Prove that the class of dense linear orders cannot be axiomatized by purely universal sentenes

Question

That is, prove that there is no set $T$ of purely universal sentences such that for every structure $A$ over the signature $\{\leq\}$, $A$ is a dense linear order iff $A\models T$.

Answer 1

Suppose there was such a set $T$. Clearly $(\mathbb{Q}, \le) \models T$, as it is a dense linear order.

$(\mathbb{Z}, \le)$ is a substructure of $(\mathbb{Q}, \le)$ so $T$ also holds in this, because purely universal sentences stay true on subsets: $(\mathbb{Z}, \le) \models T$.

But $(\mathbb{Z}, \le)$ is not a dense linear order, which contradicts the assumption on $T$.