Foundations with Imagination

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 $ZFC<sup>{-}$ and ZFC^{-}+\existsκ(Inacc(κ)\wedge\forall x\ |x|\leqκ)$ also fail to eliminate imaginaries.

• The paper shows that a foundational theory of countable set theory — $ZFC^{-}_{count}$ proves Gamma(U) — cannot eliminate imaginaries, specifically witnessing equivalent relationships by showing it is non-definable in arithmetic.
• The main impetus behind implementing the 'automatically moving names' issue was to prevent generation of mutually constructible sets.
• The author's extension theorem's generally accepted targeting of flexibility in names as true.

Overview

This paper, by Toby Meadows, establishes that several natural foundational theories cannot eliminate imaginaries. A theory $T$ eliminates imaginaries if whenever $T$ proves that a formula $E$ defines an equivalence relation, there is a formula $F$ such that $T$ proves $F$ defines a function respecting $E$: for all $\bar{x}, \bar{y}$, $\bar{x}E\bar{y}$ if and only if $F(\bar{x}) = F(\bar{y})$. The main result is that countable set theory — denoted $T$0, obtained from $T$1 by dropping the Powerset Axiom and adding the axiom $T$2 — fails to eliminate imaginaries. As corollaries, the paper shows that $T$3 and $T$4 also fail. Since $T$5 is bi-interpretable with second order arithmetic, these results bear directly on frameworks used in reverse mathematics.

The significance of elimination of imaginaries in foundational contexts is well established: it underwrites Scott's trick (defining cardinals without choice), ultrapower constructions from large cardinals, and quotient-free interpretations of other theories. In the terminology of Barrett and Halvorson, failure to eliminate imaginaries entails failure of Morita completeness and hence of delivering quotient sorts. It also interacts with recent work on categoricity: Enayat and Łełyk showed that internally categorical, sequential theories that eliminate imaginaries are tight, where tightness means any pair of bi-interpretable extensions must coincide. The paper's results show that internal categoricity does not require elimination of imaginaries: $T$6 is internally categorical yet still tight while failing to eliminate imaginaries.

A warmup: second order arithmetic

The paper first proves that $T$7 — second order arithmetic with full comprehension and definable choice ($T$8, often denoted $T$9) — cannot eliminate imaginaries. The proof uses forcing over $E$0 with $E$1 and an automorphism argument. The equivalence relation is mutual constructibility on reals, $E$2 iff $E$3, which is $E$4 and hence definable in second order arithmetic. The key technical ingredient is a lemma due to Asaf Karagila: given a $E$5-generic filter $E$6 over $E$7, any new set $E$8 has names whose denotations can be moved by ground-model automorphisms of $E$9 that are $F$0-based (determined by permutations of the first coordinate) while fixing a prescribed condition. Applying this to a hypothetical definable $F$1 respecting mutual constructibility yields two mutually constructible reals sent to distinct reals, contradicting respect.

A softer indirect argument is also recorded, credited to Ali Enayat and the author independently. Friedman and Visser proved that sequential theories bi-interpretable via identity-preserving interpretations are definitionally equivalent; since $F$2 and $F$3 are bi-interpretable but not definitionally equivalent, their witnessing interpretations cannot both be identity-preserving. Indeed, $F$4 interprets $F$5 only via a quotient interpretation, so if $F$6 eliminated imaginaries it could convert this to an identity-preserving interpretation, contradicting non-definitional-equivalence. The author notes this strategy cannot be reversed to prove the corresponding result for $F$7, since the latter already interprets $F$8 identity-preservingly.

Why naive strategies fail

The direct generalization to $F$9 is not straightforward, and the paper is careful to explain why. The obvious plan — coding hereditarily countable sets by well-founded extensional relations on $T$0 and transporting the warmup argument through such codes — breaks down because there is no uniform way to define a function selecting a code for each hereditarily countable set. For instance, after forcing over a model of $T$1 to add a Cohen real $T$2, the structure $T$3 cannot define a linear ordering of its hereditarily countable sets, though it can linearly order its reals; hence no injection from $T$4 into $T$5 is definable there.

The choice of equivalence relation also matters, and the paper documents two instructive failures:

• Equivalence relations with set-sized cells fail trivially: if every cell $T$6 is a set, then $T$7 itself eliminates imaginaries. This rules out relations like eventual agreement $T$8 on $T$9, whose classes are countable.
• Isomorphism of coded binary relations also fails: Friedman's theorem gives a definable function $F$0 respecting isomorphism, namely the Scott sentence of the coded structure in $F$1, uniformly computable from structural properties alone.

Mutual constructibility, a $F$2 relation considerably more complex than the analytic, non-Borel isomorphism relation, is what makes the argument succeed. The author remarks candidly that the claim "feels obviously true" — without powerset, Scott's trick is unavailable and nothing looks likely to replace it — but that the details are substantial.

The main theorem

The main theorem states that $F$3 cannot eliminate imaginaries. The witness model is $F$4, where $F$5 is generic over $F$6 for $F$7, the completion of Cohen forcing, and the equivalence relation is again mutual constructibility $F$8. The proof proceeds by contradiction: assuming a definable $F$9 respects $E$0, a case analysis over the constructibility degree of the input real $E$1 (whether $E$2, $E$3 with output in $E$4 or not, or $E$5) produces, via automorphisms fixing a relevant condition, two mutually constructible reals with distinct $E$6-values.

