I'd appreciate your help with proofing one or both of the following statements:
1) let $M$ be an infinite countable structure. We want to show that there's an uncountable structure in $Mod(Th(M))$, whereby $Th(M)$ is the theory of $M$ (i.e. the set of all closed formulas $\phi$, which are fulfilled in every $\mathcal{M} \in M$).
Hint: The completeness theorem and the compactness theorem do hold for uncountable languages, too. Use uncountably many constant symbols.
2) let $\Sigma$ be a theory (i.e. a set of closed formulas) with the following property: For every natural number n there's a natural number n' > n and a $\mathcal{M} \in Mod(\Sigma)$ with exactly n' elements. (whereby $Mod(\Sigma)$ is the class of all structures $\mathcal{M}$ (of a given language) that fulfill $\mathcal{M} \vDash \Sigma$). Show that there's an infinite structure in $Mod(\Sigma)$.
For 1) I don't know what to do.
For 2) I have the following idea: Assume there was no infinite structure in $Mod(\Sigma)$. Then there would have to be a maximal structure $\sigma \in Mod(\mathcal{M})$ (i.e. a structure with more elements than every other strucutre in $Mod(\Sigma)$ has). Let m be the number of its elements. Then there would be no structure $\sigma'$ in $Mod(\Sigma)$ with exactly m+1 elements, which is a contradiction to the property of $\Sigma$.
[Short question: Is the whole expression for $Mod(\Sigma)$, "Model for $\Sigma$" or what's it called?]
Best Answer
I don't think you can say that because there's no infinite structure, there is a maximal structure. In fact, that is pretty much what you're trying to prove. It's certainly conceivable at the outset that there could be a structure of every finite size, no matter how big, but no infinite structure.
The key to both problems is a compactness argument where you add a set of new constant symbols $\{c_i:i\in I\},$ as well as the axioms $c_i\ne c_j$ for all $i,j\in I.$ Then you show that every finite subtheory of this extended theory has a model by finding a model with more elements than $c_i$ that appear in the subtheory and assigning distinct elements to be interpretations of those $c_i.$ Then by compactness, the theory has a model, which by construction must have cardinality $\ge |I|.$
The notation $Mod(\Sigma)$ is not something I've seen very much (though even if you hadn't defined it, it would have been pretty clear from context what it was.) The most effecient expression I can think of is "the class of all models of $\Sigma$".
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I will explain the compactness argument in more detail, for the particular case of problem 2 (as I've said, 1 is very similar). Let $\Sigma$ be a $L$-theory that has models of every finite size. Now, let $\{c_i: i\in \mathbb N\}$ be a countably infinite collection of distinct constant symbols that do not appear in $L.$ Consider the extended language $L' = L\cup \{c_i: i\in \mathbb N\}$ and the $L'$-theory $$ \Sigma'=\Sigma \cup \{c_i\ne c_j: \mbox{$i,j\in \mathbb N$ and $i\ne j$}\}.$$
We claim $\Sigma'$ has a model. By compactness, it suffices to show that any finite subtheory of $\Sigma'$ is satisfiable. So let $\Sigma_0$ be such a finite subtheory. Then only a finite number $$ \{c_{i_k}:k\in \{1,2,\ldots, n\}\}$$ of the new constant symbols occur in $\Sigma_0.$ Then let $M$ be a model of $\Sigma$ with $n$ or more elements (which exists by hypothesis). Then $M$ becomes a model of $\Sigma_0$ when we interpret each of the $c_{i_k}$ as distinct elements of $M,$ since distinctness means it will satisfy any statement of the form $c_{i_k} \ne c_{i_{k'}}$ that might appear in $\Sigma_0.$
So let $N$ be a model of $\Sigma'.$ Clearly $N$ is a model of $\Sigma.$ We can also see that $N$ is infinite, since if it were finite, then one of the $c_i\ne c_j$ axioms in $\Sigma'$ would have to fail by pigeonhole.