Prove that class of all structures that are isomorphic structure of the form $\mathbb{A}=\langle\mathcal{P}(A),\cup^{\mathbb{A}},\cap^{\mathbb{A}},\subseteq^{\mathbb{A}}\rangle, $ is not axiomatizable.$$\mathbb{A}=\langle\mathcal{P}(A),\cup^{\mathbb{A}},\cap^{\mathbb{A}},\subseteq^{\mathbb{A}}\rangle$$ should be understand as "normal" operations on set.
It can be proved by Lowenheim-Skolem's theorem. If there existed a such set $S$ thet $|\mathcal{P}(S)| = \aleph_0$ then Lowenheim-Skolem's theorem wouldn't apply here, right?
Yes, that's right.
In a bit more detail: by Lowenheim-Skolem, we would have to have a countably infinite model of that theory. But then the underlying set of that model, $M$, would have the same cardinality as some powerset - that is, for some $B$ we'd have $\vert M\vert=\vert\mathcal{P}(B)\vert$.
But then $B$ must be of cardinality strictly smaller than that of $M$. What sets have cardinality less than $\aleph_0$? The finite ones! So $B$ is finite. But then $M$ is finite too, contradiction.