# Example of language $\mathcal{L}_1$ and set $\Gamma_1$ s.t. $\Gamma_1$ is Henkin but not consistent

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 $\psi$

• Presumably $Fml_{\mathcal{L}_1}$ means the set of all formulas in the language? If so, then it is is not difficult to see that it is inconsistent - but you need a definition of consistency that works for sets of formulas. – Rob Arthan May 27 '15 at 17:00
• @RobArthan Can you explain why it is not difficult to see that it is inconsistent, thanks. – Sam Houston May 28 '15 at 15:26
• A set $X$ of formulas is consistent if there is no formula $\psi$ such that $X \vdash \psi$ and $X \vdash \lnot\psi$, where $X \vdash \phi$ means that for some $\phi_1, \ldots, \phi_n \in X$, $\phi_1 \land \ldots \land \phi_n \vdash \phi$. But if $X$ contains every formula, $X \vdash \phi$ for any formula $\phi$ (because $\phi \vdash \phi$ and $\phi \in X$). – Rob Arthan May 28 '15 at 16:26