Foundations with Imagination

TL;DR

The paper investigates whether countable-set theories, notably $ZFC_{count}^{-}$ (i.e., $ZFC$ without Powerset plus the axiom that every set is countable), can eliminate imaginaries. It develops forcing-automorphism techniques, starting from a warmup with $SOA$ and extending to $ZFC_{count}^{-}$, to produce a definable equivalence relation on $\mathbb{R}$—mutual constructibility $xEy\iff L[x]=L[y]$—that cannot be witnessed by any definable function $F$ on $\mathbb{R}$. Central to the argument are two lemmas (rangeOutOfL and MovingNames) and an inductive/base-case framework built via a Boolean-algebra automorphism approach, including Sikorski-type lifting results. The main result shows that $ZFC_{count}^{-}$ cannot eliminate imaginaries, and the same conclusion extends to $ZFC^{-}$ and to variants with an inaccessible κ; the work clarifies limitations of quotients and Morita completeness in countable-set theories and demonstrates that internal categoricity need not entail elimination of imaginaries.

Abstract

We show that countable set theory, $ZFC^{-}+\forall x\ |x|\leqω$, 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^{-}+\existsκ(Inacc(κ)\wedge\forall x\ |x|\leqκ)$ also fail to eliminate imaginaries.

Theorems & Definitions (30)

• Definition 1
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 5
• Lemma 6
• Lemma 7
• proof