Invariant Elementary Definability

• Invariant Elementary Definability is the study of when indirect invariance conditions allow a class, relation, or witness to be recovered by an explicit formula across various logical settings.
• The framework differentiates between syntactic and semantic definability, using methods like permutation invariance, elementary comparison, and naturality to bridge indirect specifications with explicit definitions.
• Key implications include the definability of truth-like predicates, canonical witness selection in HOD, and the calibration of logical complexity in areas ranging from set theory to modal and finite model theory.

Searching arXiv for the cited papers to ground the synthesis in current arXiv records. arxiv_search query: (Dorais et al., 2017) invariant elementary definability (Łełyk et al., 2023, Semenov et al., 2013, Lindell et al., 16 Jul 2025) Invariant elementary definability studies when a class, relation, witness, or construction that is specified only indirectly—by automorphism lifting, elementary equivalence, admissible presentations, preservation under polymorphisms, or the existence of a definable nonempty set—can be recovered by an explicit formula, or at least by a sharply delimited logical fragment. The literature suggests that the phrase does not denote a single uniform program. In set theory it calibrates definable witness selection against $V=\mathrm{HOD}$; in model theory it appears as “elementary-invariance-to-definability”; in modal and finite model theory it is tied to closure and reflection under bisimulations, bounded morphisms, ultrafilter extensions, and Hanf-type locality; and in category-theoretic work it is expressed through naturality and weak naturality of constructions (Dorais et al., 2017, Łełyk et al., 2023, Lindell et al., 16 Jul 2025).

1. Conceptual scope and terminological boundaries

A recurrent distinction is between explicit definability and invariance principles that only determine a notion up to semantic behavior. In sequential theories, Łełyk and Wcisło distinguish syntactic definability, semantic definability, and parameter-free semantic definability, and then show that uniform elementary comparison properties can force explicit definitions of truth-like predicates (Łełyk et al., 2023). In the classical invariance tradition, Feferman-style and Krasner-style correspondences connect preservation under permutations or similarities with definability in $L_{\infty\infty}$ or $L^-_{\infty\infty}$, rather than with first-order definability itself (Bonnay et al., 2013). Semenov and Soprunov give a first-order counterpart by replacing permutations of elementary extensions with permutations of $A^{\mathbb N}$ that almost preserve the basic relations; a relation is first-order definable from a given family exactly when every such permutation that almost preserves the family also almost preserves the relation (Semenov et al., 2013).

The phrase “elementary” is also not always model-theoretic. In Raphaël Fino’s work on Vishik’s Elementary Discrete Invariant, “elementary” refers to a cycle-theoretic sub-invariant, and the paper explicitly states that it does not mean first-order definability or invariance under elementary equivalence (Fino, 2017). That warning is important because several strands of the literature use “definability” and “invariance” in technically different senses: first-order definability of relations, external definability of subsets of a monster model, primitive positive definability, modal definability of frame classes, or uniform definability of a class function on structures.

A second boundary concerns logic strength. Invariance under arbitrary subgroups of the full permutation group corresponds, in general, to $L_{\infty\infty}$, not to elementary logic; first-order elementary definability appears there only as the special case of closed subgroups in the topology of pointwise convergence (Bonnay et al., 2013). Dually, Engström’s theorem shows that under pure second-order logic with Henkin semantics, implicit definability of generalized quantifiers collapses to explicit first-order definability, so the semantic setting can drastically shrink the gap between invariance or implicitness and elementary definability (Engström, 2014).

2. Definable witnesses, ordinal definability, and the $V=\mathrm{HOD}$ boundary

In set theory, the central invariant-definability problem is whether every definable nonempty set has a definable element. Dorais and Hamkins work with external, model-theoretic definability: $a\in M$ is definable if some formula $\varphi(x)$ uniquely characterizes it in $M$, while ordinal-definability is definability from ordinal parameters. Their main theorem states that, over $\mathrm{ZF}$, the following are equivalent: $L_{\infty\infty}$0, existence of a definable well-order of the universe, every definable nonempty set having a definable element, every definable nonempty set having an ordinal-definable element, every ordinal-definable nonempty set having an ordinal-definable element, and every $L_{\infty\infty}$1-definable nonempty set having an ordinal-definable element (Dorais et al., 2017).

