One knows that the sequent calculus over the set of sequents $\Gamma \varphi$ is correct and complete, meaning that the derivable sequents are precisely the correct ones.
However, adding just one axiom $\tfrac{}{\Gamma \varphi}$ (where $\Gamma \varphi$ is not correct) to the rules of the sequent calculus, can one now derive every sequent?
To prove or disprove this, it might be important to know that I am talking about the specific sequent calculus introduced in "Mathematical Logic" by Ebbinghaus, Thomas and Flum.
Best Answer
Not in general, no.
Consider adding the sequent $\{ \} P$ where $P$ is a specific logic statement that is not a contradiction.
Then all statements that you can derive (as a sequent $\{ \} \phi$) are going to be either tautologies, or statements that are logically implied by $\phi$... which means you cannot derive the sequent $\{ \} \neg P$
The only time that adding a specific sequent would allow you to infer all sequents is when the sequent you add is equivalent to a contradiction, e.g. if you add the sequent $\{ \} P \land \neg P$ for some specific $P$.