Does Mathematics Need New Axioms?

You might even ask, What do we mean by "does"? Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest difference lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Some logicians might protest this distinction since they identify themselves as (working) mathematicians who happen to specialize in mathematical logic. 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 as The Joumal of Symbolic Logic and the Annals of Pure and Applied Logic, whose contents, from a cursory inspection, look just like those of other mathematical journals, setting subjects aside. Looking even closer, you can pick up a paper on, say, the semi-lattice of degrees of unsolvability or the model theory of fields and not see it as any different in general character from a paper on combinatorial graph theory or cohomology of groups; they belong to the same big frame of mind, so to speak. But if you pick up G6del'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, we're in a different 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 axiomatic systems. And it's that stance I want to distinguish from that of the mathematician working on analysis or algebra or topology or degrees of unsolvability, and so on. It's awkward to keep talking about the logician as metamathematician, and I won't keep qualifying it that way, but that's what I intend. Though I won't at all neglect the viewpoint of the working mathematician, for most of this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem? [12], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. In recent years logicians have learned a great deal that is relevant to G6del's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and