This identifies a precise threshold. The passage from “there exists an element of a definable set” to “there exists a definable element of that set” is not a weak closure property of definability; it is exactly the global invariant condition $L_{\infty\infty}$2. The striking point is that the restriction to $L_{\infty\infty}$3-definable sets already suffices. By contrast, the threshold is sharp: every model of $L_{\infty\infty}$4 has a forcing extension with $L_{\infty\infty}$5 in which every $L_{\infty\infty}$6-definable nonempty set has an ordinal-definable element (Dorais et al., 2017).

The same pattern extends to $L_{\infty\infty}$7 when $L_{\infty\infty}$8 is $L_{\infty\infty}$9-definable. Then $L^-_{\infty\infty}$0 is equivalent to every definable nonempty set having an $L^-_{\infty\infty}$1-definable element, to every definable nonempty set having an $L^-_{\infty\infty}$2-definable element, and to every $L^-_{\infty\infty}$3-definable nonempty set having a member in $L^-_{\infty\infty}$4. The important cases include $L^-_{\infty\infty}$5 and $L^-_{\infty\infty}$6 (Dorais et al., 2017). In this strand, invariant elementary definability is therefore the problem of when definable existence can be upgraded to a canonical witness, and the answer is controlled exactly by the gap between $L^-_{\infty\infty}$7 and its canonical definability core.

3. Elementary comparison and the definability of truth, satisfaction, and unique definition

Łełyk and Wcisło analyze a different but closely related paradigm. Let $L^-_{\infty\infty}$8 be an r.e. sequential theory in a finite language, let $L^-_{\infty\infty}$9 be a fixed sublanguage, and consider the truth theory $A^{\mathbb N}$0, the uniform satisfaction theory $A^{\mathbb N}$1, and the definability theory $A^{\mathbb N}$2. The paper studies abstract model-theoretic properties such as imposing $A^{\mathbb N}$3-elementarity, imposing $A^{\mathbb N}$4-elementary equivalence, preserving $A^{\mathbb N}$5-definability, and imposing equality of $A^{\mathbb N}$6-definables. Its basic forward theorem states that definable truth-like predicates automatically yield these invariance properties: $A^{\mathbb N}$7 uniformly imposes $A^{\mathbb N}$8-elementary equivalence, $A^{\mathbb N}$9 uniformly preserves $L_{\infty\infty}$0-definability, and $L_{\infty\infty}$1 uniformly imposes $L_{\infty\infty}$2-elementarity (Łełyk et al., 2023).

The decisive result is the converse. If $L_{\infty\infty}$3 uniformly imposes $L_{\infty\infty}$4-elementarity, then $L_{\infty\infty}$5 syntactically defines $L_{\infty\infty}$6 modulo $L_{\infty\infty}$7; if $L_{\infty\infty}$8 uniformly preserves $L_{\infty\infty}$9-definability, then $V=\mathrm{HOD}$0 syntactically defines $V=\mathrm{HOD}$1 modulo $V=\mathrm{HOD}$2; and if $V=\mathrm{HOD}$3 uniformly imposes $V=\mathrm{HOD}$4-elementary equivalence, then $V=\mathrm{HOD}$5 syntactically defines $V=\mathrm{HOD}$6 modulo $V=\mathrm{HOD}$7 (Łełyk et al., 2023). There are also semantic analogues: non-uniform versions yield semantic definability or parameter-free semantic definability rather than one fixed formula.

This is a direct “elementary-invariance-to-definability principle.” The relevant invariance is not mainly automorphism-invariance inside one model, but invariance under $V=\mathrm{HOD}$8-elementary extension, $V=\mathrm{HOD}$9-elementary equivalence, and preservation of low-complexity definables across models. The proof mechanism is explicit: a uniform invariance property yields a bounded-complexity reduction of every $a\in M$0-formula to an $a\in M$1-formula; partial satisfaction predicates $a\in M$2 available in sequential theories then convert that bounded reduction into a full definable truth or satisfaction predicate (Łełyk et al., 2023).

The same paper also delineates strict limits. $a\in M$3 and $a\in M$4, but $a\in M$5 does not semantically define $a\in M$6, $a\in M$7 does not semantically define $a\in M$8, and under mild assumptions $a\in M$9 does not semantically define $\varphi(x)$0. In the arithmetic setting, $\varphi(x)$1 both syntactically and semantically (Łełyk et al., 2023). Thus invariance may force explicit definability, but only at the correct strength and only for the corresponding semantic notion.

