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Structural results on idealistic equivalence relations

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Pith's one-line read Under analytic determinacy, this paper exhibits continuum many idealistic analytic equivalence relations that are not classwise Borel isomorphic to any orbit equivalence relation.

desk verdict A serious, carefully written paper showing idealistic analytic equivalence relations are much richer than orbit equivalence relations, but the central non-embeddability proof rests on an unpublished Becker/Steel lemma that needs more transparency before I would sign off. read the letter →

arxiv 2506.08217 v1 pith:QD2CKPOK submitted 2025-06-09 math.LO

Filippo Calderoni , Luca Motto Ros This is my paper · ORCID
classification math.LO MSC 03E1503E60
keywords idealisticequivalencerelationsBorelreducibilityorbitclasswiseisomorphismUlminvariantsabelianp-groupsanalyticdeterminacytagged-treeforcing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to establish that idealistic analytic equivalence relations are not all classwise Borel isomorphic to orbit equivalence relations, and that the exceptions form a rich part of the Borel-reducibility order. The main theorem, conditional on $\Sigma^1_1$-determinacy, embeds the structure $(\mathcal{O},\le_B)$ of Borel orbit equivalence relations with uncountably many orbits into $(\mathcal{I},\le_B)$, where $\mathcal{I}$ is the collection of idealistic analytic equivalence relations with only Borel classes that are not classwise Borel isomorphic to any orbit equivalence relation. Consequently there are $2^{\aleph_0}$ many $\le_B$-incomparable idealistic equivalence relations of this kind. The witness is an equivalence relation $E_B$ on countable abelian $p$-groups, classified by Ulm invariants; it is idealistic via a Borel map that selects an isomorphism class inside every class, yet cannot be classwise Borel embedded into an orbit relation under the determinacy assumption. The paper also gives an elementary counterexample to a 1997 form of the $E_1$ conjecture.

What carries the argument

The central objects are the equivalence relation $E_B$ on codes of certain abelian $p$-group structures, the Ulm invariants and Ulm length that classify such groups, and the relation of classwise Borel isomorphism $\simeq_{cB}$, which is finer than Borel bireducibility when classes have different sizes. Three devices carry the proof: a Borel selection map that picks one isomorphism class inside each $E_B$-class, turning the meager ideal on the symmetric group into a witness that $E_B$ is idealistic; the logic action of the infinite symmetric group together with a transfer lemma, used to control Borel ranks; and forcing with tagged trees, whose retagging lemma provides the separation principle that rules out classwise Borel embeddings into orbit relations.

What would settle it

Exhibit a classwise Borel embedding of $E_B$ into some orbit equivalence relation: Lemma 3.13 shows such an embedding forces $\widetilde{H}_\beta$ to be $\Pi^0_{\beta+1}$ for club many $\beta$, while Lemma 3.11 asserts the opposite under $\Sigma^1_1$-determinacy, so finding the embedding (or proving the $\Pi^0_{\beta+1}$ bound without determinacy) would refute the main result.

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Extended reading notes

Core claim

The central claim is that, assuming $\Sigma^1_1$-determinacy, there is an order-preserving embedding of $(\mathcal{O},\le_B)$ into $(\mathcal{I},\le_B)$, where $\mathcal{O}$ is the class of Borel orbit equivalence relations with uncountably many orbits and $\mathcal{I}$ is the class of idealistic analytic equivalence relations whose classes are Borel and which are not classwise Borel isomorphic to an orbit equivalence relation. In particular, there are $2^{\aleph_0}$ many $\le_B$-incomparable equivalence relations in $\mathcal{I}$. The relation that carries the construction is $E_B$, defined on codes of countable structures expanding an abelian $p$-group by constants for an infinite-rank divisible subgroup; $x\,E_B\,y$ holds exactly when the underlying $p$-groups are isomorphic. $E_B$ is idealistic because a Borel map sends each $E_B$-class to a single isomorphism class inside it, so the meager ideal on the group action can be pulled back; its classes are Borel; and it is not classwise Borel embeddable into any orbit relation, because any such embedding would force each $\widetilde{H}_\beta$ to be $\Pi^0_{\beta+1}$ for club many $\beta$, contradicting the lower bound obtained from tagged-tree forcing.

Load-bearing premise

The proof that $E_B$ is not classwise Borel embeddable into any orbit relation depends on an unpublished lemma asserting that, under $\Sigma^1_1$-determinacy, the sets $\widetilde{H}_\beta$ are not $\Pi^0_{\beta+1}$ for club many $\beta$; if that lemma or the determinacy assumption fails, the main theorem collapses.

