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!
 A: 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.
