THE DIALETHIC MINI-HISTORY OF SET THEORY

In the one hundred years since 1902, mathematicians have struggled to rehabilitate the theory of sets. There are plenty of versions: Zermelo-Frankel (ZF), ZF with Choice (ZFC), von Neumann-Bernays-Gödel (NBG), Kelley-Morse (KM), Math Logic (ML), New Foundations (NF), Set Theory and Its Logic (STL), and others. They are not equivalent, but each does roughly as well as any other in mimicking the integers. These modern theories are, so far as we can see, consistent and free of the Russell Paradox. Each is a development of the iterative conception of set and each is an articulation of the cumulative hierarchy. For all its post-Paradox centrality, the hierarchical approach to sets has attracted an ambush of its own in the dialethist's insistence that the approach is entirely mistaken. Here, then, is the Dialethist's Tale.