, I have decided to omit the proof and to be
content with referring you to Young's Fundamental Concepts (pp. 12, I43) where the defectiveness in question is
demonstrated simply and clearly.