Two lemmas drive the argument:

• No definable representatives landing in $E$7 (Lemma 3.3): in $E$8, no definable function $E$9 satisfies $\bar{x}, \bar{y}$0 iff $\bar{x}, \bar{y}$1. The proof uses Abraham–Shore's observation that in a Cohen extension of $\bar{x}, \bar{y}$2, no constructibility degree other than the ground-model degree and the degree of the generic real is definable — established by swapping mutually generic factors of $\bar{x}, \bar{y}$3 — whence the degrees admit no definable well-ordering, whereas such an $\bar{x}, \bar{y}$4 would induce one via the canonical well-ordering of $\bar{x}, \bar{y}$5.
• Moving names (Lemma 3.4): every new element of $\bar{x}, \bar{y}$6 has a name that can be moved by automorphisms of $\bar{x}, \bar{y}$7 fixing any prescribed condition, with the movement forced rather than merely realized in one extension.

The second lemma is the technical heart. The author introduces the notion of a flexible name: a name $\bar{x}, \bar{y}$8 for $\bar{x}, \bar{y}$9 such that below every condition there exist uncountably many pairwise order-2 automorphisms fixing the condition and forcing pairwise distinct shifts of $\bar{x}E\bar{y}$0. The theorem that every new hereditarily countable set has a flexible name is proved by induction on a rank stratifying $\bar{x}E\bar{y}$1, using a base lemma (for subsets of $\bar{x}E\bar{y}$2) and a successor lemma (for countable sets containing an element with a flexible name).

The successor lemma's proof is a counting argument: if $\bar{x}E\bar{y}$3 were not flexible, a maximal sequence of pairwise-forced-distinct shifts would be countable; but flexibility of $\bar{x}E\bar{y}$4 supplies uncountably many automorphisms, and since $\bar{x}E\bar{y}$5 is countable, uncountably many must share a single condition forcing $\bar{x}E\bar{y}$6 into some fixed $\bar{x}E\bar{y}$7 — impossible when all those shifts of $\bar{x}E\bar{y}$8 are forced distinct and $\bar{x}E\bar{y}$9 is forced countable.

The base lemma constructs, inside $F(\bar{x}) = F(\bar{y})$0, a countable atomless dense subalgebra generated by an independent sequence $F(\bar{x}) = F(\bar{y})$1 built along an infinite binary tree: even stages split conditions to force disagreement about membership in $F(\bar{x}) = F(\bar{y})$2 across all flip patterns of even coordinates; odd stages ensure density in $F(\bar{x}) = F(\bar{y})$3 by refining toward an enumeration of $F(\bar{x}) = F(\bar{y})$4. A supporting lifting fact — proved via Sikorski's extension theorem — shows that every automorphism of a countable atomless Boolean algebra dense in $F(\bar{x}) = F(\bar{y})$5 lifts to an automorphism of $F(\bar{x}) = F(\bar{y})$6. The density argument then shows that any two distinct flip patterns force distinct shifted names, yielding continuum many automorphisms as required. The author notes this lifting fact appears not to be recorded in the literature, though he expects it to be folklore.

Corollaries and generalization

Two consequences follow. First, since $F(\bar{x}) = F(\bar{y})$7 of any model of $F(\bar{x}) = F(\bar{y})$8 is a model of $F(\bar{x}) = F(\bar{y})$9, the argument yields that $T$00 itself cannot eliminate imaginaries. Second, the entire framework transfers to $T$01 plus the existence of an inaccessible cardinal $T$02 with every set injectible into $T$03: one forces with $T$04, replaces the rank induction by a $T$05-length induction, works with a $T$06-sized atomless Boolean algebra closed under $T$07 meets and joins (isomorphic to the Lindenbaum algebra of the infinitary propositional logic $T$08), and builds the tree of automorphisms over functions $T$09. Only regularity of $T$10 is used.

Limitations and open questions

Several caveats are explicit in the paper. The negative results are witnessed by a single equivalence relation, mutual constructibility, whose $T$11 complexity appears essential; the paper shows that simpler candidates (eventual agreement, isomorphism) provably do admit eliminators, but it does not characterize which equivalence relations are eliminable in $T$12. The main theorem is established relative to specific generic models ($T$13 of Cohen extensions of $T$14); the transfer from "fails in some model" to "the theory cannot eliminate imaginaries" relies on the definition quantifying over what $T$15 proves, and the paper does not address whether every model of $T$16 exhibits such failures. The lifting fact for Boolean algebra embeddings is presented without a literature precedent, and the inaccessible-cardinal generalization is sketched rather than given in full detail. Finally, the relationship between eliminability of imaginaries and tightness remains only partially mapped: the paper shows tightness without eliminability is possible, but leaves open precisely which categoricity-theoretic properties entail or presuppose elimination of imaginaries.

Conclusion

The paper demonstrates that the ability to eliminate imaginaries, long taken for granted in foundational theories admitting Scott's trick or definable well-orderings, fails for the natural theory of countable sets, for $T$17, and for its inaccessible-strengthening. The proofs combine classical techniques — Karagila-style automorphism arguments, Abraham–Shore analysis of constructibility degrees, Sikorski's extension theorem — with a new flexible-names machinery for hereditarily countable sets. Beyond the specific negative results, the work clarifies the logical geography surrounding internal categoricity and tightness, showing that the former two properties can hold together without elimination of imaginaries.