Foundations of Mathematics

Axiomatic System

What is an axiomatic system?

An axiomatic system is a logical system that has a definite set of axioms; from these axioms, theorems can be derived. In each system, propositions (statements) are proved on the basis of a limited number of axioms or postulates—all with a few undefined terms. The other terms are defined on the basis of the undefined terms. One of the first axiomatic systems was Euclidean geometry.

Overall, an axiomatic system has several basic components: the undefined terms of the system (primitives); well-formed formulas, or how symbols are put into the system based on certain allowed rules, sometimes called defined terms; axioms, or what is also known as “self-evident truths” of the system; theorems, or statements that are proved based on axioms or other proven theorems; and finally, the rules of inference, or those that allow moves from certain formulas to other formulas.


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