Scott Analysis below the Vaught Ordinal

Abstract: We develop new tools for determining the existence of models of specific Scott ranks under countability conditions. Using these, we improve a result of Sacks by showing that any counterexample to Vaught's conjecture must have at least two models of every parameterized Scott rank -- a result that contrasts with the unparameterized case, where minimal counterexamples have only one model at many ranks. We further prove that theories with fewer than continuum many models have trivial Scott spectra and provide a general, systematic classification of low Scott rank models when only countably many $Σ_α$-types are realized. Additionally, we classify the Scott complexity spectra for many Ehrenfeucht theories, and prove the $ω$-Vaught's conjecture in this setting, answering an infinitary strengthening of a question of Pillay and Tanović. We demonstrate that the Scott complexity of prime models for $ω$-stable first-order theories is commensurate with the complexity of the theory itself. Along the way, we apply our methods to concrete theories like p-groups, trees, and Boolean algebras, answering questions of Harris--Montalbán and Alvir--Csima--MacLean regarding specific structures.

• The paper develops a framework showing that type-counting constraints below the Vaught ordinal restrict Scott invariants, including a Henkin construction that produces low-rank models under suitable smallness assumptions.
• The authors prove that theories with fewer than continuum many countable models realize every Scott rank below their Vaught ordinal, while counterexamples to Vaught’s conjecture have at least two models at every parameterized Scott rank.
• The paper proves the omega-Vaught conjecture for Pi-alpha Ehrenfeucht theories, improves the bound for first-order Ehrenfeucht theories to vo(T) < omega·2, and bounds the Scott complexity of prime models of omega-stable theories.

This paper, by Gonzalez, Rossegger, and Turetsky, develops a general framework for Scott analysis in the region of the quantifier hierarchy where a theory has only countably many $\Sigma_\alpha$-types realized among its models — that is, strictly below the Vaught ordinal. The framework yields several concrete advances: a strengthening of a theorem of Sacks on the model profile of counterexamples to Vaught's conjecture, a proof that such counterexamples must have models of every Scott rank up to their Vaught ordinal, a resolution of an infinitary strengthening of a question of Pillay and Tanović concerning Ehrenfeucht theories, and a determination of the Scott complexity of prime models of $\omega$-stable first-order theories. The unifying theme is that countability assumptions on types impose strong structural constraints on which Scott invariants can be realized, constraints that fail in general.

Background and setting

The paper works in $L_{\omega_1\omega}$, where every countable structure has a Scott sentence characterizing it up to isomorphism among countable structures (2606.15205). The authors use Montalbán's Scott rank $SR(\mathcal{M})$ (least $\alpha$ with a $\Pi_\alpha$ Scott sentence), the parameterized rank $SR_p(\mathcal{M})$ (least $\alpha$ with a $\Sigma_{\alpha+2}$ Scott sentence over finitely many parameters), and the finer Scott complexity $SC(\mathcal{M})$, which records whether the simplest Scott sentence is $\omega$0, $\omega$1, or $\omega$2 for some $\omega$3. These distinctions matter: a central contribution is that the parameterized and unparameterized notions behave qualitatively differently below the Vaught ordinal, unlike in most prior work.

The Vaught ordinal $\omega$4 is the least ordinal at which either continuum many $\omega$5-types appear among models of $\omega$6, or $\omega$7 has only countably many models, all of bounded Scott rank. A theory satisfies (infinitary) Vaught's conjecture exactly when $\omega$8; a counterexample is $\omega$9-small for every countable $L_{\omega_1\omega}$0. The paper's technical core is a set of type-omitting-style constructions showing that smallness forces the existence of low-rank models, and a back-and-forth game argument showing that each $L_{\omega_1\omega}$1-class contains at most one structure of Scott rank $L_{\omega_1\omega}$2.

Three technical tools

The first tool is a Henkin construction, in flavor analogous to the existence of prime models rather than to classical type omission. If $L_{\omega_1\omega}$3 is consistent, $L_{\omega_1\omega}$4-small, and $L_{\omega_1\omega}$5, then $L_{\omega_1\omega}$6 has a model $L_{\omega_1\omega}$7 with $L_{\omega_1\omega}$8. The construction enumerates the countably many $L_{\omega_1\omega}$9-types realized among models of $SR(\mathcal{M})$0 and builds a Henkin set in which every realized $SR(\mathcal{M})$1-type is isolated by a $SR(\mathcal{M})$2-formula, which is precisely the condition for Scott rank at most $SR(\mathcal{M})$3. The parameterized analogue holds one level higher: a $SR(\mathcal{M})$4-small $SR(\mathcal{M})$5 theory has a model with $SR(\mathcal{M})$6. Notably, the analogous statement for Scott complexity $SR(\mathcal{M})$7 fails: the authors exhibit a $SR(\mathcal{M})$8 theory with two models of complexities $SR(\mathcal{M})$9 and $\alpha$0, neither of complexity $\alpha$1. They also record limitations near limit levels: there is a $\alpha$2 theory that is $\alpha$3-small for every $\alpha$4 yet has no model of Scott complexity at most $\alpha$5.

