Vaught's Conjecture in Model Theory

• Vaught's Conjecture is a central open problem in model theory, addressing the spectrum of non-isomorphic countable models for complete theories in a countable language.
• It asserts that for any such theory, the number of countable models is either countable or has the cardinality of the continuum.
• Research leverages combinatorial, algebraic, and computability approaches, verifying the conjecture in classes like almost chainable and monomorphic theories.

Vaught’s Conjecture (VC) is a central open problem in model theory concerning the cardinality spectrum of non-isomorphic countable models of complete first-order (and, in later generalizations, infinitary) theories. The conjecture asserts that for any complete theory in a countable language, the number of non-isomorphic countable models must be countable or continuum, with no intermediate cardinalities. Research on VC spans combinatorial, structural, and computability-theoretic approaches, yielding several verification schemes in broad classes and introducing multiple sharp formulations and algebraic frameworks.

1. Definition and Classical Formulation

Let $\mathcal{T}$ be a complete first-order theory in a countable language and $I(\mathcal{T},\omega)$ the number of non-isomorphic countable models of $\mathcal{T}$.

• Classical Vaught’s Conjecture:

$$I(\mathcal{T},\omega)\in\{\leq\aleph_0, 2^{\aleph_0}\}$$

That is, for every such $\mathcal{T}$, either there are countably many or continuum many non-isomorphic countable models.

The sharp form (VC$^\#$) further restricts the possible spectrum to $\{0,1,2^{\aleph_0}\}$, prohibiting any complete theory from having exactly $\aleph_0$ non-isomorphic countable models (Kurilić, 2022).

2. Reduction to Structural Profiles: Almost Chainable and Monomorphic Theories

A significant verification of VC for broad classes uses structural “profile” invariants and chainability criteria:

• Almost Chainable Structures: An $L$-structure ${\mathbb Y}$ is almost chainable iff there exist a finite set $F\subset Y$ and a linear order $<$ on $Y\setminus F$ such that for each partial automorphism $\varphi$ of $\langle Y\setminus F, <\rangle$, the mapping $\mathrm{id}_F \cup \varphi$ is a partial automorphism of $\mathbb Y$.
• Fraïssé’s Profile Boundedness: For finite $L$, ${\mathbb Y}$ is almost chainable if and only if the profile $\rho_{\mathbb Y}(n)$—the number of non-isomorphic $n$-element substructures—is uniformly bounded in $n$ (Kurilić, 2019).
• Main Results:
  • For any countable, complete almost chainable theory with infinite models, $I(\mathcal{T},\omega)\in\{1,2^{\aleph_0}\}$. Hence, Vaught’s Conjecture holds in this class (Kurilić, 2019).
  • The same dichotomy holds for monomorphic theories (all models have all $n$-element substructures isomorphic): $I(\mathcal{T},\omega)\in\{1,\mathfrak{c}\}$, with $I(\mathcal{T},\omega)=1$ exactly when some countable model is simply definable by an $\omega$-categorical linear order (Kurilić, 2018).

3. Theories of Partial Orders: Decomposition and Sharp VC

Recent advances characterize the model spectrum for large classes of theories of partial orders using lexicographic and product decompositions.

• Finite Lexicographic Decomposition (FLD$_1$-theory): A complete theory $\mathcal T$ of partial order is an FLD$_1$-theory if any (equivalently, some) model ${\mathbb X}$ admits a decomposition

${\mathbb X} =\sum_{\mathbb I}{\mathbb X}_i$

where $\mathbb I$ is a finite poset and each ${\mathbb X}_i$ has a largest element.

• VC-Decomposition and VC$^\#$-Decomposition: Such a decomposition is a VC-decomposition if every summand satisfies VC; a VC$^\#$-decomposition requires the summands to satisfy VC$^\#$ ($I({\mathbb X}_i)\in\{1,\mathfrak{c}\}$).
• Key Results (Kurilić, 6 Jan 2026, Kurilić, 2022):
  • VC holds for $\mathcal{T}$ iff $\mathcal{T}$ is large or its atomic model has a VC-decomposition.
  • VC$^\#$ holds for any theory admitting a VC$^\#$-decomposition.
  • The closure under finite lexicographic sums, disjoint unions, and finite products of classes of linear orders, Boolean algebras, and finite monomorphic trees preserves both VC and VC$^\#$.
  • For infinite disjoint unions of linear orders, the spectrum is always either $1$ (if all components are finite or $\omega$-categorical and only finitely many isomorphism types appear) or continuum (Kurilić, 2022).

