Foundations with Imagination

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