Spectrum of a theory

In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of

In this section T is a countable complete theory and κ is a cardinal.

The Löwenheim–Skolem theorem shows that if I(T,κ) is nonzero for one infinite cardinal then it is nonzero for all of them.

Morley's problem was a conjecture (now a theorem) first proposed by Michael D. Morley that I(T,κ) is nondecreasing in κ for uncountable κ. This was proved by Saharon 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 theory T, either I(T,κ) = 2^κ for all uncountable cardinals κ, or ${\displaystyle \textstyle I(T,\aleph _{\xi })<\beth _{\omega _{1}}(|\xi |+\aleph _{0})}$ for all ordinals ξ (See Aleph number and Beth 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 and Michael C. Laskowski gave the following complete solution to the spectrum problem for countable theories in uncountable cardinalities. If T 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 the Rado graph.
2. ${\displaystyle \beth _{d+1}(|\alpha +\omega |)}$ for some countable infinite ordinal d. (For finite d see case 8.) Examples: The theory with equivalence relations E_β for all β with β+1<d, such that every E_γ class is a union of infinitely many E_β classes, and each E₀ class is infinite.
3. ${\displaystyle \beth _{d-1}(|\alpha +\omega |^{2^{\aleph _{0}}})}$ for some finite positive ordinal d. Example (for d=1): the theory of countably many independent unary predicates.
4. ${\displaystyle \beth _{d-1}(|\alpha +\omega |^{\aleph _{0}}+\beth _{2})}$ for some finite positive ordinal d.
5. ${\displaystyle \beth _{d-1}(|\alpha +\omega |+\beth _{2})}$ for some finite positive ordinal d;
6. ${\displaystyle \beth _{d-1}(|\alpha +\omega |^{\aleph _{0}})}$ for some finite positive ordinal d. Example (for d=1): the theory of countable many disjoint unary predicates.
7. ${\displaystyle \beth _{d-1}(|\alpha +\omega |+\beth _{1})}$ for some finite ordinal d≥2;
8. ${\displaystyle \beth _{d-1}(|\alpha +\omega |)}$ for some finite positive ordinal d;
9. ${\displaystyle \beth _{d-2}(|\alpha +\omega |^{|\alpha +1|})}$ for some finite ordinal d≥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, where G is some subgroup of the symmetric group on n ≥ 2 elements. Here, we identify αⁿ with the set of sequences of length n of elements of a set of size α. G
    acts 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 the wreath product
    of G 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.

• Spectrum of a sentence

• 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.
  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