Inmodel theory, a branch ofmathematical logic, thespectrum of a theoryis given by the number ofisomorphism classes ofmodels in variouscardinalities. More precisely, for anycomplete theoryT in a language we writeI(T,κ) for the number of models ofT (up to isomorphism) of cardinalityκ. Thespectrum problem is to describe the possible behaviors ofI(T,κ) as a function ofκ. It has been almost completely solved for the case of acountable theoryT.

In this sectionT is a countable complete theory andκ is a cardinal.

TheLöwenheim–Skolem theorem shows that ifI(T,κ) is nonzero for one infinite cardinal then it is nonzero for all of them.

Morley's categoricity theorem was the first main step in solving the spectrum problem: it states that ifI(T,κ) is 1 for some uncountableκ then it is 1 for all uncountableκ.

Robert Vaught showed thatI(T,ℵ₀) cannot be 2. It is easy to find examples where it is any given non-negative integer other than 2. Morley proved that ifI(T,ℵ₀) is infinite then it must be ℵ₀ or ℵ₁ or 2^ℵ₀. It is not known if it can be ℵ₁ if thecontinuum hypothesis is false: this is called theVaught conjecture and is the main remaining open problem (in 2005) in the theory of the spectrum.

Morley's problem was aconjecture (now a theorem) first proposed byMichael D. Morley thatI(T,κ) isnondecreasing inκ for uncountableκ. This was proved bySaharon Shelah. For this, he proved a very deep dichotomy theorem.

Saharon Shelah gave an almost complete solution to the spectrum problem. For a given complete theoryT, eitherI(T,κ) = 2^κ for all uncountable cardinalsκ, or for all ordinals ξ (SeeAleph number andBeth number for an explanation of the notation), which is usually much smaller than the bound in the first case. Roughly speaking this means that either there are the maximum possible number of models in all uncountable cardinalities, or there are only "few" models in all uncountable cardinalities. Shelah also gave a description of the possible spectra in the case when there are few models.

By extending Shelah's work, Bradd Hart,Ehud Hrushovski andMichael C. Laskowski gave the following complete solution to the spectrum problem for countable theories in uncountable cardinalities. IfT is a countable complete theory, then the number I(T, ℵ_α) of isomorphism classes of models is given for ordinals α>0 by the minimum of 2^ℵ_α and one of the following maps:

1. 2^ℵ_α. Examples: there are many examples, in particular any unclassifiable or deep theory, such as the theory of theRado graph.
2. ${\displaystyle {\beth }_{d+1}(|\alpha +\omega |)}$ for some countable infinite ordinald. (For finited see case 8.) Examples: The theory with equivalence relationsE_β for all β with β+1<d, such that everyE_γ class is a union of infinitely manyE_β classes, and eachE₀ class is infinite.
3. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega {|}^{2^{{\mathrm{\aleph }}_0}})}$ for some finite positive ordinald. Example (ford=1): the theory of countably many independent unary predicates.
4. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega {|}^{{\mathrm{\aleph }}_0}+{\beth }_2)}$ for some finite positive ordinald.
5. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega |+{\beth }_2)}$ for some finite positive ordinald;
6. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega {|}^{{\mathrm{\aleph }}_0})}$ for some finite positive ordinald. Example (ford=1): the theory of countable many disjoint unary predicates.
7. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega |+{\beth }_1)}$ for some finite ordinald≥2;
8. ${\displaystyle {\beth }_{d-1}(|\alpha +\omega |)}$ for some finite positive ordinald;
9. ${\displaystyle {\beth }_{d-2}(|\alpha +\omega {|}^{|\alpha +1|})}$ for some finite ordinald≥2; Examples: similar to case 2.
10. ${\displaystyle {\beth }_2}$. Example: the theory of the integers viewed as an abelian group.
11. ${\displaystyle |(\alpha +1{)}^n/G|-|{\alpha }^n/G|}$ for finite α, and |α| for infinite α, whereG is some subgroup of the symmetric group onn ≥ 2 elements. Here, we identify αⁿ with the set of sequences of lengthn of elements of a set of size α.Gacts on αⁿ by permuting the sequence elements, and |αⁿ/G| denotes the number of orbits of this action. Examples: the theory of the set ω×n acted on by thewreath product ofG with all permutations of ω.
12. ${\displaystyle 1}$. Examples: theories that are categorical in uncountable cardinals, such as the theory of algebraically closed fields in a given characteristic.
13. ${\displaystyle 0}$. Examples: theories with a finite model, and the inconsistent theory.

Moreover, all possibilities above occur as the spectrum of some countable complete theory.

The numberd in the list above is the depth of the theory.IfT is a theory we define a new theory 2^T to be the theory with an equivalence relation such that there are infinitely many equivalence classes each of which is a model ofT. We also define theories${\displaystyle {\beth }_n(T)}$ by${\displaystyle {\beth }_0(T)=T}$,${\displaystyle {\beth }_{n+1}(T)=2^{{\beth }_n(T)}}$. Then${\displaystyle I({\beth }_n(T),\lambda )=min({\beth }_n(I(T,\lambda )),2^{\lambda })}$ . This can be used to construct examples of theories with spectra in the list above for non-minimal values ofd from examples for the minimal value ofd.

• Spectrum of a sentence

• C. C. Chang,H. J. Keisler,Model Theory.ISBN 0-7204-0692-7
• Saharon Shelah, "Classification theory and the number of nonisomorphic models",Studies in Logic and the Foundations of Mathematics, vol. 92, IX, 1.19, p.49 (North Holland, 1990).
• Hart, Bradd; Hrushovski, Ehud; Laskowski, Michael C. (2000). "The Uncountable Spectra of Countable Theories".The Annals of Mathematics.152 (1):207–257.arXiv:math/0007199.Bibcode:2000math......7199H.doi:10.2307/2661382.JSTOR 2661382.
• Bradd Hart, Michael C. Laskowski, "A survey of the uncountable spectra of countable theories",Algebraic Model Theory, edited by Hart, Lachlan, Valeriote (Springer, 1997).ISBN 0-7923-4666-1

• Model theory

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