A formal system is a structured framework used in mathematics and logic to derive conclusions from a set of axioms and rules. It consists of symbols, syntax, and rules for manipulating these symbols to create statements or proofs. The goal is to ensure that every statement can be logically deduced from the established axioms. In a formal system, the axioms are accepted as true without proof, while the rules dictate how new statements can be formed. This approach is foundational in areas like mathematics, computer science, and philosophy, allowing for rigorous reasoning and the exploration of complex ideas.