# Prove that if $\Gamma$ is inconsistent, then $\Gamma$ doesn’t have a model.

The task is to prove that If set of sentences $$\Gamma$$ is inconsistent, then $$\Gamma$$ doesn't have a model. It is a corollary of Soundness Theorem: If $$\Gamma\vdash A$$, then $$\Gamma\models A$$. Could anyone explain how to prove it?

• What have you tried? Where did you get stuck? What are some things you already know about models and the definition of $\models$? (BTW I've TeXified your question; see here, and if there's a symbol you don't know the LaTeX code for you can usually find it via detexify.) Commented Apr 22, 2020 at 17:34

Suppose $$\Gamma$$ is inconsistent, then for some formula $$\phi$$, $$\Gamma\vdash\phi,\neg\phi$$. By the soundness theorem, $$\Gamma\vDash\phi,\neg\phi$$. Any model for $$\Gamma$$ thus satisfies $$\phi$$, but then it cannot satisfy $$\neg\phi$$, which is a contradiction, so $$\Gamma$$ has no model.
• Yeah, note that the definition of satisfaction means $\mathcal{M} \models \varphi$ iff $\mathcal{M} \not\models \neg\varphi$ and $\mathcal{M} \models \neg\varphi$ iff $\mathcal{M} \not\models \varphi$ when $\varphi$ is a sentence.