How to prove that there does not exist a term such that $Γ \vdash $, where $Γ=\{\lnot \mid \in \} \cup \{\exists \mid \}$

Question

I didn't get correct answers so I post here again. Let $L$ be a logic language with one unary predicate symbol , no constant symbols, no function symbols. Let $V$ be a set of variables. We define the set of wffs $Γ=\{\lnot \mid \in \} \cup \{\exists \mid \}$. Prove that there does not exist a term such that $Γ\vdash$. Could someone please give me some hints? Much appreciated. Thanks.

Answer 1

You cannot prove it.

If we have $Γ=\{ ¬ |∈ \} \cup \{ ∃ \} ⊢ Rt$ we have also, by Completeness : $Γ \vDash Rt$.

This means that $Γ \cup \{ \lnot Rt \}$ is unsatisfiable.

But has been already shown that it is satisfiable, due to the fact that the language has no constants and no functions symbols, but only variables.