The third tool is parameter-free and general: if $\alpha$6 and $\alpha$7 (equivalence in the back-and-forth relations), then $\alpha$8. The proof uses that automorphism orbits of $\alpha$9 are $\Pi_\alpha$0-definable to convert a winning strategy in the $\Pi_\alpha$1 game into one for the $\Pi_\alpha$2 game. The immediate corollary is that a $\Pi_\alpha$3-class contains at most one structure of Scott rank exactly $\Pi_\alpha$4, which licenses the notion of a labeled back-and-forth class: under $\Pi_\alpha$5-smallness, every $\Pi_\alpha$6-class of models of $\Pi_\alpha$7 contains a unique Scott-rank-$\Pi_\alpha$8 representative, and the collection of labels is the minimal $\Pi_\alpha$9-universal class. This retroactively explains, in a uniform way, the classification of Scott-rank-2 linear orderings of Gonzalez and Rossegger, which had previously been proved without reference to these counting facts.

Consequences for Vaught's conjecture

Two results frame the paper's contribution to Vaught's conjecture, and they pull in opposite directions.

On one side, combining the existence lemmas with a chaining argument, the authors prove that a $SR_p(\mathcal{M})$0 theory with fewer than continuum many countable models has a model of Scott rank $SR_p(\mathcal{M})$1 for every $SR_p(\mathcal{M})$2. In particular, below the Vaught ordinal there is no Scott skipping: the Scott spectrum of such a theory is an initial segment, in sharp contrast to Harrison-Trainor's examples of sentences with arbitrarily long gaps in their Scott spectra — all of which necessarily have continuum many models.

On the other side, the parameterized/unparameterized distinction produces a dichotomy. The authors prove that a counterexample to Vaught's conjecture must have at least two models of every parameterized Scott rank, strengthening Sacks's 1983 result, which guaranteed two models only at $SR_p(\mathcal{M})$3-admissible ranks. However, they also prove a limiting result: assuming Vaught's conjecture fails, there is a minimal counterexample (which exists by Harnik and Makkai) with exactly one model of unparameterized Scott rank $SR_p(\mathcal{M})$4 for $SR_p(\mathcal{M})$5-many ordinals $SR_p(\mathcal{M})$6, where $SR_p(\mathcal{M})$7 is constructed from fixed points of an explicitly defined function on the unique uncountable $SR_p(\mathcal{M})$8-classes. The proof of this theorem is the paper's most delicate argument: minimality forces a unique uncountable $SR_p(\mathcal{M})$9-class at each level, and the fixed-point iteration ensures all models outside that class have Scott rank strictly below the limit point.

The authors are explicit that these two results nearly compose into a proof of Vaught's conjecture: strengthening either result — two models at every unparameterized rank, or one model at every parameterized rank — would yield an immediate contradiction. They note that the only obstruction is the influence of finitely many parameters, and that no intermediate rank notion currently removes it. This is an open problem the paper leaves precisely stated.

Models where parameters matter and applications

A sharpening of the counting analysis shows that if $\alpha$0 is $\alpha$1-small with a model of Scott complexity at least $\alpha$2, then either $\alpha$3 has a model of Scott complexity exactly $\alpha$4 or infinitely many models of complexity exactly $\alpha$5. Both alternatives are necessary, as witnessed by the Ehrenfeucht example above and by the theory of linear orderings, which has no $\alpha$6 model but countably many $\alpha$7 models. A consequence is that any counterexample to Vaught's conjecture has, for every $\alpha$8, a model with $\alpha$9 — parameters genuinely matter in counterexamples.

This yields a non-constructive answer to a question of Alvir, Csima, and MacLean: since Abelian p-groups are $\Sigma_{\alpha+2}$0-small (Khisamiev) and have models of arbitrarily high Scott rank, there is an Abelian p-group of Scott complexity $\Sigma_{\alpha+2}$1, or infinitely many of complexity $\Sigma_{\alpha+2}$2. The authors concede this is an unsatisfactory resolution — it gives no explicit group — and pose as an open question the explicit construction of such a group and the determination of which non-$\Sigma_{\alpha+2}$3 complexities are attainable.

Two further applications close this section. For order-theoretic trees (shown $\Sigma_{\alpha+2}$4-small by Richter), the authors classify the Scott-rank-1 trees via a map $\Sigma_{\alpha+2}$5 sending a tree to the finite antichain of embeddability-minimal trees absent from its age, using Kruskal's tree theorem and Fraïssé's theorem; a purely combinatorial question about which such antichains have the joint embedding property remains open. For Boolean algebras, they confirm in full a conjecture of Harris and Montalbán: every non-splitting $\Sigma_{\alpha+2}$6-type of Boolean algebras (a type with exactly one $\Sigma_{\alpha+2}$7 descendant) is an isomorphism type. Harris and Montalbán had verified this only for $\Sigma_{\alpha+2}$8; the abstract argument via labeled back-and-forth classes handles all finite $\Sigma_{\alpha+2}$9 at once.

