How do axioms function in a formal system?

In a formal system, axioms function as the foundational building blocks from which all other theorems are derived. They are specific, well-formed formulas within the system that are singled out as basic truths or starting points.12

Their role within the system includes the following:

• Starting Points for Deduction: Axioms serve as the initial premises in a formal sequence. A theorem is defined as any formula that is either an axiom itself or a conclusion reached by applying formal rules of inference to previously established theorems.12
• Syntactic Specification: Axioms are identified through purely formal, syntactic criteria rather than by their intuitive meaning. Because they are part of the system's syntax, their selection allows for the mechanical generation of theorems, a process studied in proof theory.23
• Structural Foundation: By providing a set of axioms and inference rules, a formal system creates a structure where the validity of a derivation can be checked mechanically, independent of the specific interpretation or truth-value of the symbols involved.123

In essence, axioms provide the necessary constraints to define the deductive structure of the system.1