Spectrum of a theory
model theory concept
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.
Early results
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.
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