maths document

May 12th, 2015
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Does mathematics need new axioms?Solomon FefermanThe question, \Does mathematics need new axioms?", is ambiguous inpractically every respect.What do we mean by `mathematics'?What do we mean by `need'?What do we mean by `axioms'?You might even ask, What do we mean by `does'?Part of the ambiguity lies in the various points of view from which thisquestion might be considered. The crudest dierence lies between the pointof view of the working mathematician and that of the logician concernedwith the foundations of mathematics. Now some of my fellow mathematicallogicians might protest this distinction, since they consider themselves to bejust more of those \working mathematicians". Certainly, modern logic hasestablished itself as a very respectable branch of mathematics, and there arequite a few highly technical journals in logic, such as The Journal of Symbolic Logic and the Annals of Pure and Applied Logic, whose contents, froma 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 eldsand not see it as any dierent in general character from a paper on combinatorial graph theory or cohomology of groups; they belong to the samebig frame of mind, so to speak. But if you pick up Godel's paper on theincompleteness of axiom systems for mathematics, or his work and that ofCohen on the consisten

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