4. Automorphisms, infinitary logics, external traces, and weakened definability

Invariance-definability correspondences often require stronger logics than first-order logic. Feferman, McGee, and the generalized Krasner correspondence show that invariance under arbitrary groups of permutations yields definability in $\varphi(x)$2, and in the equality-free setting the correct semantic transformations are similarities rather than permutations. For generalized quantifiers, $\varphi(x)$3 iff $\varphi(x)$4 is definable in $\varphi(x)$5; without equality, one only gets definability of the restricted quantifier $\varphi(x)$6 in $\varphi(x)$7 (Bonnay et al., 2013). The first-order case reappears there only as a topological special case: automorphism groups of first-order structures are exactly the closed subgroups of the full symmetric group.

Semenov and Soprunov recover a first-order invariance theorem by changing the semantic arena. For a countable structure $\varphi(x)$8 and countable definability space $\varphi(x)$9, a permutation $M$0 of $M$1 almost preserves a relation $M$2 if, for every sequence of arguments, the truth values of $M$3 and its image disagree at only finitely many coordinates. Their theorem states that $M$4 is first-order definable from $M$5 iff every permutation of $M$6 that almost preserves all relations from $M$7 also almost preserves $M$8 (Semenov et al., 2013). This is a combinatorial Svenonius theorem that avoids explicit reference to elementary extensions in the statement.

Several papers show that the target notion of definability may have to be weakened. Zambella studies the class $M$9 and proves that if $\mathrm{ZF}$0 has finite VC-dimension, then $\mathrm{ZF}$1 is approximable from below and from above, hence externally definable; in NIP theories, external definability is equivalent to finite VC-dimension of $\mathrm{ZF}$2 (Zambella, 2014). Khanaki proves that for a countable structure $\mathrm{ZF}$3, a formula $\mathrm{ZF}$4 is NIP on $\mathrm{ZF}$5 iff every finitely satisfiable $\mathrm{ZF}$6 is Baire $\mathrm{ZF}$7 definable over $\mathrm{ZF}$8, so the right invariant substitute for ordinary definability of types in the NIP setting is topological rather than elementary definability (Khanaki, 2017).

The same theme appears in infinitary syntax. A class of structures is both pseudo-elementary $\mathrm{ZF}$9 and $L_{\infty\infty}$00-elementary exactly when it is definable by a $L_{\infty\infty}$01-sentence, namely an $L_{\infty\infty}$02-sentence with countable conjunctions but no countable disjunctions (Boney et al., 2018). By contrast, Beth-style implicit-to-explicit transfer can fail in nonclassical settings: in the super-relevant logic $L_{\infty\infty}$03, epimorphisms need not be surjective in the corresponding class of algebras, and the Beth Definability Property fails (Garber, 2020).

5. Specialized domains: valuations, convex polymorphisms, and modal frame definability

In valued-field model theory, the invariant object is often a valuation singled out by elementary information about the value group. Jahnke and Simon show that if $L_{\infty\infty}$04 is henselian and $L_{\infty\infty}$05 is not closed in its divisible hull, then $L_{\infty\infty}$06 is definable in the language of rings with one parameter. They also prove that one parameter is, in general, optimal: there are ordered henselian valued fields with the prescribed value group for which the only parameter-free $L_{\infty\infty}$07-definable henselian valuation is the trivial valuation (Krapp et al., 2021). The obstruction is elementary-equivalence-theoretic: a $L_{\infty\infty}$08-henselian non-henselian field elementarily equivalent to a henselian field blocks $L_{\infty\infty}$09-definability.

In universal-algebraic CSP-style analysis over $L_{\infty\infty}$10, Meyer studies structures $L_{\infty\infty}$11 with $L_{\infty\infty}$12 convex. The paper identifies infinitary primitive positive definability as the intermediate notion between pp-definability and polymorphism invariance. Depending on $L_{\infty\infty}$13, the infinitary pp-definable relations are exactly one out of six possible families, and in this convex setting infinitary pp-definability coincides with invariance under all countable-arity polymorphisms (Meyer, 2024). One consequence is that there is no locally closed clone between the clone of affine combinations and the clone of convex combinations.

