For every natural number $m$, the existentially closed models of the theory of fields with $m$ commuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a m
odel-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.
This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of nat
ural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature S, a class ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.
