Foundations with Imagination

TL;DR

This paper demonstrates that certain foundational set theories, including variants of ZFC, cannot eliminate imaginaries, meaning they cannot always provide canonical representatives for definable equivalence relations.

Contribution

It proves the limitations of specific countable set theories in eliminating imaginaries, highlighting fundamental constraints in their expressive power.

Findings

Countable set theory with certain axioms cannot eliminate imaginaries.

ZFC^{-} and related theories also fail to eliminate imaginaries.

Theories examined do not provide representatives for all definable equivalence relations.

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.