Editorial extensions

If this is right

  • If Theorem 1.4 is correct, the folklore question of whether every idealistic relation is Borel bireducible with an orbit relation has a negative answer for analytic relations with Borel classes, at least under $\Sigma^1_1$-determinacy and with respect to classwise Borel isomorphism.
  • Proposition 1.5 shows the new idealistic examples cannot be found below Borel orbit relations: any idealistic relation $\le_B$-reducible to a Borel orbit relation is classwise Borel isomorphic to one, and the same holds if it is classifiable by countable structures (Corollary 1.6).
  • The counterexample of Proposition 1.7 refutes the original $E_1$ conjecture and its weakening: there is a Borel $E$ with $E_1\not\le_B E$ that is not Borel bireducible with any idealistic relation, so the $E_1$-dichotomy must be revised as in Conjecture 4.4.
  • Because $E_B$ is Ulm classifiable, $E_0\not\le_B E_B$, so the disjoint-union amplification of Theorem 3.2 embeds all of $\mathcal{O}$ into $\mathcal{I}$, giving $2^{\aleph_0}$ incomparable idealistic relations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the $\Sigma^1_1$-determinacy assumption could be removed, the same $E_B$ construction might produce a ZFC proof that idealistic analytic relations are not all classwise Borel isomorphic to orbit relations, settling the folklore Question 1.1 negatively in a strong sense (the paper lists this as Question 4.1).
  • Because $E_B$ is itself Borel reducible to an orbit equivalence relation (the isomorphism relation on countable structures), the paper suggests classwise Borel isomorphism, rather than Borel bireducibility, is the threshold where the idealistic/orbit distinction becomes observable; a natural test case is whether the phenomenon can occur among Borel idealistic relations (Question 4.2).
  • The counterexample to the $E_1$ conjecture uses only a Borel relation that is reducible to a countable Borel relation but not bireducible with one; this raises the possibility that many other 'hard' Borel relations below countable Borel relations are likewise non-idealistic, yielding further ZFC counterexamples to variants of the conjecture.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper studies the structure of idealistic analytic equivalence relations. Its main theorem (Theorem 1.4), under Sigma_1^1-determinacy, embeds the partial order of Borel orbit equivalence relations with uncountably many orbits into the partial order of idealistic analytic equivalence relations that have only Borel classes and are not classwise Borel isomorphic to an orbit equivalence relation; in particular, it yields 2^{aleph_0} pairwise <=_B-incomparable such relations. The construction uses Becker's equivalence relation E_B on codes for countable abelian p-groups with a distinguished divisible subgroup. The paper also proves Proposition 1.7, a counterexample to the Hjorth--Kechris Conjecture 1.3, using Hjorth's equivalence relation E_H, and it discusses several remaining questions. The proof that E_B is idealistic and does not classwise Borel embed into an orbit equivalence relation occupies Section 3 and relies on a game argument (Lemma 3.11) and on Steel's forcing, which is developed in Appendix A.

Significance. If the main theorem is correct, it is a substantial structural result: under analytic determinacy, idealistic analytic equivalence relations are strictly richer than orbit equivalence relations up to classwise Borel isomorphism, with a rich embedding of the orbit-equivalence-relation order. The paper also gives a clean counterexample to a long-standing conjecture variant and usefully reorganizes the surrounding open problems. The exposition is detailed, including an appendix on Steel's forcing, and the authors are explicit about which parts come from Becker's unpublished notes. The main non-embeddability proof is intricate and the decisive Lemma 3.11 rests on several external or unpublished inputs; as it stands, the manuscript does not allow an independent check of all load-bearing steps, which limits the verifiability of the main claim.

major comments (3)
  1. [Section 3.3, Claim 3.11.3 (p. 15)] The claim that the set {beta < omega_1 : omega[x]_1 < beta for all x in union_{alpha<beta} fM(alpha,omega)} is club is justified only by "standard arguments" and no proof is supplied. This is load-bearing: it is exactly what guarantees that the generic play F(w) belongs to fM(beta,omega) rather than to fM(alpha,omega) for some alpha<beta, and without this the separation argument using Proposition A.10 fails. Please provide a complete proof (for example, a Skolem-hull/closure argument in L(sigma)) or a precise citation with the necessary hypotheses verified.
  2. [Appendix A, Fact A.9] Fact A.9, used in Claim 3.11.3 to conclude omega^{t(g)}_1 = omega^{<t(g),sigma>}_1 = beta for P_beta-generics over L(sigma), is cited to Harrington [16, Theorems 2.9 and 2.10] but is not proved in the appendix, despite the appendix being described as almost self-contained. Since the correctness of this fact for merely sigma-admissible beta (rather than, say, limits of sigma-admissibles) is essential to the lower and upper bounds on omega[F(w)]_1, the authors should state the precise hypotheses and give a proof or a detailed reference with those hypotheses explicitly checked.
  3. [Section 3.3, Lemma 3.11] The proof of Lemma 3.11 is taken from Becker's unpublished notes [5] and parts are attributed to Steel. The manuscript does not include those notes, and the proof as written relies on the unproved club claim and the black-box Fact A.9. Because this lemma is the only step that makes E_B non-classwise-Borel-embeddable into an orbit equivalence relation, the central theorem is not independently checkable from the manuscript. I request that the authors either include the relevant arguments from [5] or make the present proof fully self-contained at these points.
minor comments (6)
  1. [Definition 2.5] The phrase "if if there are Borel reductions" contains a duplicated "if" and should be corrected.
  2. [Proof of Lemma 3.8(c)] The game is called "Ehrenfeuch-Fraisse" but should be "Ehrenfeucht-Fraisse".
  3. [Appendix A, after Definition A.3] The sentence "we say say that a tree-formula" appears to have a duplicated word; it should read "we say that a tree-formula".
  4. [Proof of Proposition A.8] The phrase "there are at most 2 |P_alpha|-many dense subsets" is typographically broken; the exponent should be typeset as 2^{|P_alpha|}. Additionally, the argument would benefit from a short explanation of why the number of dense subsets in L(r) is less than omega_1^V.
  5. [Section 3.3, paragraph before Lemma 3.11] In the sentence "each C_n is a ~=-equivalence class", the relation should be explicitly identified as the isomorphism relation on L-structures (the logic action), to avoid confusion with E_B.
  6. [Claim 3.11.1] The phrase "the logic action j_L is actually recursive" would be clearer as "the logic action is recursive in the effective descriptive set theory sense" or "computable", since the intended meaning is not immediately transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the construction of E_B is explicit and the cited external results are independent, so any gaps are matters of verifiability rather than circularity.

