Spectrum of a theory

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. The spectrum problem is to describe the possible behaviors of I(T, κ) as a function of 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. 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 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. Morley proved that if I(T,א0) is infinite then it must be א0 or א1 or 2א0. 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. 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) = 2κ for all uncountable cardinals κ, or 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.