axiomatised

The term "axiomatised" refers to the process of defining a system, theory, or concept using a set of fundamental assumptions or axioms. These axioms are statements that are accepted as true without proof, serving as the bedrock upon which the rest of the system is built. In an axiomatised system, all other propositions, theorems, or statements are derived logically from these initial axioms. This approach is prevalent in various fields, notably in mathematics, logic, and theoretical computer science.

An axiomatised system aims for rigor and completeness. By starting with a minimal set of self-evident truths, it seeks to avoid ambiguity and ensure that all conclusions can be traced back to these foundational principles. The goal is to create a consistent and coherent structure where every element is demonstrably true within the framework established by the axioms. This method allows for the formal verification of statements and the exploration of the logical consequences of the initial assumptions.

The process of axiomatisation involves carefully selecting axioms that are independent, meaning none can be derived from the others, and sufficient, meaning they are enough to deduce all other desired statements within the system. Prominent examples of axiomatised systems include Euclidean geometry, Zermelo-Fraenkel set theory (ZFC), and Peano arithmetic. These systems demonstrate how a complex body of knowledge can be elegantly and rigorously constructed from a small number of unproven starting points.