A set of postulates for ordinary complex algebra

The well-known algebra which forrús one of the main branches of elementary mathematics is a body of propositions expressible in terms of five fundamental concepts—(the class of complex numbers, with the operations of addition and of multiplication, and the subclass of real numbers, with the relation of order)— and deducible from a small number of fundamental propositions, or hypotheses. The object of the present paper is to analyze these fundamental propositions, as far as may be, into their simplest component statements, and to present a list which shall not only be free from redundancies, and sufficient to determine the algebra uniquely, but shall also bring out clearly the relative importance of the several fundamental concepts in the logical structure of the algebra. A more precise statement of the problem is the following : we consider two undefined classes, A" and C; two undefined operations, which we may denote by © and © ; and an undefined relation, which we may denote by © ; and we impose upon these five (undefined) fundamental concepts certain arbitrary conditions, or postulates, to serve as the fundamental propositions of an abstract deductive theory (the other propositions of the theory being all the propositions which are deducible from the fundamental propositions by purely logical processes) ; the problem then is, to choose these fundamental propositions so that all the theorems of algebra, regarded as formal or abstract propositions, shall be deducible from them—the class K and C corresponding to the classes of complex and real numbers, respectively, and the symbols ©, ©, and © to the ordinary +, x , and <. Furthermore, the set of postulates, to be satisfactory, must determine the algebra uniquely ; in other words, the set of postulates adopted must be such that any two systems ( K, C, ©, ©, © ) which satisfy them all shall be simply isomorphic with respect to the fundamental concepts — that is, shall be capable of being brought into one-to-one correspondence in such a way that correspond-