Axioms

The parallel postulate (Postulate 5): If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Euclidean geometry is an axiomatic system, in which all theorems ("true statements") are derived from a small number of axioms. Near the beginning of the first book of the *Elements*, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath)

"Let the following be postulated":

1) "To draw a straight line from any point to any point."

2) "To produce [extend] a finite straight line continuously in a straight line."

3) "To describe a circle with any centre and distance [radius]."

4) "That all right angles are equal to one another."

*5) The parallel postulate*: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also taken to be unique.

1) The *Elements* also include the following five "common notions":

2) Things that are equal to the same thing are also equal to one another (formally the Euclidean property of equality, but may be considered a consequence of the transitivity property of equality).

If equals are added to equals, then the wholes are equal (Addition property of equality).

3) If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).

4) Things that coincide with one another are equal to one another (Reflexive Property).

5) The whole is greater than the part.

An axiom is a statement that is considered to be true; however, it cannot be proven or demonstrated because it is simply considered as self-evident. Basically, anything declared to be true and accepted, but does not have any proof or has some practical way of proving it, is an axiom. It is also sometimes referred to as a postulate, or an assumption.

Axioms can be categorized as logical or non-logical. Logical axioms are universally accepted and valid statements, while non-logical axioms are usually logical expressions used in building mathematical theories.

Axioms serve as the starting point of other mathematical statements. These statements, which are derived from axioms, are called theorems.

A theorem, by definition, is a statement proven based on axioms, other theorems, and some set of logical connectives. Theorems are often proven through rigorous mathematical and logical reasoning, and the process towards the proof will, of course, involve one or more axioms and other statements which are already accepted to be true.

Theorems are often expressed to be derived, and these derivations are considered to be the proof of the expression. The two components of the theorem’s proof are called the hypothesis and the conclusion. It should be noted that theorems are more often challenged than axioms, because they are subject to more interpretations, and various derivation methods.

It is not difficult to consider some theorems as axioms, since there are other statements that are intuitively assumed to be true. However, they are more appropriately considered as theorems, due to the fact that they can be derived via principles of deduction.

1. An axiom is a statement that is assumed to be true without any proof, while a theory is subject to be proven before it is considered to be true or false.

2. An axiom is often self-evident, while a theory will often need other statements, such as other theories and axioms, to become valid.

3. Theorems are naturally challenged more than axioms.

4. Basically, theorems are derived from axioms and a set of logical connectives.

5. Axioms are the basic building blocks of logical or mathematical statements, as they serve as the starting points of theorems.

6. Axioms can be categorized as logical or non-logical.

7. The two components of the theorem’s proof are called the hypothesis and the conclusion.