Ehrenfeucht theories and the $SC(\mathcal{M})$0-Vaught conjecture

The paper's finest-calibrated results concern Ehrenfeucht theories — $SC(\mathcal{M})$1 theories with finitely many models — the setting in which Vaught originally posed his conjecture. The main theorem answers affirmatively an infinitary strengthening of Martin's conjecture as posed by Pillay and Tanović: every $SC(\mathcal{M})$2 Ehrenfeucht theory satisfies $SC(\mathcal{M})$3, i.e., satisfies the $SC(\mathcal{M})$4-Vaught conjecture of Gonzalez and Montalbán. Since every first-order theory is $SC(\mathcal{M})$5, this gives $SC(\mathcal{M})$6 for first-order Ehrenfeucht theories, improving Wagner's bound of $SC(\mathcal{M})$7.

The proof proceeds through a structural analysis of Scott complexity spectra — the full behavior of the counting function $SC(\mathcal{M})$8 recording how many models realize each complexity $SC(\mathcal{M})$9. The key structural facts are: every finitely-many-models theory has a $\omega$00 model; a model of complexity above $\omega$01 forces the existence of both $\omega$02 and $\omega$03 models; and, for Ehrenfeucht theories, $\omega$04 for all $\omega$05 — every $\omega$06 model is $\omega$07-back-and-forth equivalent to a distinct $\omega$08 model, though never to another $\omega$09 model. This inequality is asymmetric, as witnessed by explicit theories $\omega$10 with exactly $\omega$11 models realizing $\omega$12, $\omega$13, and $\omega$14.

The authors completely characterize the Scott complexity spectra supported strictly below $\omega$15: these are exactly those satisfying the three constraints above, and all are realized by explicit constructions using two composition operations on theories (one adding a $\omega$16 model, one combining theories while preserving a unique $\omega$17 model). This yields $\omega$18 and $\omega$19 for the spectrum counting function $\omega$20, and $\omega$21, with the authors suspecting $\omega$22. They exhibit a $\omega$23 theory with exactly six models, one of complexity $\omega$24, showing that Ehrenfeucht theories can realize the complexity needed for the eighth spectrum. The exact value of $\omega$25 for $\omega$26 remains open, as does the question of which finite Scott-rank complexities are attainable by models of $\omega$27 Ehrenfeucht theories.

Prime models of first-order theories

The final section establishes that the Scott complexity of a prime model is commensurate with the complexity of its theory. If $\omega$28 is a complete $\omega$29 first-order theory that is $\omega$30-stable or has fewer than continuum many countable models, its prime model has Scott complexity at most $\omega$31. The proof Morleyizes the vocabulary to obtain a $\omega$32 theory $\omega$33, applies the base case — where $\omega$34-smallness plus the type-counting hypothesis yields a $\omega$35 model, which the elementary embedding from the prime model forces to be isomorphic to the prime model — and then unwinds the definitions, at a cost of one quantifier alternation per level.

Two corollaries limit the coding power of complete first-order theories: such a theory has at most one model of Scott complexity $\omega$36, and consequently is not on top for effective bi-interpretation or any reduction preserving unparameterized Scott rank. The $\omega$37-stability hypothesis is genuinely needed: the authors give a complete $\omega$38 theory over infinitely many unary predicates, admitting quantifier elimination, with no $\omega$39 model at all — orbits cannot be isolated because each element's isolating formula leaves some predicate undetermined, and the theory realizes both polarities. They also note the bound is sharp in general: prime models of non-standard completions of Peano arithmetic attain Scott rank $\omega$40. Independently, Jason Block has proven the $\omega$41 base case and its converse (that a $\omega$42 Scott complexity implies primality).

Limitations and open questions

The paper is candid about the boundaries of its methods. The smallness hypotheses are essential, not incidental: the limit-level counterexamples and the quantifier-elimination example above show that dropping or weakening them invalidates the existence lemmas. The dichotomy in the parameter-mattering proposition is irreducible, with both alternatives realized by concrete theories. The resolution of the Alvir–Csima–MacLean question is purely abstract and non-effective. The proof strategy against Vaught's conjecture via combining the two model-profile theorems stalls on the parameter issue, for which the authors offer no remedy. Open problems stated in the paper include: the value of $\omega$43 for $\omega$44 (with $\omega$45 conjectured to be 8); the attainable Scott complexities of models of $\omega$46 Ehrenfeucht theories; an explicit construction of an Abelian p-group with $\omega$47; the combinatorial characterization of Scott-rank-1 trees via joint embedding; and the existence of a rank notion interpolating between the parameterized and unparameterized Scott ranks.

Conclusion

This paper consolidates Scott analysis below the Vaught ordinal into a systematic theory organized around type-counting hypotheses and the fine structure of back-and-forth equivalence classes. Its main results — the two-model theorem for counterexamples to Vaught's conjecture, the triviality of Scott spectra below the Vaught ordinal, the $\omega$48-Vaught conjecture for Ehrenfeucht theories, and the tight bound on prime model complexity — each derive from a small set of reusable tools whose scope extends well beyond the specific applications given. The paper also clarifies, through the parameterized/unparameterized dichotomy, exactly where a promising route to Vaught's conjecture currently fails, and leaves that failure point as a sharply formulated open problem.