#### 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)**

**Proof that the base angles of an isosceles**

**triangle are equal (Bridge of Asses)**

**Peano Axioms of Arithmetic**

**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