# Axiom Systems

#### Axiom System

• An axiom system is a set of axioms (or postulates) from which theorems (or logical consequences) are logically derived.
• The classic example is Euclid’s system of Plane Geometry in the Elements, which consists of:
• Axioms / Postulates
• Theorems
• Proofs of Theorems.

#### Reasons for Axiom Systems

• To systematize a body of knowledge and have a standard way of proving established truths. The objective is the simplest set of axioms from which all (and only) the truths can be derived. For example:
• Axiom systems for branches of mathematics.
• To set forth self-evident truths so that proving a theorem establishes its truth. For example, the systems set forth by philosophers:
• Rene Descartes (Cogito Ergo Sum)
• Benedict de Spinoza (who derived the existence of God from his axioms).
• To explain certain kinds of phenomena and provide a means of supporting or disproving the postulates.
• Scientific theories

#### Axiom System for Euclidean Plane Geometry

• Axioms (Postulates)
• For any two different points, (a) there exists a line containing these two points, and (b) this line is unique.
• A straight line segment can be prolonged indefinitely.
• A circle can be constructed when a point for its centre and a distance for its radius are given.
• All right angles are equal.
• For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique.
• Sample Theorems (Logical Consequences)
• Two sides of a triangle are equal if and only if the angles opposite them are equal.
• The sum of the angles of a triangle is 180 degrees
• The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse (Pythagorean Theorem)