Question: Give an example of language $\mathcal{L}_1$ and set $\Gamma_1$ of $\mathcal{L}_1$-formulae such that $\Gamma_1$ is Henkin but not consistent
Answer: Let $\mathcal{L}_1$ be arbitrary and $\Gamma_1$ be equal to $Fml_{\mathcal{L}_1}$.
$\exists x\varphi\rightarrow\varphi\frac{x}{x}\in\Gamma_1$ for every $\varphi\in Fml_\mathcal{L}$ is Henkin since for every formula of $\exists x\varphi\in Fml_\mathcal{L}$ there is $t\in Tm_\mathcal{L}$ s.t. $t$ is substitutable for $x$ in $\varphi$ and $\Gamma\vdash\exists x\varphi\rightarrow \varphi\frac{t}{x}$
$\varphi$ is consistent iff there is no $\psi\in Fml_\mathcal{L}$ with $\varphi\vdash\psi$ and $\varphi\vdash \neg\psi$
What is the $\psi$ that makes this set not consistent, how do I show $\varphi\vdash\psi$ and $\varphi\vdash \neg\psi$ for some $\psi$