full rationale

The paper's central derivation—showing that Becker's equivalence relation E_B lies in the class I and then embedding (O, ≤_B) into (I, ≤_B)—does not reduce to its own inputs. E_B is defined explicitly from the theory T of abelian p-groups with constants naming Z(p^∞)^(ω), and its idealistic character is proved via Proposition 2.4 using a Borel selector θ from Lemma 3.8(b); this is a genuine construction, not a definitional equivalence. The non-embeddability into orbit equivalence relations is obtained by combining Lemma 3.11 (for club many β, eH_β is not Π^0_{β+1}) with Lemma 3.13 (classwise Borel embeddability into an orbit relation would force club many such sets to be Π^0_{β+1}); these are incompatible conclusions, so the pair is not circular. The proof of Lemma 3.11 relies on external results—Lemma 3.11 is attributed to Becker/Steel, Fact A.9 to Harrington, Proposition A.8 to folklore, and the club claim in Claim 3.11.3 on [4, Theorem 0.11] plus 'standard arguments'—but these are independent mathematical facts, not the paper's own fitted values or definitions. The authors disclose that Section 3.3 builds on Becker's unpublished notes [5] and thank Becker for sharing them; while this creates a verifiability gap, it is not circularity. Citations to the authors' own prior work ([31] for classwise Borel isomorphism and Proposition 2.9) support auxiliary structural facts and are not the load-bearing step that produces the new counterexample. Hence no step of the derivation is, by the paper's own equations, equivalent to its inputs; score 0.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No free parameters. The paper's claims rest on Sigma_1^1-determinacy and on several external results, including unpublished notes of Becker. E_B is an explicit construction, not an invented entity.

assumptions (7)
  • domain assumption Sigma_1^1-determinacy (analytic determinacy)
    Assumed in Theorem 1.4, Theorem 3.1, Lemma 3.11, and Proposition A.8. It is not proved in ZFC.
  • standard math Borel Glimm-Effros dichotomy and Silver's dichotomy
    Used in Theorem 3.2 to conclude F <=_B id_R and id_R <=_B F_i.
  • standard math Hjorth's theorem (existence of E_H)
    Used in Proposition 1.7 to obtain a Borel equivalence relation not bireducible with any countable Borel one.
  • standard math Kechris-Louveau theorem: E_1 does not reduce to any idealistic Borel equivalence relation
    Used in Proposition 1.7 and throughout the discussion of the E_1 dichotomy.
  • standard math Ulm's theorem and Zippin's theorem on countable abelian p-groups
    Foundational for the definition and analysis of E_B in Section 3.
  • standard math Sami's Transfer Lemma and Becker-Kechris [6, Corollary 5.1.10]
    Used in Lemma 3.13 to transfer Borel ranks and find a low-rank orbit class.
  • domain assumption Becker's unpublished notes [5]
    The main construction and non-embeddability proof are modeled on these notes; they are not publicly available.

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Cite this review

Pith. "Pith review of Structural results on idealistic equivalence relations." pith.science (2026). https://pith.science/paper/QD2CKPOK

@misc{pith2026250608217,
  author       = {Pith},
  title        = {Pith review of: Structural results on idealistic equivalence relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QD2CKPOK}},
  note         = {Machine review of arXiv:2506.08217}
}
read the original abstract

We address some fundamental problems concerning the structure of idealistic equivalence relations. In particular, we show that, under analytic determinacy, there are continuum many idealistic analytic equivalence relations that are not classwise Borel isomorphic to an orbit equivalence relation. Moreover, we also discuss alternative formulations of the E1 conjecture, and present an elementary counterexample to one of its earliest variants proposed by Hjorth and Kechris in 1997.

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Forward citations

Cited by 1 Pith paper

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Reference graph

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