How does a theorem differ from an axiom in a formal system?

In a formal system, the distinction between an axiom and a theorem is fundamental to the structure of deduction:

• Axioms are specific formulas within the system that are singled out as starting points. They are accepted as the basic, unproved foundations of the system.123
• Theorems are formulas that are derived from the axioms through a finite sequence of steps, following established rules of inference. A theorem is essentially any formula that can be proven to follow from the axioms.123

In short, while axioms serve as the initial premises of the system, theorems are the conclusions reached by applying the system's rules to those axioms (or to previously established theorems). Every axiom is considered a basic theorem, but not every theorem is an axiom, as most theorems require a formal proof to be established.234