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)

Proof that the base angles of an isosceles
triangle are equal (Bridge of Asses)

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