Conclusion

The history of the Axiom of Choice is the history of how an assumption’s status can change. At a given point in time, each assumption in mathematics forms part of a nexus of suppositions with varying degrees of explicitness. Throughout history, mathematicians have operated within conceptual frameworks in which certain assumptions were stated but in which, on the other hand, certain assumptions were tacit or even unconscious. Euclid, among others, presupposed properties of the continuum that were not recognized explicitly as necessary for geometry until the nineteenth century. During the late nineteenth century, Frege emphasized the need to make all one’s assumptions explicit, and Hubert’s axiomatization of geometry furthered this process by using a formal axiomatic method as distinct from that of Euclid. Zermelo’s Axiom of Choice is best viewed as a further attempt to formulate an implicit assumption explicitly.