How are axioms used to derive theorems in a formal system?

In a formal system, axioms serve as the foundational starting points from which other formulas, known as theorems, are derived. The process of deriving these theorems relies on a structured, mechanical approach rather than the specific meaning of the symbols involved.123

The derivation process functions as follows:

• Rules of Inference: A formal system includes specific rules that dictate how one or more formulas (premises) can be used to infer a new formula (conclusion).13
• Proof Construction: A theorem is defined as any formula that can be proven through a finite sequence of well-formed formulas. In this sequence, every formula must either be an axiom or be correctly inferred from preceding formulas in the sequence using the system's established rules.13
• Mechanical Verification: Because the system is formal, it is possible to verify mechanically whether a given sequence of formulas constitutes a valid proof by checking that each step adheres to the defined axioms and rules of inference.3

Ultimately, this axiomatic structure allows for the analysis of deductive reasoning by focusing on the logical form and structure of the system rather than the interpretation of its symbols.12