Let ψ be a first order formula. Wikipedia defines the spectrum of the formula ψ as follows: The spectrum of ψ is the set of natural numbers n such that there is a finite model for ψ with n elements.
In this wikipedia article, it says:
In full generality, it is not known if the set of spectra of a theory is closed by complementation, it is the so-called Asser's Problem.
So, there is talk of the spectrum of a theory, that is a set of formulas. Why is the spectrum previously only defined for a single formula instead of a whole set of formulas? Are the spectra of theories the same as the spectra of formulas? Is the spectrum problem for sets of formulas really open?
To improve this question, let me add some explanations:
Above I have given the definition of the spectrum of a first order sentence. One can also define the spectrum of a set $X$ of first order sentences: Let the spectrum of $X$ be the set of all cardinalities of finite structures that satisfy all the sentences in $S$.
Now I am wondering:
Given a set $X\subseteq \mathbb N_+$, is $X$ the spectrum of some first order sentence ψ if and only if $X$ is the spectrum of some set $S$ of first order formulas?
Let $S$ be a set of first order sentences. Is the spectrum of $S$ always decidable? (If we replace "set of first order formulas" by a single "first order formula" then the answer is yes, of course)