Axioms and Theorem

Definition of Axiom

Axiom is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary definition. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other truths.

Definition of Theorem

Theorem is a statement which has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms.

Brief Background of Axiom

The Early Greeks developed a logico-deductive method whereby conclusions (new knowledge) follow from premises (old knowledge).

Common notions (very basic, self-evident assertions) by Euclid:

• Things which are equal to the same thing are also equal to one another.
• If equals be added to equals, the wholes are equal.
• If equals be subtracted from equals, the remainders are equal.
• Things which coincide with one another are equal to one another.
• The whole is greater than part.

Euclid

Definition of Axiomatic System

Axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derived theorems.
Properties of Axiomatic System

Axiomatic system has three properties:

• Consistent if it lacks contradiction.
• Complete if for every statement, either itself or its negation or contradiction is derivable.
• Independent if it is not a theorem that can be derived from other axoms in the system.

Indicative Conditional

Many theorems are of the form of indicative conditional:

If A, then B.

In this case A is called the hypothesis (antecedent) of the theorem and B the conclusion (consequent).

Relation of Theorem to Proof

Theorems are precisely true in the sense that they possess proofs. Therefore, to establish a theorem, the existence of a line of reasoning from the axioms in the system to the given statement must be demonstrated.