Does mathematics need new axioms?

The question, \Does mathematics need new axioms?", is ambiguous in practically every respect. What do we m e a n b y `mathematics'? What do we m e a n b y `need'? What do we m e a n b y `axioms'? You might e v en ask, What do we m e a n b y `does'? Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest diierence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now s o m e o f m y fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there are quite a few highly technical journals in logic, such a s The Journal of Symbolic Logic and the Annals of Pure and Applied L ogic, whose contents, from a cursory inspection, look just like those of other mathematical journals, setting subjects aside. Looking even closer, you can pick u p a p a p e r o n , say, the semi-lattice of degrees of unsolvability or the model theory of elds and not see it as any diierent in general character from a paper on com-binatorial graph theory or cohomology of groupss they belong to the same big frame of mind, so to speak. But if you pick u p G odel's paper on the incompleteness of axiom systems for mathematics, or his work and that of Cohen on the consistency and independence of the Axiom of Choice relative to the axioms of set theory, w e're in a diierent frame of mind, because we are doing what Hilbert called metamathematics: the study of mathematics itself by the means of mathematical logic through its formalization in ax-iomatic systems. And it's that stance I want to distinguish from that of 1