# 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, thereby having 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, the axiom systems for branches of mathematics, such as:
• Euclidean Plane Geometry
• Arithmetic
• Set Theory
• To set forth self-evident truths, so that proving a theorem establishes its truth. For example, the philosophic systems set forth by:
• Rene Descartes, who endeavored to derive the existence of God and an external world from the self-evident proposition that he had thoughts.
• Benedict de Spinoza, who endeavored to derive the existence of God from his axioms.
• To formulate a scientific theory from which testable predictions can be derived, thus providing a means of supporting or disproving the theory

#### 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)

#### Peano Axioms of Arithmetic

• Axioms (Postulates)
• Zero is a natural number.
• Every natural number has a successor in the natural numbers.
• Zero is not the successor of any natural number.
• If the successor of two natural numbers is the same, then the two original numbers are the same.
• If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
• Sample Theorems (Logical Consequences)
• For any natural numbers x and y:  x + y = y + x
• For any natural numbers x, y, and z:  x+(y+z) = (x+y)+z.
• For any natural numbers x, y, and z:  x(y+z) = xy+xz