How do axioms and theorems form a mathematical system?

In mathematics, a mathematical system is constructed using the axiomatic method, which organizes terms and relationships to analyze deduction. This process begins with a set of primitive, undefined terms and a collection of basic propositions known as axioms or postulates.123

The system is developed through the following components:

• Formal Language: A set of primitive symbols and rules of formation that dictate how formulas are constructed within the system.3
• Axioms: Fundamental assumptions that serve as the starting point for the system. These are often chosen based on criteria such as consistency, plausibility, and richness.23
• Inference Rules: Logical procedures used to derive new statements from the axioms or previously established formulas.23
• Theorems: Statements that are proven to be true within the system. A theorem is established through a proof—a finite sequence of well-formed formulas where each step is either an axiom or a result of valid inference.34

Together, these elements form a formal system that allows mathematicians to explore the deductive structure of thought.35