First-Order Language without any Constant, Function, or Relation recently I saw an interesting problem from a textbook and wondering if there is any neat and elementary solution for it:
For a language without any function, constant, or relation, how do we get a theory $\Sigma$ such that none of the finite theories has identical logical consequences as $\Sigma$?
I am thinking about using the theory $\Sigma$ as $\{\lambda_i \mid i \geq 2\}$, where $\lambda_i$ means that "there are at least $i$ elements in the universe", which is easy to construct.
Does it work? If so, how to prove that no other finite theory  has the same logical consequence as it?
Thanks!

You're on the right track with your $\Sigma$. For proving that it has the desired property, I think the crucial fact is that any finite theory is equivalent to a single sentence, and that no single sentences can treat more than finitely many universe sizes differently from an infinite universe.
Proving the latter will need a bit of footwork, but I think you should be able to get through by proving by structural induction that every wff $\varphi$ is equivalent to one of the form
$$ (\psi_0 \land \tau_0) \lor \cdots \lor (\psi_n \land \tau_n) $$
where there is one disjunct for each possible equivalence relation on the free variables of $\varphi$, and $\psi_k$ is a conjunction of literals $x_i=x_j$ or $x_i\ne x_j$ that forces the variables to have precisely that relation to each other, and $\tau_k$ is some (finite) propositional combination of your $\lambda_i$s.
Then, in particular if $\varphi$ is a sentence, it will be equivalent to just a $\tau_0$, and if $i$ is the largest $i$ such that $\lambda_i$ occurs in $\tau_0$, $\varphi$ must have the same truth value in an $i$-element universe as in an infinite universe.
