In model theory, a branch of mathematical logic, the spectrum of a theory is given by the number of isomorphism classes of models in various cardinalities.
More precisely, for any complete theory T in a language we write I(T, κ) for the number of models of T (up to isomorphism) of cardinality κ.
It has been almost completely solved for the case of a countable theory T. In this section T is a countable complete theory and κ is a cardinal.
Morley's categoricity theorem was the first main step in solving the spectrum problem: it states that if I(T,κ) is 1 for some uncountable κ then it is 1 for all uncountable κ. Robert Vaught showed that I(T,ℵ0) cannot be 2.
It is easy to find examples where it is any given non-negative integer other than 2.
It is not known if it can be ℵ1 if the continuum hypothesis is false: this is called the Vaught conjecture and is the main remaining open problem (in 2005) in the theory of the spectrum.
For this, he proved a very deep dichotomy theorem.
Saharon Shelah gave an almost complete solution to the spectrum problem.
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, ℵα) of isomorphism classes of models is given for ordinals α>0 by the minimum of 2ℵα and one of the following maps: Moreover, all possibilities above occur as the spectrum of some countable complete theory.
The number d in the list above is the depth of the theory.
This can be used to construct examples of theories with spectra in the list above for non-minimal values of d from examples for the minimal value of d.