Axiomatizability of a relative complement

Question

Given a fixed first order lexicon $\mathcal{L}$, suppose $\mathcal{K}$ is an axiomatizable class such that $\mathcal{K}\subseteq Mod(\varphi)$ for some sentence $\varphi$. If $Mod(\varphi)-\mathcal{K}$ is axiomatizable, Does $\mathcal{K}$ necessarily to be finitely axiomatizable?

I proved the converse but I don't know how to deal with this direction. I've already proved using compactness that if $\mathcal{K}$ and $\mathcal{K}^{c}$ are both axiomatizable classes, then $\mathcal{K}$ need to be finitely axiomatizable. At this point, I tried to use last fact but this is not the case because $Mod(\varphi)-\mathcal{K}$ is a relative complement, whereas $\mathcal{K}^{c}$ is the universal complement.

I would appreciate some hint, ¡Thanks in advance!

Answer 1

One way to do this is to shift attention away from $\mathcal{K}$ itself, so that we don't get stuck inside $Mod(\varphi)$.

Let $$\mathcal{K}_\varphi=\mathcal{K}\cup\{\mathfrak{A}:\mathfrak{A}\models\neg\varphi\}.$$

Now $Mod(\varphi)-\mathcal{K}=(\mathcal{K}_\varphi)^c$, and we're assuming that that's axiomatizable. Our natural next step is to show that $\mathcal{K}_\varphi$ is axiomatizable:

Suppose $\mathcal{K}=Mod(\Gamma)$ and let $\Gamma'=\{\gamma\vee\neg\varphi:\gamma\in\Gamma\}$.

In fact, a minor tweak of this shows that the union of finitely many axiomatizable classes is axiomatizable:

Given $\Gamma_1,...,\Gamma_k$ we have $$Mod(\{\gamma_1\vee...\vee\gamma_k: \gamma_1\in\Gamma_1,...,\gamma_k\in\Gamma_k\})= \bigcup_{1\le i\le k}Mod(\Gamma_i).$$

OK, so we have that $\mathcal{K}_\varphi$ and $(\mathcal{K}_\varphi)^c$ are each axiomatizable. So we know that $\mathcal{K}_\varphi$ is in fact finitely axiomatizable. We now want to turn a finite axiomatization of $\mathcal{K}_\varphi$ into a finite axiomatization of $\mathcal{K}$:

If $\mathcal{K}_\varphi=Mod(\Theta)$, think about $Mod(\{\theta\wedge\varphi:\theta\in\Theta\})$.