Whether there exists a stable Ehrenfeucht theory - a countable, complete, stable first-order theory T with finitely many but more than one countable model, 1<I(ℵ0,T)<ℵ01<I(\aleph_0,T)<\aleph_0 - is a classical open question of model theory that has circulated since the 1970s as a weak form of the conjecture that every Ehrenfeucht theory has the strict order property. The dichotomy "one or infinitely many countable models" is known for superstable theories (Lachlan 1973), supersimple theories (Kim 1999), and several adjacent stable classes (finite-weight unions, finitely based, 1-based), and Kim's 2019 theorem on NSOP1_1 theories forces any stable example to contain a type of infinite self-preweight. All known Ehrenfeucht constructions have the strict order property. No example and no impossibility theorem is known for the general stable (necessarily non-superstable) case, which is the precise open core; the question is stated as open in the Pillay–Tanović survey of August 2026.

Background

For a countable, complete first-order theory TT with infinite models, write I(ℵ0,T)I(\aleph_0,T) for the number of pairwise non-isomorphic countable models of TT; TT is called ℵ0\aleph_0-categorical when I(ℵ0,T)=1I(\aleph_0,T)=1. The possible values of I(ℵ0,T)I(\aleph_0,T) are strongly constrained: the value 22 never occurs (Vaught), and for countable first-order theories the only possible infinite values are ℵ0\aleph_0, ℵ1\aleph_1, and 2ℵ02^{\aleph_0} (Morley). Theories with 1<I(ℵ0,T)<ℵ01<I(\aleph_0,T)<\aleph_0 - finitely many, but more than one, countable models - are called Ehrenfeucht theories, after the first example, recorded by Vaught: the theory of dense linear orders without endpoints together with an increasing sequence of constants c0<c1<c2<⋯c_0<c_1<c_2<\cdots, whose three countable models differ according to whether the constant sequence is unbounded, has a supremum, or is bounded without a supremum. Every Ehrenfeucht theory is small, meaning that for each nn there are only countably many complete nn-types over the empty set (every type is realized in some countable model by downward Loewenheim-Skolem, and finitely many countable models jointly realize only countably many types). All known Ehrenfeucht theories are variations of this example - obtained from ℵ0\aleph_0-categorical theories that define dense linear orders or meet-trees (partial orders in which the predecessors of any element form a dense linear order without endpoints and any two elements have an infimum) by adding axioms for infinitely many constants or unary predicates - and every one of them has the strict order property (SOP): some formula defines a partial order with infinite chains.

A theory is stable if it lacks the order property: no formula φ(xˉ,yˉ)\varphi(\bar x,\bar y) admits tuples aˉi,bˉj\bar a_i,\bar b_j in a model with φ(aˉi,bˉj)\varphi(\bar a_i,\bar b_j) holding exactly when i<ji<j. Stability is the fundamental dividing line of Shelah's classification theory: unstable theories have the maximal number of models in every uncountable cardinality, while stable theories admit a dimension-theoretic analysis of their models through forking independence, the notion of independence under which a type forks over a set when it implies a formula that divides, i.e., that becomes kk-inconsistent along an indiscernible sequence (a sequence whose finite tuples of entries all satisfy the same formulas). Superstable theories form a proper subclass of the stable theories in which every type does not fork over some finite set; superstability is strictly stronger than stability. On the simple side, where forking is still well behaved but stability may fail, the analogous class is the supersimple theories. Since the strict order property implies instability, a stable Ehrenfeucht theory would in particular be an Ehrenfeucht theory without SOP, so the question below is a weak form of the conjecture - circulating since the 1970s and still open (Conjecture 1 of the 2026 Pillay–Tanović survey) - that every Ehrenfeucht theory has the strict order property.

The evidence for a negative answer is substantial. Lachlan proved that a countable superstable theory is never Ehrenfeucht: it has exactly one or at least ℵ0\aleph_0 countable models. The same dichotomy holds for countable increasing unions of stable theories of finite weight (Tsuboi), for stable finitely based theories, in which every set of indiscernibles is based on a finite subset (Hrushovski), and for stable 1-based theories, in which forking is trivial in the sense that every tuple is independent from any parameter set over the algebraic closure of the tuple intersected with the algebraic closure of the parameter set (Pillay). Kim extended Lachlan's theorem to all supersimple theories, his proof using only the symmetry and transitivity of nonforking available in simple theories. Most recently, Kim proved that a non-ℵ0\aleph_0-categorical NSOP1_1 theory with nonforking existence cannot have finitely many countable models unless some finite tuple has preweight ω\omega with respect to itself; here NSOP1_1 is the weakest of Shelah's strict-order hierarchy above simplicity, nonforking existence is the axiom that every type has a nonforking extension, and the preweight of a tuple counts how many pairwise independent tuples it can fork with simultaneously. Because stable theories are NSOP1_1 and satisfy nonforking existence, this constrains any stable Ehrenfeucht theory: it must contain a type of infinite self-preweight. Benda proved that every Ehrenfeucht theory contains a powerful type - a non-isolated complete type pp such that the prime model (the model elementarily embeddable into every model) over a single realization of pp already realizes all complete types of the theory - so a stable Ehrenfeucht theory would have to combine a powerful type with infinite self-preweight. Herwig constructed a small stable theory with a type of infinite self-preweight, showing that these necessary conditions are consistent with stability and smallness individually; but nobody has converted such a configuration into an Ehrenfeucht theory, and no impossibility theorem is known for the stable, necessarily non-superstable, case.

Problem Statement

Does there exist a countable, complete, stable first-order theory TT, with infinite models, such that

1<I(ℵ0,T)<ℵ0,1<I(\aleph_0,T)<\aleph_0,

that is, a stable Ehrenfeucht theory? Equivalently: must every countable, complete, stable theory be either ℵ0\aleph_0-categorical or have infinitely many pairwise non-isomorphic countable models?

Here 'stable' is the standard notion (no formula with the order property), not a stronger condition such as superstability or ω\omega-stability. A positive answer must exhibit an explicit theory together with a proof of its stability and a determination of the exact finite number of isomorphism types of its countable models; a negative answer must prove the dichotomy for all countable complete stable theories. Since the superstable case is settled by Lachlan's theorem, any example is necessarily stable but not superstable, and by the known partial results it must also fail to be 1-based and finitely based, must not be a countable increasing union of stable theories of finite weight, and must contain a powerful type of infinite self-preweight (a type whose preweight with respect to itself is ω\omega).

References

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