Proof in First-Order Logic using Compactness Theorem

Question

If we have $\Sigma$ and $T$ as two first-order theories such they do not have any common models.

How can I prove that there is a sentence φ such that Σ ⊨ φ and Τ ⊨ ¬ φ? Does Compactness Theorem help?

Answer 1

HINT: Suppose that there is no such sentence. Let $\Phi$ be any finite subset of $\Sigma$, and let $\varphi=\bigwedge\Phi$; clearly $\Sigma\vDash\varphi$. Now show that there is a model of $T\cup\Phi$, and use the compactness theorem to get a common model.