The two preceding chapters were devoted to an outline of the
foundations of an elementary mathematical theory which consti-
tutes a fragment of arithmetic. In the present chapter we shall
proceed to considerations of a methodological nature, concern-
ing the system of axioms and primitive terms upon which that
theory is based.

We shall begin with concrete examples illustrating the remarks
of Section 39 concerning such problems as arbitrariness in the
selection of axioms and primitive terms, the possible omission of
superfluous axioms, and so on.

Let us start out with the question whether our system of Axioms
1-11--it will briefly be referred to as SYSTEM U--possibly contains
any superfluous axioms, that is, axioms which can be derived from
the remaining axioms of the system. We shall see at once that it
is easy to answer this question, and, moreover, affirmatively. In
fact, we have:

Three of the axioms of System U, namely, one of the Axioms 4 or 5,
Axiom 6, and one of the Axioms 10 or 11, can be derived from the
remaining axioms.

PROOF. We show first that

(I) either of the Axioms 4 or 5 can be derived from the other
with the help of Axioms 1-3.

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