I’m currently following the book “Mathematical logic” of Ebbinghaus. He says that a sequent $\Gamma\varphi$ is derivable when exists a derivation of this sequent in the sequent calculus $\mathfrak{S}$ and we note it as $\vdash\Gamma\varphi$ or even $\Gamma\vdash\varphi$. I understood this as “from $\Gamma\varphi$ you can get another sequent using the rules of the calculus” but probably I’m not understanding well the motivation of this definition because is difficult for me to imagine a sequent that would be not derivable by means of some of the rules, specially some like Antecedent rule or Assumption rule. Could someone bring me a sequent that is not derivable in this sense?.
In fact, the definition fits better for me if I think about the derivability of $\ \Gamma\varphi$ as “from another sequents you can reach $\Gamma\varphi$ using the rules of the calculus” instead as I said before.