Equivalence infinite Spectrum problem and finite spectrum problem Definitions:
For a given first order sentence $\phi$ define $\text{spectrum}(\phi)$ to be the set of all cardinalities of the finite models of $\phi$. A set $S\subseteq\mathbb N_+$ is said to be a spectrum, if there is a first order sentence $\phi$ with $\text{spectrum}(\phi)=S$.
For a given first order sentence $\phi$ define $\infty\text{-spectrum}(\phi)$ to be the class of all cardinal numbers $\kappa$ for which there is a model $M$ of $\phi$ such that the cardinality of $M$ is $\kappa$. Let $\text{Card}$ be the class of all cardinal numbers. Set $\text{Card}_+:=\text{Card}\setminus \{0\}$. A subclass $T$ of $\text{Card}_+$ is said to be an $\infty$-spectrum if there is a first order sentence $\phi$ with $\infty\text{-spectrum}(\phi)=T$.
Question: Are the following assertions equivalent?
(1) For every spectrum $S$, the set $\mathbb N_+\setminus S$ is also a spectrum.
(2) For every $\infty$-spectrum $T$, the class $\mathrm{Card}_+\setminus T$ is also an $\infty$-spectrum.
How to prove this? Am I too stupid or am I too stupid?
 A: (2) is false: if a sentence $\varphi$ has any infinite models, then $\infty$-spectrum($\varphi$) contains every infinite cardinal (this is Lowenheim-Skolem) - so $Card_+\setminus T$ contains no infinite cardinal. Thus, any sentence $\varphi$ such that


*

*$\varphi$ has an infinite model, but

*spectrum$(\varphi$) is not a cofinite subset of $\mathbb{N}$
is a counterexample to (2) (using the fact that, by Compactness, if a sentence $\psi$ has arbitrarily large finite models then it has an infinite model). For example, let $\varphi$ be the sentence saying that the structure is a field.
Meanwhile, (1) is open (called the spectrum problem, see http://www.diku.dk/hjemmesider/ansatte/neil/SpectraSubmitted.pdf). So the two principles are certainly not easily proved to be equivalent; and if (1) is true, then they are not equivalent.
Certainly, they are not equivalent "morally" - the disproof of (2) is a straightforward application of basic results about first-order logic (compactness and Lowenheim-Skolem), while even if (1) is false, it would be false for a much deeper reason (based purely on the amount of work which has already gone into its study, and the interesting partial results gotten along the way).
An important thing to keep in mind is that (2) is a statement about infinite model theory, while (1) is a statement about finite model theory; and these two aspects of model theory are extremely different. For instance, finite analogues of compactness and Lowenheim-Skolem are easily disproved; and while the relation "$\theta\models \psi$" is c.e. by the Completeness Theorem, the relation "$\theta\models\psi$ on finite models" is co-c.e.. So you should not expect questions about finite model theory to have anything to do with their classical model theoretic counterparts. (Note that this is part of what makes finite model theory so cool!)

By the way, this is essentially just the combination of answers your other questions have already received (Infinite Spectrum Problem and Is T an infinity spectrum whenever T is a spectrum?).
