Clippings

Cantor became aware of such antinomies in 1895, and over the next decades the mathematical community tried to find a way to patch up the logical breach they had created. The final resolution of this affair required the formal axiomatization of set theory — even as Euclid had provided his axiomatic approach to geometry — in which the carefully chosen axioms prohibited just such paradoxes as these. Logically, this was no easy matter. But, in the end, the newly created “axiomatic set theory” more carefully controlled precisely what was and what was not a “set.” Under this system, the “universal set” was not a set at all; it was exclued from the collection of objects that the axioms of set theory addressed. Thus, almost by magic, the paradox dissolved.

This resolution was, obviously, a compromise measure, an axiomatic attempt to carve away, with surgical precision, the troubling features of set theory while retaining all of the good points of Cantor’s creation. Cantor’s own, more informal approach is now called “naive set theory,” to contrast it with the logical superstructure of axiomatic set theory. The latter now stands as a satisfactory, albeit rather abstruse and technical, foundation for the theory of sets. It represents a triumph of the sentiments expressed by mathematician David Hilbert, who vowed, “No one will expel us from the paradise that Cantor has created.”