Compactness Theorem for Non-elementary classes

Question

Let $\mathcal{L}$ be a first-order language and let $\mathcal{C}$ be a class of $\mathcal{L}$-structures which is not necessarily an elementary class (i.e. there does not exist a theory $T$ such that $\mathcal{C}=mod(T)$).

Let $\Sigma$ be a set of $\mathcal{L}$-sentences such that for every finite subset $\Sigma_0$ of $\Sigma$ there is a member $\mathcal{M}_{}\in\mathcal{C}$ so that $\mathcal{M}\models \Sigma_0$. Then, can we conclude that there is a member $\mathcal{N}\in\mathcal{C}$ such that $\mathcal{N}\models \Sigma$?

Answer 1

No, we cannot. For example, take $\mathcal{C}$ to be the class of all finite structures in our language, let $\varphi_n$ be the sentence asserting that there exist at least $n$ elements in the universe - that is, $\varphi_n$ is the sentence $$\exists x_1,..., x_n(\bigwedge_{1\le i<j\le n}\neg x_i=x_j),$$ which is clearly first-order - and let $\Sigma=\{\varphi_n:n\in\mathbb{N}\}$. Every finite subset of $\Sigma$ is satisfied in all sufficiently large finite structures, but $\Sigma$ itself is only satisfied in infinite structures.

---

Note that we can do a similar trick with larger cardinalities, assuming a large enough language. For any infinite cardinal $\kappa$, let $C_\kappa$ be the language consisting of $\kappa$-many distinct constant symbols $c_\eta$ ($\eta<\kappa$), take $\Sigma=\{\neg c_\eta=c_\theta: \eta<\theta<\kappa\}$, and take $\mathcal{C}$ to be the class of all structures of cardinality $<\kappa$. Then every subset of $\Sigma$ with cardinality $<\kappa$ (not just every finite subset) is satisfied in some element of $\mathcal{C}$, but no element of $\mathcal{C}$ satisfies all of $\Sigma$.