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 $A\models T$.


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    $\begingroup$ Given a model of such a theory, consider whether a substructure must be a model as well. $\endgroup$ – Jishin Noben Jan 26 at 9:36

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 $T$.

  • $\begingroup$ Why the purely universal sentences are true on subsets? $\endgroup$ – Gonzalo Jan 26 at 11:44
  • $\begingroup$ @Gonzalo $\forall x \phi(x)$ holds for the larger set, so also for the smaller. Induct on the structure of the formula. $\endgroup$ – Henno Brandsma Jan 26 at 11:49
  • $\begingroup$ Thank you for the clarification $\endgroup$ – Gonzalo Jan 26 at 12:33

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