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In article <1903t2INNanu@cs.utexas.edu> turpin@cs.utexas.edu (Russell Turpin)
 writes:

   >For some theories, there are classical axiomatizations that stand
   >apart from set theory, e.g., arithmetic, real algebra, and plane
   >geometry.  Curiously, these all have intuitive axiomatizations
   >that do not rely on sets, and at least in the case of arithmetic
   >and plane geometry, axioms that were stated before set theory was
   >developed.  But the "natural" axioms for groups seem to rely on
   >sets: "a group is a *set* with a binary operator + such that ..."

  The concept of a group involves the notion of set, but the
elementary group axioms don't rely on sets any more than the axioms of
any other first order theory. If you want to talk about all models of
the axioms of elementary arithmetic, or about all real-closed fields
you will again speak of sets. In ordinary mathematics, one is very
much concerned with general models of the elementary group axioms,
which were introduced precisely to describe a structure common to many
mathematical objects, but not with general models of the axioms of
elementary arithmetic.

  Most group theory of course cannot be formulated within the elementary
theory of groups, for the reason indicated above. If you want a theory
weaker than ZFC within which e.g. the study of finite groups can be carried
out, such theories have been formulated, and somebody more knowledgeable
will be able to give a suitable reference.

  However, I wish to take issue with your statement that the classical
axiomatizations of arithmetic, real algebra, or elementary geometry do
not rely on sets. In each case the intuitive axiomatization does rely on
some notion of set (of natural numbers, of real numbers, of points) -
namely in the induction axiom or some continuity axiom (such as the
completeness axiom for ordered fields). The corresponding first order
theories are another matter. In the case of arithmetic, the first
order theory is as we know incomplete, in the case of the real numbers,
it turns out to be complete. But in both cases, the classical and
intuitive axiomatization is the second order one.