Foundations with Imagination

Abstract

We show that countable set theory, $ZFC^{-}+\forall x\ |x|\leq\omega$, is unable to eliminate imaginaries. In other words, this theory cannot provide representatives for arbitrary definable equivalence relations. We also see that $ZFC^{-}$ and ZFC^{-}+\exists\kappa(Inacc(\kappa)\wedge\forall x\ |x|\leq\kappa)$ also fail to eliminate imaginaries.