Vaught's Conjecture Research Articles

Nearly Countable Dense Homogeneous Spaces

AbstractWe study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely n types of countable dense sets: such a space contains a subset S of size at most n−1 such that S is invariant under all homeomorphisms of X and X ∖ S is countable dense homogeneous. We prove that every Borel space having fewer than c types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or c many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

Isomorphism of Computable Structures and Vaught's Conjecture

AbstractThe following question is open: Does there exist a hyperarithmetic class of computable structures with exactly one non-hyperarithmetic isomorphism-type? Given any oracle α ∈ 2ω, we can ask the same question relativized to α. A negative answer for every α implies Vaught's Conjecture for Lω1ω.

Vaught's Theorem on Axiomatizability by a Scheme

AbstractIn his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has decidable extensions. We also explore the ramifications of our work on finite axiomatizability of schemes in the presence of suitable comprehension principles.

On Vaught’s Conjecture and finitely valued MV algebras

AbstractWe show that the complete first order theory of an MV algebra has \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2^{\aleph _0}$\end{document} countable models unless the MV algebra is finitely valued. So, Vaught's Conjecture holds for all MV algebras except, possibly, for finitely valued ones. Additionally, we show that the complete theories of finitely valued MV algebras are \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2^{\aleph _0}$\end{document} and that all ω‐categorical complete theories of MV algebras are finitely axiomatizable and decidable. As a final result we prove that the free algebra on countably many generators of any locally finite variety of MV algebras is ω‐categorical.

Some variants of Vaught's conjecture from the perspective of algebraic logic

Vaught’s Conjecture states that if T is a complete first order theory in a countable language such that T has uncountably many pairwise non-isomorphic countably infinite models, then T has 2^ℵ_0 many pairwise non-isomorphic countably infinite models. Continuing investigations initiated in S´agi, we apply methods of algebraic logic to study some variants of Vaught’s conjecture. More concretely, let S be a σ-compact monoid of selfmaps of the the natural numbers. We prove, among other things, that if a complete first order theory T has at least ℵ1 many countable models that cannot be elementarily embedded into each other by elements of S, then, in fact, T has continuum many such models. We also study-related questions in the context of equality free logics and obtain similar results. Our proofs are based on the representation theory of cylindric and quasi-polyadic algebras (for details see Henkin, Monk and Tarski (cylindric Algebras Part 1 and Part 2)) and topological properties of the Stone spaces of these algebras.

Independently axiomatizable ℒω1,ω theories

AbstractIn partial answer to a question posed by Arnie Miller [4] and X. Caicedo [2] we obtain sufficient conditions for an ℒω1,ω theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every ℒω1,ω theory in a countable language has an independent axiomatization. The second, this time outright in ZFC, every intersection of a family of Borel sets can be formed as the intersection of a family of independent Borel sets.

Comparing Borel Reducibility and Depth of an ω-Stable Theory

In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories ( T α ) α < ω 1 with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility.

Relative Vaught's Conjecture for Some Meager Groups

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.

The Vaught Conjecture: Do Uncountable Models Count?

We give a model theoretic proof, replacing admissible set theory by the LopezEscobar theorem, of Makkai’s theorem: Every counterexample to Vaught’s conjecture has an uncountable model which realizes only countably many Lω1,ω-types. The following result is new. Theorem. If a first order theory is a counterexample to the Vaught conjecture then it has 2א1 models of cardinality א1. In this paper we prove several properties of putative counterexamples to the Vaught conjecture. Specifically, these results concern the number of models the counterexample has in power א1. One of these results was proved 30 years ago using admissible model theory; we give a more straightforward argument. The following question guides our discussion. Is the Vaught Conjecture model theory? Here are some possible ways in which this question would have a clear answer. Shelah, Buechler, Newelski have shown using rather difficult techniques from stability theory that the conjecture holds for first order theories that are ‘simple’ from the stability theoretic standpoint: ω-stable or superstable with finite U -rank. If a counterexample were found for a first order theory of slightly greater complexity (e.g. a stable but not superstable first order theory), this would indicate the issue was a model theoretic one. If on the other hand, a uniform proof for sentences of Lω1,ω were given using methods of descriptive set theory, then it would not be a model theoretic problem. The results below give partial answers to the following methodological questions. What specific model theoretic as opposed to descriptive set theoretic techniques can attack the problem? Can one use more direct model theoretic arguments to obtain some result of admissible model theory? I would argue the problem is model theoretic if its solution is different for Lω,ω and Lω1,ω. So we will investigate the differences between properties known about counterexamples to

The Borel Complexity of Isomorphism for Theories with Many Types

During the Notre Dame workshop on Vaught's Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for countable models of a first-order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types.

Categories of Topological Spaces and Scattered Theories

We offer a topological treatment of scattered theories intended to help to explain the parallelism between, on the one hand, the theorems provable using Descriptive Set Theory by analysis of the space of countable models and, on the other, those provable by studying a tree of theories in a hierarchy of fragments of infinintary logic. We state some theorems which are, we hope, a step on the road to fully understanding counterexamples to Vaught's Conjecture. This framework is in the early stages of development, and one area for future exploration is the possibility of extending it to a setting in which the spaces of types of a theory are uncountable.

Vaught's Conjecture Workshop Participants

Vaught's Conjecture Workshop Participants

Introduction to the Special Issue on Vaught's Conjecture

Introduction to the Special Issue on Vaught's Conjecture

The Number of Countable Differentially Closed Fields

We outline the Hrushovsk-Sokolovic proof of Vaught's Conjecture for differentially closed fields, focusing on the use of dimensions to code graphs.

A Note on Counterexamples to the Vaught Conjecture

If some infinitary sentence provides a counterexample to Vaught's Conjecture, then there is an infinitary sentence which also provides a counterexample but has no model of cardinality bigger than ℵ₁.

Vaught's Conjecture and Group Rings

ABSTRACT We discuss Vaught's conjecture for complete theories of modules over some group rings over the integers and other Dedekind domains, and over pullback rings of Dedekind domains.

Elementary Regular Rings. II

We extend the well-known result by Burris and Werner on existence of defining sequences for elementary products of models to arbitrary enrichments of Boolean algebras (we obtain a complete analog of the Feferman–Vaught theorem). This enables us to establish decidability of the elementary theory of a classical object of number theory, the ring of adeles.

Algorithmic uses of the Feferman–Vaught Theorem

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.

Very simple theories without forking

We prove Vaught's conjecture for minimal trivial simple theories satisfying the generalized independence theorem.

Computability-Theoretic Complexity of Countable Structures

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures.