The Vaught conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

in the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

model theory In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

originally proposed by

Robert Lawson Vaught Robert Lawson Vaught (April 4, 1926 – April 2, 2002) was a mathematical logician and one of the founders of model theory.first-order complete theory in a countable language is finite or ℵ₀ or 2.

Morley Morley may refer to: Places England * Morley, Norfolk, a civil parish * Morley, Derbyshire, a civil parish * Morley, Cheshire, a village * Morley, County Durham, a village * Morley, West Yorkshire, a suburban town of Leeds and civil parish * ...

showed that the number of countable models is finite or ℵ₀ or ℵ₁ or 2, which solves the conjecture except for the case of ℵ₁ models when the

continuum hypothesis In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent ...

fails. For this remaining case, has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture. As of 2021, the counterexample has not been verified.

Statement of the conjecture

Let

T

be a first-order, countable, complete theory with infinite models. Let

I(T, \alpha)

denote the number of models of ''T'' of cardinality

\alpha

up to isomorphism, the

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of color ...

of the theory

T

. Morley proved that if ''I''(''T'', ℵ₀) is infinite then it must be ℵ₀ or ℵ₁ or the cardinality of the continuum. The Vaught conjecture is the statement that it is not possible for

\aleph_ < I(T,\aleph_) < 2^

. The conjecture is a trivial consequence of the

; so this axiom is often excluded in work on the conjecture. Alternatively there is a sharper form of the conjecture that states that any countable complete ''T'' with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel, in "On Vaught's conjecture". Cabal Seminar 76—77 (Proc. Caltech-UCLA Logic Sem., 1976—77), pp. 193–208, Lecture Notes in Math., 689, Springer, Berlin, 1978, this form of the Vaught conjecture is equiprovable with the original).

Original formulation

The original formulation by Vaught was not stated as a conjecture, but as a problem: ''Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly'' ℵ₁ ''non-isomorphic denumerable models?'' By the result by Morley mentioned at the beginning, a positive solution to the conjecture essentially corresponds to a negative answer to Vaught's problem as originally stated.

Vaught's theorem

Vaught proved that the number of countable models of a complete theory cannot be 2. It can be any finite number other than 2, for example: *Any complete theory with a finite model has no countable models. *The theories with just one countable model are the ω-categorical theories. There are many examples of these, such as the theory of an infinite set, or the theory of a

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

unbounded

total order In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexiv ...

. *

Ehrenfeucht Andrzej Ehrenfeucht (, born 8 August 1932) is a Polish-American mathematician and computer scientist. Life Andrzej Ehrenfeucht formulated the Ehrenfeucht–Fraïssé game, using the back-and-forth method given in Roland Fraïssé's PhD thesis. ...

gave the following example of a theory with 3 countable models: the language has a relation ≥ and a countable number of constants ''c''₀, ''c''₁, ... with axioms stating that ≥ is a dense unbounded total order, and ''c''₀ < ''c''₁ < ''c''₂ < ... The three models differ according to whether this

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

is unbounded, or converges, or is bounded but does not converge. *Ehrenfeucht's example can be modified to give a theory with any finite number ''n'' ≥ 3 of models by adding ''n'' − 2 unary relations ''P''_''i'' to the language, with axioms stating that for every ''x'' exactly one of the ''P''_''i'' is true, the values of ''y'' for which ''P''_''i''(''y'') is true are dense, and ''P''₁ is true for all ''c''_''i''. Then the models for which the sequence of elements ''c''_''i'' converge to a limit ''c'' split into ''n'' − 2 cases depending on for which ''i'' the relation ''P''_''i''(''c'') is true. The idea of the proof of Vaught's theorem is as follows. If there are at most countably many countable models, then there is a smallest one: the

atomic model Atomic theory is the scientific theory that matter is composed of particles called atoms. Atomic theory traces its origins to an ancient philosophical tradition known as atomism. According to this idea, if one were to take a lump of matter a ...

, and a largest one, the

saturated model In mathematical logic, and particularly in its subfield model theory, a saturated model ''M'' is one that realizes as many complete types as may be "reasonably expected" given its size. For example, an ultrapower model of the hyperreals is \al ...

, which are different if there is more than one model. If they are different, the saturated model must realize some ''n''-type omitted by the atomic model. Then one can show that an atomic model of the theory of structures realizing this ''n''-type (in a language expanded by finitely many constants) is a third model, not isomorphic to either the atomic or the saturated model. In the example above with 3 models, the atomic model is the one where the sequence is unbounded, the saturated model is the one where the sequence converges, and an example of a type not realized by the atomic model is an element greater than all elements of the sequence.

Topological Vaught conjecture

The topological Vaught conjecture is the statement that whenever a

Polish group Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles, people from Poland or of Polish descent * Polish chicken * Polish brothers (Mark Polish and Michael Polish, born 1970), American twin scree ...

acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...

continuously on a

Polish space In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named ...

, there are either countably many

orbits In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...

or continuum many orbits. The topological Vaught conjecture is more general than the original Vaught conjecture: Given a countable language we can form the space of all structures on the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

s for that language. If we equip this with the

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

generated by first-order formulas, then it is known from A. Gregorczyk,

A. Mostowski Andrzej Mostowski (1 November 1913 – 22 August 1975) was a Polish mathematician. He is perhaps best remembered for the Mostowski collapse lemma. Biography Born in Lemberg, Austria-Hungary, Mostowski entered University of Warsaw in 1931. He ...

, C. Ryll-Nardzewski, "Definability of sets of models of axiomatic theories" (''Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics)'', vol. 9 (1961), pp. 163–7) that the resulting space is Polish. There is a continuous action of the infinite

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

(the collection of all permutations of the natural numbers with the topology of point-wise convergence) that gives rise to the

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...

of isomorphism. Given a complete first-order theory ''T'', the set of structures satisfying ''T'' is a minimal, closed invariant set, and hence Polish in its own right.

References

* * * R. Vaught, "Denumerable models of complete theories", Infinitistic Methods (Proc. Symp. Foundations Math., Warsaw, 1959) Warsaw/Pergamon Press (1961) pp. 303–321 * * {{DEFAULTSORT:Vaught Conjecture Conjectures Unsolved problems in mathematics Model theory

Statement of the conjecture

Original formulation

Vaught's theorem

Topological Vaught conjecture

See also

References