Modal logic supplies a parallel family of invariant elementary definability theorems. For $L_{\infty\infty}$14, the fragment of modal logic with positive universal modality, a class of Kripke models is definable iff it is elementary and closed under surjective bisimulations; for elementary frame classes, definability is characterized by closure under generated subframes and bounded morphic images together with reflection of ultrafilter extensions and finitely generated subframes (Sano et al., 2015). Kikot proves a dichotomy for frame conditions of the form $L_{\infty\infty}$15: when every undirected cycle passes through the root, the associated modal logic is modally definable and generalized-Sahlqvist tame; when a minimal rooted diagram has a cycle not passing through the root, modal definability, finite axiomatizability, canonicity, and elementarity of the validating-frame class all fail together (Kikot, 2014).

Predicate modal logic provides a sharp negative boundary. Rybakov and Shkatov construct a normal predicate modal logic that is recursively enumerable and Kripke complete but not complete with respect to any elementary class of Kripke frames, and they also produce a logic that is not complete with respect to any elementary class of rooted frames although it is complete with respect to an elementary class of non-rooted frames (Rybakov et al., 2019). Thus recursively enumerable Kripke completeness does not guarantee elementary frame definability.

6. Naturality, presentations, and current directions

A categorical version of invariant elementary definability appears in the study of natural constructions. In “Naturality and Definability III,” a construction $L_{\infty\infty}$16 is encoded by a two-sorted structure $L_{\infty\infty}$17 with $L_{\infty\infty}$18. Naturality means that the natural restriction homomorphism $L_{\infty\infty}$19 splits; weak naturality means it weakly splits modulo the center. The main uniformity theorem states that if $L_{\infty\infty}$20 is a class of $L_{\infty\infty}$21-models first-order definable from a parameter $L_{\infty\infty}$22, and for every $L_{\infty\infty}$23 the map $L_{\infty\infty}$24 splits, then there exists a class function $L_{\infty\infty}$25 from first sorts to $L_{\infty\infty}$26 that is uniformly definable from $L_{\infty\infty}$27 and satisfies $L_{\infty\infty}$28 (Asgharzadeh et al., 2023). This is a direct passage from naturality to uniform first-order definability with no extra parameters beyond those already defining the class.

The same paper establishes the converse direction as a consistency phenomenon. In a suitable GCH-preserving class generic extension, every uniformisable uni-construction problem is weakly natural (Asgharzadeh et al., 2023). Together with the forcing theorem that absence of lifting can be forced to destroy $L_{\infty\infty}$29-solvability, this shows that definability and invariance are closely linked but not equivalent in bare ZFC.

The most recent finite-model-theoretic extension generalizes order-invariant definability to presentations that depend on the structure’s own relations, not only on an arbitrary order of the universe. A presentation scheme $L_{\infty\infty}$30 is elementary if admissibility is first-order definable in the expanded signature, neighborhood bounded if there is $L_{\infty\infty}$31 such that $L_{\infty\infty}$32, and local if it satisfies localization and disjoint local amalgamation. Under these hypotheses, two locality theorems hold. First, every $L_{\infty\infty}$33-invariant elementary boolean query on locally finite structures is Hanf local for full Hanf equivalence. Second, on bounded-degree structures, every $L_{\infty\infty}$34-invariant elementary boolean query is Hanf $L_{\infty\infty}$35-threshold local, non-uniformly in the degree bound (Lindell et al., 16 Jul 2025). The examples include linear orders, traversal presentations, and local orders on neighborhoods; connectivity is $L_{\infty\infty}$36-invariant elementary over finite simple graphs, while local orders are neighborhood bounded and linear orders are not (Lindell et al., 16 Jul 2025).

These developments suggest a common conclusion. Invariant elementary definability is not one theorem but a family of exact calibrations. In some settings invariance already forces first-order definability; in others it yields only semantic definability, Baire $L_{\infty\infty}$37 definability, infinitary definability, pp-type definability, or consistency-level weak naturality. The decisive parameters are the comparison relation being respected—automorphisms, elementary equivalence, presentations, polymorphisms, frame constructions, or inner-model quotients—the complexity of the formulas allowed, and the degree to which arbitrary choices can be eliminated canonically.