The Bridge That Only Holds the Finite
In the world of logic, there is a bridge. It spans the chasm between a rule's admissibility and its derivability from axioms. As long as rules involve only a finite number of premises, the bridge holds firm — for classical logic, anything that doesn't violate truth is already contained in the syntax. But add infinite premises, and the supports start to tremble. The bridge, alas, is not all-powerful.