How does an axiomatic system build theories from basic concepts and propositions?

In the philosophy of mathematics, an axiomatic system builds theories through a rigorous, formal procedure known as the axiomatic method. This process organizes knowledge by deriving conclusions from a foundational set of principles using specific logical rules.12

The construction of such a system typically involves four essential components:

• Primitive Symbols: A set of undefined terms that serve as the basic building blocks of the system. All other symbols and concepts are defined using these primitives.23
• Formation Rules: Guidelines that specify how symbols can be combined to create well-formed formulas (wffs), ensuring the language of the system is precise.23
• Axioms: A selected list of well-formed formulas accepted as starting points without proof. These represent the fundamental propositions of the theory.31
• Transformation Rules: Logical procedures that allow for the derivation of new formulas, known as theorems, from the axioms or previously established theorems.3

A proof is a finite sequence of formulas where each step is either an axiom or a result of applying transformation rules. By abstracting away from specific meanings, these systems allow for the analysis of deductive structures, focusing on validity and logical consistency.23