Why Are All the Sets All the Sets?

Abstract

Necessitists about set theory think that the pure sets exists, and are the way they are, as a matter of necessity. They cannot explain why the sets ( de rebus ) are all the sets. This constitutes the Ur‐Objection against necessitism; it is the primary motivation cited by potentialists about set theory. At least three families of potentialism draw motivation from the Ur‐Objection. Contingentists think that any things could form a set even if they actually did not. Prioritists think that sets hyperintensionally depend upon their members. Structural‐potentialists think that any possible set‐hierarchy could be extended. However, once we have disentangled these three versions of potentialism, we see that the Ur‐Objection should not motivate anyone.