4. Proof Techniques: Algebraic and Descriptive-Set-Theoretic Approaches

A robust methodology deploys algebraic logic and descriptive-set theory.

• Algebraic encodings: Models of a theory are represented as ultrafilters in a suitable locally finite algebra (e.g., cylindric or quasi-polyadic algebra). The isomorphism or distinguishability relation is Borel (even analytic) on a Polish space, making them amenable to classical dichotomy theorems.
• Glimm-Effros and Silver Dichotomies: Any Borel (analytic) equivalence relation on a Polish space has either countably many or continuum many classes. Thus, the model spectrum for such definable equivalence classes is restricted to these two possibilities (Assem et al., 2013, Assem et al., 2018).
• Type-omitting and closure arguments: The construction of models omitting specified types, together with back-and-forth arguments and combinatorial decompositions, facilitates the reduction from structural or syntactic conditions to the VC dichotomy in concrete classes.

5. Computability-Theoretic and Infinitary Generalizations

VC has been connected to problems in computability theory and generalized to infinitary logics.

• Computability-Theoretic Equivalents: For $L_{\omega_1,\omega}$ sentences, the absence of an intermediate cardinal model spectrum is equivalent to hyperarithmetic-is-recursive on a cone of Turing degrees, with an exact description of degree spectra of models in counterexample scenarios (Montalban, 2012).
• Infinitary and $\omega$-Vaught’s Conjecture: The infinitary generalization posits the same dichotomy for complete $L_{\omega_1,\omega}$ theories. The $\omega$-VC strengthens this by asserting a uniform bound on the Vaught ordinal: for sentence $S$ and any $\Pi^{{\rm in}}_\alpha$-extension $T$, ${\rm vo}(T)\leq \alpha+\omega$ (Gonzalez et al., 2022).

6. Notable Special Cases and Open Directions

Verification for Specific Classes:

• Theories of linear orders with unary predicates (Rubin), rooted trees admitting finite monomorphic decompositions, partial orders closed under finite products and disjoint unions, reticles, Boolean algebras—all satisfy VC and, in many cases, the sharp VC$^\#$ (Kurilić, 2022, Kurilić, 6 Jan 2026).
• Theories with a definable infinite discrete linear order have continuum many countable models (Tanović, 2022).

Research Frontiers:

• Open cases for arbitrary countable theories remain; no ZFC proof exists for the global conjecture.
• The structural approach via profile bounds or decompositions and the computability-theoretic reductions suggest further lines of attack for both VC and sharper forms such as $\omega$-VC and VC$^\#$.
• Generalization to classes with finite profile, or the possibility of decompositions into components for which VC is known, remains a leading research area.

Summary Table: Model Spectrum for Key Theory Classes

Class	Model Spectrum	Source
Almost chainable theories	$\{1,2^{\aleph_0}\}$	(Kurilić, 2019)
Monomorphic theories	$\{1,\mathfrak{c}\}$	(Kurilić, 2018)
FLD$_1$ partial order theories (VC)	$\{\leq\aleph_0,\mathfrak{c}\}$	(Kurilić, 6 Jan 2026)
FLD$_1$ partial order theories (VC$^\#$)	$\{1,\mathfrak{c}\}$	(Kurilić, 6 Jan 2026)
Disjoint unions of linear orders	$\{1,2^{\aleph_0}\}$	(Kurilić, 2022)
Theories with definable discrete order	$2^{\aleph_0}$	(Tanović, 2022)

7. Contextual and Methodological Significance

VC is confirmed for a wide range of natural theory classes—including, but not limited to, almost chainable, monomorphic, many partial order theories, and theories with strong symmetry or bounded profile. The unifying principle in these verifications is the reduction, via interpretability, type profiles, decomposability, or computable presentation, to scenarios already governed by dichotomy theorems or rigid combinatorial invariants.

Further work seeks to identify new classes and closure properties, sharpen the exact dividing lines, and clarify whether all failures of VC (should they exist) must encode “complex” combinatorial or recursion-theoretic phenomena beyond the reach of these structural criteria.